Question 3 2 pts A widget factory produces n widgets in t hours of a single day. The number of widgets the factory produces is given by the formula n(t) = 10,000t - 25t2, 0≤t≤9. The cost, c, in dollars of producing n widgets is given by the formula c(n) = 2040 + 1.74n. Find the cost c as a function of time t that the factory is producing widgets.
A) c(t) = 2040 + 17,400t - 43.5t²
B) c(t) = 2045 +17,400t - 42.5t²
C) c(t) = 2045 +17,480t - 42.5t²
D) c(t) = 2040 + 17,480t - 43.5t²

The z-transform of the finite duration signal x(n) = (2, 4, 5, 7, 0, 1) is O2 + 4z + 5z² + 7z³ + z⁴. the z-transform is a mathematical tool used to analyze discrete-time signals in the frequency domain.

It converts a sequence of numbers, in this case, x(n), into a function of a complex variable z. The z-transform is defined as the sum of the sequence elements multiplied by z raised to the power of the corresponding index.

Given the finite duration signal x(n) = (2, 4, 5, 7, 0, 1), we can directly apply the definition of the z-transform to obtain its expression. Each element of the sequence is multiplied by z raised to the power of its index, and the results are summed up.

x(0) = 2 * z^0 = 2

x(1) = 4 * z^1 = 4z

x(2) = 5 * z^2 = 5z^2

x(3) = 7 * z^3 = 7z^3

x(4) = 0 * z^4 = 0

x(5) = 1 * z^5 = z^5

Adding up these terms, we get the z-transform of x(n) as O2 + 4z + 5z² + 7z³ + z⁴.

Learn more about the: z-transform

brainly.com/question/32622869

#SPJ11

The integral [infinity]∫02e−√ˣ dx converges.the value of the integral [infinity]∫02e−√ˣ dx is 2.

Now let's explain the steps to calculate the integral. We start by observing that the integrand, e−√ˣ, is a decreasing function as x increases. We can compare it to another function, 1/x, which is also a decreasing function. Taking the limit as x approaches infinity, we find that e−√ˣ is dominated by 1/x, meaning that 1/x grows faster than e−√ˣ. Therefore, we can conclude that the integral converges.
To evaluate the integral, we can use a substitution. Let u = √ˣ, then du = (1/2√x) dx. The limits of integration become u = 0 when x = 0 and u = ∞ when x = ∞. Making the substitution, the integral becomes [infinity]∫02(2e^(-u)) du.
Now we can evaluate this integral by using the limits of integration. As we integrate 2e^(-u) with respect to u from 0 to ∞, the result is 2. Therefore, the value of the integral [infinity]∫02e−√ˣ dx is 2.
In conclusion, the integral [infinity]∫02e−√ˣ dx converges and its value is 2.

Learn more about integral here
https://brainly.com/question/31433890



#SPJ11

False. It's important to note that the Gibbs phenomenon is a characteristic of the Fourier series approximation and not a property of the original signal itself.

The Gibbs phenomenon can occur even in signals without discontinuities. The Gibbs phenomenon is a phenomenon observed in the Fourier series representation of a signal. It refers to the phenomenon where overshoots or ringing artifacts occur near a discontinuity or sharp change in a signal. However, the presence of a discontinuity is not a necessary condition for the Gibbs phenomenon to occur.

The Gibbs phenomenon arises due to the inherent nature of the Fourier series approximation. The Fourier series represents a periodic signal as a sum of sinusoidal components with different frequencies and amplitudes. When the signal has a discontinuity or sharp change, the Fourier series struggles to accurately represent the rapid transition, leading to overshoots or ringing artifacts in the vicinity of the discontinuity. These artifacts occur even if the signal is continuous but has a rapid change in its slope.

It can be mitigated by using alternative signal representations or by considering higher-frequency components in the approximation.

Learn more about discontinuity here:
brainly.com/question/28914808

#SPJ11

Given : Sahib falls off a 52.7 m high bridge into a river, and we need to calculate the time of the jump in seconds.

To calculate how long the jump lasts, we can use the equations of motion for free fall. Let's assume that Sahib falls vertically downward, neglecting air resistance.

The key equation to use is the equation for the vertical displacement of an object in free fall:

y = (1/2)gt^2

where y is the vertical displacement, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.

In this case, Sahib falls from a height of 52.7 m, so we can set y = -52.7 m (taking downward as the negative direction). Plugging in the values, we have:

-52.7 = (1/2)(9.8)t^2

To find the time duration of the jump, we can rearrange the equation and solve for t:

t^2 = (-52.7) * 2 / 9.8 t^2 = -107.4 / 9.8 t^2 ≈ -10.95

Since time cannot be negative, we disregard the negative sign. Taking the square root, we find:

t ≈ √10.95 t ≈ 3.31 s

Therefore, the jump lasts approximately 3.31 seconds.

To know more about seconds, visit

https://brainly.com/question/2437806

#SPJ11

A state space representation for the system can be obtained using the Partial Fraction Expansion (PFE) method. A state feedback controller can be designed to achieve 20.79% overshoot and a settling time of 4 seconds, with the third closed loop pole at s = -6. The range of the third closed loop pole should be chosen to approximate the system's response to that of a second-order system. The closed-loop transfer function of the system can be determined. The steady-state error due to a unit step input can be calculated.

(a) To obtain a state space representation using PFE, we express the open-loop transfer function G(s) in partial fraction form, and then determine the matrices A, B, C, and D for the state space representation.

(b) To design a state feedback controller for 20.79% overshoot and a settling time of 4 seconds, we can use pole placement techniques. By placing the third closed-loop pole at s = -6, we can calculate the desired feedback gain matrix K to achieve the desired response.

(c) The range of the third closed-loop pole can be determined by analyzing the desired system response characteristics. Generally, for a second-order system approximation, the damping ratio and natural frequency are crucial. By choosing appropriate values for the third closed-loop pole, we can approximate the system response to that of a second-order system.

(d) The closed-loop transfer function of the system can be obtained by combining the open-loop transfer function G(s) with the feedback controller transfer function.

(e) The steady-state error due to a unit step input can be calculated using the final value theorem. By evaluating the limit of the closed-loop transfer function as s approaches zero, the steady-state error can be determined.

LEARN MORE ABOUT Partial Fraction Expansion here: brainly.com/question/31707489

#SPJ11

The values that p could take on the number line are given as follows:

-2.5, -4, -6.

The inequality on the number line is given by the numbers that are equal and to the left of p = -2, hence it is given as follows:

p ≤ -2.

Hence the solution is composed by values that are of -2 or less than -2.

Thus the values that p could take on the number line are given as follows:

-2.5, -4, -6.

More can be learned about inequalities at brainly.com/question/25275758

#SPJ1

For part a, we calculate the THD of the feeder current by finding the rms values of the harmonic components and the fundamental component. For part b, we consider the addition of a linear heating load and calculate the THD of the new feeder current.  

a) To calculate the THD of the feeder current, we need to find the rms values of the harmonic components and the fundamental component. The given equation represents the feeder current as a sum of sinusoidal components. We can determine the rms values of each component by dividing their amplitudes by the square root of 2. Then, we calculate the THD using the formula: THD = (sqrt(harmonic1^2 + harmonic2^2 + ... + harmonicn^2) / fundamental) * 100%. Plugging in the values for the given harmonic components, we can compute the THD.

b) When the linear heating load of 100 A (rms) is connected to the feeder, the new feeder current will include the fundamental component and additional harmonics generated by the heating load. We calculate the rms values of these harmonics and the fundamental component, similar to part a. Then, we use the THD formula to determine the THD of the new feeder current.

Learn more about harmonic components here:

https://brainly.com/question/33364188

#SPJ11

The divergence theorem holds for the vector field F on the given region E. The flux of F across the surface of the cube is 12.

The divergence theorem states that the flux of a vector field across a closed surface is equal to the volume integral of the divergence of that field over the region enclosed by the surface. In this case, the region E is a cube bounded by the planes x=0, x=2, y=0, y=2, z=0, and z=2. The vector field F(x,y,z) = 4xi + xyj + 2xzk is defined in three dimensions.

To calculate the flux, we need to find the divergence of F and integrate it over the volume of the cube. The divergence of F is given by div(F) = ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z.

Calculating the partial derivatives, we have:

∂Fx/∂x = 4

∂Fy/∂y = x

∂Fz/∂z = 2x

Therefore, div(F) = 4 + x + 2x = 3x + 4.

Integrating the divergence over the volume of the cube, we have:

∫∫∫ div(F) dV = ∫∫∫ (3x + 4) dV = ∫[0,2]∫[0,2]∫[0,2] (3x + 4) dxdydz.

Evaluating this triple integral, we get:

∫[0,2] (3x + 4) dx = [[tex]3/2x^2[/tex] + 4x] from 0 to 2 = (3/2 * [tex]2^2[/tex]+ 4*2) - (3/2 *[tex]0^2[/tex] + 4*0) = 12.

Therefore, the flux of F across the surface of the cube is 12.

LEARN MORE ABOUT divergence theorem here: brainly.com/question/28155645

#SPJ11

An algorithm is a collection of instructions that perform a specific task. It is a step-by-step method for solving a problem. To implement a reduction for the dynamic system blocks, the following algorithm can be used:

Step 1: Write the system equations in the transfer function form.

Step 2: Reduce the transfer function to its simplest form using algebraic manipulations.

Step 3: Design the controller using the reduced transfer function.

Step 4: Verify the performance of the system using simulation.

The given system blocks are dynamic blocks. It can be represented in transfer function form as below.

G1(s) = 5G2(s)

= 4/(2s + 1)Km

= 1Gs(s) = 1/(s - 1)

The transfer function for the system is

G1(s) * G2(s) * Gs(s) = [5 * 4]/[(2s + 1) * (s - 1)] = 20/(2s² - s - 4)

To reduce the transfer function to its simplest form, factorize the denominator.

2s² - s - 4 = (2s + 4)(s - 1)

Therefore, the transfer function can be written as

G(s) = 20/[(2s + 4)(s - 1)]

The controller can be designed using the reduced transfer function. After that, the performance of the system can be verified using simulation. Thus, the computer algorithm can be used to implement the reduction for the given dynamic system blocks.

To learn more about algorithm follow the given link

https://brainly.com/question/29674035

#SPJ11

The area of the region enclosed by the graphs of y = e^x, y = e^-x, and y = 3 is approximately 4.95 square units, which is the final answer.

Given that the region enclosed by the graphs of y = e^x, y = e^-x, and y = 3.

The required area enclosed by the three given graphs can be obtained using integration.

Therefore, the expression for the area enclosed by the graphs is given by:

A = ∫_{a}^{b} (f(x) - g(x)) dx  .................(1)

where f(x) = 3, g(x) = e^-x, and g(x) = e^x.

To find the limits of integration, we equate e^x to 3 and solve for x as:

e^x = 3⇒ x = ln 3

Therefore, the limits of integration are a = −ln 3 and b = ln 3.

Substituting the given expressions into equation (1) and simplifying, we get:

A = ∫_{-ln3}^{ln3} (3 - e^x - e^-x) dx  .................(2)

Integrating the above expression by applying integration by substitution, we get:

A = [3x + e^x + e^-x]_{-ln3}^{ln3}

A = [3ln3 + e^{ln3} + e^{-ln3}] - [-3ln3 + e^{-ln3} + e^{ln3}]

A = [3ln3 + 3 + 1/3] - [-3ln3 + 1/3 + 3]

A = 3ln3 + 1/3 + 3ln3 - 1/3

A = 6ln3 = 4.95... ≈ 4.95

Therefore, the area of the region enclosed by the graphs of y = e^x, y = e^-x, and y = 3 is approximately 4.95 square units, which is the final answer.

To know more about area, visit:

https://brainly.com/question/16151549

#SPJ11

To find dz/dt using the chain rule, we need to differentiate z = sin(x/y) with respect to t.

First, let's express z in terms of t by substituting the given values for x and y:

x = 5t

y = 3 - t^2

Substituting these values into z = sin(x/y), we have:

z = sin((5t) / (3 - t^2))

Now, we can differentiate z with respect to t using the chain rule. The chain rule states that if z = f(g(t)), then dz/dt = f'(g(t)) * g'(t).

In our case, f(u) = sin(u) and g(t) = (5t) / (3 - t^2). Taking derivatives, we have:

f'(u) = cos(u)

g'(t) = (5 * (3 - t^2) - 5t * (-2t)) / (3 - t^2)^2

Now, we can substitute these derivatives into the chain rule formula:

dz/dt = f'(g(t)) * g'(t)

= cos((5t) / (3 - t^2)) * [(5 * (3 - t^2) - 5t * (-2t)) / (3 - t^2)^2]

This gives us the expression for dz/dt in terms of t.

To know more about the chain rule click here: brainly.com/question/30764359

#SPJ11

1) the critical numbers for \(f(x) = 3x^4 + 8x^3 - 48x^2\) are \(x = 0\), \(x = 2\), and \(x = -4\).

2) the critical numbers for \(f(x) = \frac{2x - 1}{x^2 + 2}\) are \(x = 2\) and \(x = -1\).

3) To find the critical numbers, we set the derivative equal to zero and solve for \(x\):

\(2\sin(x)(\cos(x) - 1) = 0\)

To find the critical numbers of a function, we need to find the values of \(x\) where the derivative of the function is either zero or undefined. Let's find the critical numbers for each function:

1) \(f(x) = 3x^4 + 8x^3 - 48x^2\)

First, we need to find the derivative of \(f(x)\):

\(f'(x) = 12x^3 + 24x^2 - 96x\)

To find the critical numbers, we set the derivative equal to zero and solve for \(x\):

\(12x^3 + 24x^2 - 96x = 0\)

Factoring out \(12x\):

\(12x(x^2 + 2x - 8) = 0\)

Using the zero product property, we have two cases:

Case 1: \(12x = 0\)

This gives us \(x = 0\) as a critical number.

Case 2: \(x^2 + 2x - 8 = 0\)

This quadratic equation can be factored as \((x - 2)(x + 4) = 0\).

So we have two additional critical numbers: \(x = 2\) and \(x = -4\).

Therefore, the critical numbers for \(f(x) = 3x^4 + 8x^3 - 48x^2\) are \(x = 0\), \(x = 2\), and \(x = -4\).

2) \(f(x) = \frac{2x - 1}{x^2 + 2}\)

First, we find the derivative of \(f(x)\) using the quotient rule:

\(f'(x) = \frac{(2)(x^2 + 2) - (2x - 1)(2x)}{(x^2 + 2)^2}\)

Simplifying:

\(f'(x) = \frac{2x^2 + 4 - 4x^2 + 2x}{(x^2 + 2)^2}\)

\(f'(x) = \frac{-2x^2 + 2x + 4}{(x^2 + 2)^2}\)

To find the critical numbers, we set the derivative equal to zero and solve for \(x\):

\(-2x^2 + 2x + 4 = 0\)

We can divide both sides by -2 to simplify the equation:

\(x^2 - x - 2 = 0\)

Factoring the quadratic equation:

\((x - 2)(x + 1) = 0\)

Using the zero product property, we have two critical numbers: \(x = 2\) and \(x = -1\).

Therefore, the critical numbers for \(f(x) = \frac{2x - 1}{x^2 + 2}\) are \(x = 2\) and \(x = -1\).

3) \(f(x) = 2\cos(x) + \sin^2(x)\)

To find the critical numbers, we need to find the derivative of \(f(x)\):

\(f'(x) = -2\sin(x) + 2\sin(x)\cos(x)\)

Simplifying:

\(f'(x) = 2\sin(x)(\cos(x) - 1)\)

To find the critical numbers, we set the derivative equal to zero and solve for \(x\):

\(2\sin(x)(\cos(x) - 1) = 0\)

Visit here to learn more about critical numbers brainly.com/question/31339061

#SPJ11

To determine the principal P that must be invested at a rate r, compounded monthly, in order to accumulate $1,000,000 for retirement in t years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where A is the desired amount, P is the principal, r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, the desired amount is $1,000,000, the interest rate is 5% (or 0.05 as a decimal), and the number of years is 45. Since the interest is compounded monthly, the compounding frequency is 12.

Using the formula, we can rearrange it to solve for P:

P = A / (1 + r/n)^(nt)

Substituting the given values, we have:

P = $1,000,000 / (1 + 0.05/12)^(12*45)

Evaluating this expression will give us the principal P needed for retirement. Rounding the answer to the nearest cent will provide the final result.

To know more about compound interest click here: brainly.com/question/14295570

#SPJ11

The correct equation to find the area under this curve using left sums is c) 0.3(f(3.5)+f(6)+f(8.5)+f(11)+f(13.5)). The left-hand sum is a method used for approximating the definite integral of a function. The value of the function is computed at the left endpoint of each subinterval and then multiplied by the width of the subinterval, after which the products are summed to estimate the total area under the curve.

In this question, we can use the given partition and left-hand sum to estimate the area under the curve using the equation below; Left Hand Sum = Δx [f(x0)+f(x1)+f(x2)+...+f(x(n-1))]

Where Δx = (b - a) / n is the width of each subinterval. Here, the partition is given as x0​=3.5, x1​=6, x2​=8.5, x3​=11, x4​=13.5, x5​=16. Hence, the width of each subinterval (Δx) can be calculated as follows;

Δx = (16 - 3.5) / 5Δx = 2.5

Using the left-hand sum and given partition, we can estimate the area under the curve of f(x) using the equation;Left Hand Sum = Δx [f(x0)+f(x1)+f(x2)+...+f(x(n-1))]

Substituting the given values into the formula; Left Hand Sum = 2.5 [f(3.5)+f(6)+f(8.5)+f(11)+f(13.5)]

Left Hand Sum = 0.3(f(3.5)+f(6)+f(8.5)+f(11)+f(13.5))

Therefore, the correct equation to find the area under this curve using left sums is c) 0.3(f(3.5)+f(6)+f(8.5)+f(11)+f(13.5)).

To know more about area under this curve visit:

https://brainly.com/question/15122151

#SPJ11

(a) The surface area of first cuboid is 27.9 cm².

(b) The surface area of second cuboid is 68.75 ft².

(c) The surface area of the cylinder is  1,570.8 in².

(d) The surface area of the triangle prism is 60 units².

The surface area of each figure is calculated by applying the following formula.

(a) The surface area of first cuboid;

S.A = 2 [ (3 cm x 2.1 cm   +   (3 cm x 1.5 cm)  +  (2.1 cm x 1.5 cm) ]

S.A = 27.9 cm²

(b) The surface area of second cuboid is calculated as;

S.A = 2 [(4.5 ft x 1.25 ft) + (4.5 ft x 5ft) + (1.25 ft x 5 ft ) ]

S.A = 68.75 ft²

(c) The surface area of the cylinder is calculated as follows;

S.A = 2πr (r + h)

S.A = 2π(10)(10 + 15)

S.A = 1,570.8 in²

(d) The surface area of the triangle prism is calculated as;

S.A = bh  + (s₁ + s₂ + s₃)l

S.A = (4 x 3) + (4 + 3 + 5)4

S.A = 60 units²

Learn more about surface area here: https://brainly.com/question/76387

#SPJ1

The value of BAC will depend on whether the triangle is acute or obtuse.

Apologies for the incorrect information provided in the previous response. Let's address the issues and provide the correct answers:

3.1 The lines BG and CF should intersect at the center of the circle. It seems there was an error in the construction steps mentioned earlier. Let's adjust the steps to ensure that the lines intersect:

1. Draw a triangle with sides measuring 56 mm, 48 mm, and 40 mm. Label the vertices as A, B, and C, respectively.

2. To find the bisector of side AB, take a compass and set its width to more than half the length of AB (28 mm in this case). Place the compass tip on point A and draw an arc that intersects AB. Without changing the compass width, place the compass tip on point B and draw another arc that intersects AB. Label the points where the arcs intersect AB as D and E.

3. With the same compass width, place the compass tip on point D and draw an arc. Without changing the compass width, place the compass tip on point E and draw another arc. These arcs will intersect each other at point F, which is the midpoint of AB.

4. Repeat steps 2 and 3 to find the midpoint of BC. Label this point as G.

5. Repeat steps 2 and 3 once again to find the midpoint of AC. Label this point as H.

6. Using a ruler, draw a line connecting point G to point F. Similarly, draw a line connecting point H to point E. These lines will intersect at the center of the circle, which we'll label as O.

7. Take a compass and set its width to the distance between point O and any of the triangle vertices (e.g., OA, OB, or OC).

8. With the compass tip on point O, draw a circle that passes through points A, B, and C.

Now, let's move on to the next question.

3.2 The angle HIG can be determined using the properties of triangles and circle angles. Since we have a circle passing through points A, B, and C, we can conclude that angle HIG is an inscribed angle subtending the same arc as angle BAC.

Inscribed angles subtending the same arc are congruent, so angle BAC and angle HIG have the same measure. To determine the measure of angle BAC, we can use the Law of Cosines:

cos(BAC) = [tex](b^2 + c^2 - a^2) / (2bc)[/tex]

Given that sides AB, BC, and AC of the triangle are 56 mm, 48 mm, and 40 mm, respectively, we can substitute these values into the equation:

cos(BAC) =[tex](48^2 + 40^2 - 56^2) / (2 * 48 * 40)[/tex]

cos(BAC) = (2304 + 1600 - 3136) / 3840

cos(BAC) = -232 / 3840

Using the inverse cosine function, we can find the measure of angle BAC:

BAC = arccos(-232 / 3840)

To know more about function visit:

brainly.com/question/31062578?

#SPJ11

The radius of convergence for the power series ∑[n=0 to ∞] 4n+1(x-3)n+1/(n+1) is 1/4, and the interval of convergence is (11/4, 13/4). The Taylor series for the function f(x) = ex centered at c = 1 is [tex]f(x) = e + e(x-1) + e(x-1)^2/2! + e(x-1)^3/3! + ...[/tex]

To find the radius and interval of convergence for the power series ∑[n=0 to ∞] 4n+1(x-3)n+1/(n+1), we can use the ratio test. The ratio test states that if the limit of |a(n+1)/a(n)| as n approaches infinity is L, then the series converges if L < 1 and diverges if L > 1.

Let's apply the ratio test to the given power series:

[tex]|a(n+1)/a(n)| = |4(n+1)+1(x-3)^(n+1+1)/(n+1+1)/(4n+1(x-3)^n/(n+1))|[/tex]

= |4(x-3)(n+2)/(n+2)| = 4|x-3|

Taking the limit as n approaches infinity:

lim(n→∞) |4(x-3)| = 4|x-3|

For the series to converge, we need 4|x-3| < 1. Solving this inequality, we have:

-1/4 < x - 3 < 1/4

11/4 < x < 13/4

Therefore, the interval of convergence is (11/4, 13/4) and the radius of convergence is 1/4.

For the function f(x) = ex, we can find its Taylor series centered at c = 1 using the definition of the Taylor series:

f(x) = f(c) + f'(c)(x-c) + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + ...

First, let's find the derivatives of f(x) = ex:

f'(x) = ex

f''(x) = ex

f'''(x) = ex

...

Now, let's evaluate these derivatives at c = 1:

[tex]f(1) = e^1 \\= e\\f'(1) = e^1 \\= e\\f''(1) = e^1 \\= e\\f'''(1) = e^1 \\= e[/tex]

...

Substituting these values into the Taylor series, we have:

[tex]f(x) = e + e(x-1) + e(x-1)^2/2! + e(x-1)^3/3! + ...[/tex]

Simplifying, we get:

[tex]f(x) = e(1 + (x-1) + (x-1)^2/2! + (x-1)^3/3! + ...)[/tex]

This is the Taylor series for f(x) = ex centered at c = 1.

To know more about radius of convergence,

https://brainly.com/question/31492060

#SPJ11

The position of the particle at which it turns around is approximately 0.995 meters.

x = 2.0 t^2 - 0.90 t^3

To find out at what position the particle turns around, we need to find the turning point or point of inflection.

This can be done by taking the second derivative of the position function and finding when it is zero.

Second derivative:

dx^2/dt^2 = 4.0 - 5.4t

At the turning point, the second derivative is zero.

dx^2/dt^2 = 0 = 4.0 - 5.4t

=> t = 0.7407 s

Substituting t = 0.7407 s in the original position function, we can find the position at which the particle turns around.

x = 2.0(0.7407)^2 - 0.90(0.7407)^3

≈ 0.995 m

Therefore, the position of the particle at which it turns around is approximately 0.995 meters.

To know more about function, visit:

https://brainly.com/question/30721594

#SPJ11

Here's the MATLAB code for each problem:

(e) Code for cos²(ωt) = (1/2) + (1/2)cos(2ωt):

matlab

Copy code

t = linspace(0, 2*pi, 1000);  % Time vector

omega = 1;  % Choose the value of omega

y = (1/2) + (1/2)*cos(2*omega*t);

(f) Code for sin²(ωt) = (1/2) - (1/2)cos(2ωt):

matlab

Copy code

t = linspace(0, 2*pi, 1000);  % Time vector

omega = 1;  % Choose the value of omega

y = (1/2) - (1/2)*cos(2*omega*t);

(g) Code for sin(nωt) * sin(mωt) = (1/2)*cos((n-m)ωt) - (1/2)*cos((n+m)ωt):

matlab

Copy code

t = linspace(0, 2*pi, 1000);  % Time vector

omega = 1;  % Choose the value of omega

n = 2;  % Choose the value of n

m = 1;  % Choose the value of m

y = (1/2)*cos((n-m)*omega*t) - (1/2)*cos((n+m)*omega*t);

(h) Code for sin²(ωt) + cos²(ωt) = 1:

matlab

Copy code

t = linspace(0, 4*pi, 1000);  % Time vector

omega = 2:6;  % Choose the values of omega

y = sin(omega.*t).^2 + cos(omega.*t).^2;

Note: In all the codes, the variable t represents the time vector and y represents the corresponding function values. Adjust the parameters (such as the time range, number of points, and the values of omega, n, and m) according to your requirements.

To know more about MATLAB, visit:

https://brainly.com/question/30763780

#SPJ11

Here is the MATLAB code for the given problems:(e) cos² (ω.*t) = (¹/2) + (¹/2)*cos(2*ω.*t):t = lin space(0, 10, 1000);omega = 1.5;figure plot(t, cos(omega .* t).^2)hold on plot(t, 0.5 + 0.5*cos(2*omega .* t))

title("Plot of cos^2(wt)")x label("t")y label("y")legend("cos^2(wt)", "0.5 + 0.5*cos(2wt)")hold off(f) sin² (ω.*t) = (¹/2) - (¹/2)*cos(2*ω.*t):t = linspace(0, 10, 1000);omega = 1.5;figure plot(t, sin(omega .* t).^2)hold on plot(t, 0.5 - 0.5*cos(2*omega .* t))title("Plot of sin^2(wt)")x label("t")y Label("y")legend("sin^2(wt)", "0.5 - 0.5*cos(2wt)")hold off(g) sin(n*ω.*t) * sin(m*ω.*t) = (1/2)*cos[(n-m)*ω.*t)] - (1/2)*cos[(n+m)*ω.*t)]:t = linspace(0, 10, 1000);omega = 1.5; n = 3; m = 2;figure plot(t, sin(n*omega.*t).*sin(m*omega.*t))hold on plot(t, 0.5*cos((n-m)*omega.*t) - 0.5*cos((n+m)*omega.*t))title("Plot of sin(wt)*sin(wt)")xlabel("t")ylabel("y")legend("sin(wt)*sin(wt)", "0.5*cos((n-m)wt) - 0.5*cos((n+m)wt)")hold off(h) sin² (ω.*t) + cos² (ω.*t) = 1:for omega = 2:6t = linspace(0, 10, 1000);

Figure plot(t, sin(omega.*t).^2 + cos(omega.*t).^2)title("Plot of sin^2(wt) + cos^2(wt)")x label("t")y label("y")end.

To know more about MATLAB, visit:

https://brainly.com/question/13974197

#SPJ11

The method used for the computation of volume created by revolving the region bounded by y = x² and y = √x about the line x = 2, is using the washers method. The summation of the volumes of each cylinder gives the volume created by revolving the region bounded by y = x² and y = √x about the line x = 2.

The volume generated by revolving the region bounded by y = x² and y = √x about the line x = 2 using the washers method is computed using the following steps:Step 1: Sketch the graphThe first step to finding the volume of the region is to sketch the graph of the given equations y = x² and y = √x. The intersection of the two equations is (0, 0) and (1, 1). The resulting graph looks like this:Graph of y = x² and y = √x.Step 2: Determine the limits of integration The limits of integration are the points at which the two functions intersect. From the graph above, the limits of integration are 0 and 1.Step 3: Determine the radius of the washer at a given xThe radius of the washer is the distance between the two curves. At any given x value, the distance between the curves is given by:r = 2 - x² - √xStep 4: Determine the height of the washerThe height of the washer is the infinitesimal change in x, which is given by:dxStep 5: Determine the volume of the washerThe volume of the washer is given by:πr²dxStep 6: Integrate to get the total volumeTo get the total volume, integrate the volume of each washer with respect to x:∫₀¹ π(2 - x² - √x)² dx= π∫₀¹ 4 - 4x² - 4x√x + x³ + 2x²√x - x dx= π(4x - 4/3 x³ - 8/15 x⁵ + 1/4 x⁴ + 2/3 x^(5/2) - 1/2 x²)₀¹= π(4 - 4/3 - 8/15 + 1/4 + 2/3 - 1/2)= π(41/30)Therefore, the volume created by revolving the region bounded by y = x² and y = √x about the line x = 2 is π(41/30).

Learn more about infinitesimal here:

https://brainly.com/question/28458704

#SPJ11

The function find(A) finds indices and * 2 points values of nonzero elements of an array A, it is true.

The first statement, "Relational operators can be applied to * 2 points only one size vectors," is not clear. It seems to be an incomplete sentence. Relational operators can be applied to vectors of any size, not just vectors with a single size.

Regarding the second statement, let's analyze it:

If `a = [1:5]`, it means that `a` is a vector with elements `[1, 2, 3, 4, 5]`.

If `b = 5 - a`, it means that each element of `b` is obtained by subtracting the corresponding element of `a` from 5. Therefore, `b` would be `[4, 3, 2, 1, 0]`.

Now, let's evaluate the given options:

- "a = 0 23 45" is false because the elements of `a` are `[1, 2, 3, 4, 5]`, not `0, 23, 45`.

- "b = 43210" is true because the elements of `b` are indeed `[4, 3, 2, 1, 0]`.

Therefore, the correct statement is: "a = 0 23 45" is false, and "b = 43210" is true.

The `find(A)` function in some programming languages, such as MATLAB or Octave, returns the indices of nonzero elements in the array `A`. It allows you to identify the positions of non-zero elements and access their values.

Learn more about matlab here: brainly.com/question/13974197

#SPJ11

If the measures of two angles of one triangle are known, the measure of the third angle can be found by subtracting their sum from 180°

(AA) states that if two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar.

What does this mean?

It means that similar triangles have their corresponding angles the same measure.

That is, the corresponding angles of the triangles have the same value or are congruent.

Example: If triangle ABC and DEF are similar triangles, then it follows that:

∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F.

Also, note that if one angle of a triangle is given then the other angles can be found using the following rule:

The sum of the angles of any triangle is 180°.

Suppose ∠B = 60° and ∠C = 30°, then ∠A = 180° - 60° - 30° = 90°.

Hence, if the measures of two angles of one triangle are known, the measure of the third angle can be found by subtracting their sum from 180°.

Learn more about congruent from this link:

https://brainly.com/question/27922830

#SPJ11

1.  You would need 48 scoops of powder to make a gallon of drink mix.

2. The actual width of the building is approximately 1,026.32 cm.

3. The actual distance between Terre Haute and St. Louis is approximately 166.48 miles.

1. To find out how much powder is needed to make a gallon of drink mix, we need to first determine the ratio of powder to water and then calculate the amount of powder required for one gallon.

The given ratio is 3 scoops of powder to 8 ounces of water. Since there are 128 ounces in a gallon, we can set up the following proportion:

3 scoops powder / 8 ounces water = x scoops powder / 128 ounces water

Cross-multiplying and solving for x, we get:

8x = 3 * 128

8x = 384

x = 384 / 8

x = 48

Therefore, you would need 48 scoops of powder to make a gallon of drink mix.

2. If the model of the building is 5 cm wide and the actual length of the building is 140.9 m, we can use the scale of the model to find the actual width of the building.

The scale is given as 5 cm represents 68.7 cm. Let's set up a proportion:

5 cm / 68.7 cm = x cm / 140.9 m

To convert 140.9 m to cm, we multiply by 100 (since there are 100 cm in a meter):

140.9 m * 100 = 14,090 cm

Now, we can solve for x:

(5 cm * 14,090 cm) / 68.7 cm = x cm

x = 1,026.32 cm

Therefore, the actual width of the building is approximately 1,026.32 cm.

3. To determine the actual distance between Terre Haute and St. Louis, given the map distance from Terre Haute to St. Louis is 1.9, we need to find the scale of the map.

The given map distance from Cincinnati to Terre Haute is 2.1, and the actual distance is 184 miles. Let's set up a proportion:

2.1 / 184 = 1.9 / x

Cross-multiplying and solving for x, we get:

2.1x = 1.9 * 184

2.1x = 349.6

x = 349.6 / 2.1

x ≈ 166.48

Therefore, the actual distance between Terre Haute and St. Louis is approximately 166.48 miles.

To learn more about actual distance visit:

brainly.com/question/358542

#SPJ11

The measure of the angle formed by a side and the angle bisector is 26 degrees.

If the measure of the given angle is 52 degrees, then the measure of the angle formed by a side and the angle bisector of that given angle can be found as follows:

The angle bisector divides the given angle into two equal angles, so each of the two resulting angles is half of the measure of the given angle.

Therefore, the measure of the angle formed by a side and the angle bisector is:

52 degrees / 2 = 26 degrees

So, the measure of the angle formed by a side and the angle bisector is 26 degrees.

Learn more about angle bisector here

https://brainly.com/question/2478436

#SPJ11

The final amount due on the loan after the partial payment is approximately $13,070.41 (rounded to the nearest cent).

To calculate the final amount due on the loan, we need to consider the principal amount, the interest accrued, and the partial payment made.

Given information:

Principal amount: $13,500

Interest rate: 11% (per year)

Loan period: 180 days

Partial payment: $1,000

Partial payment date: 50 days

First, let's calculate the interest accrued on the loan from the loan start date to the partial payment date:

Interest accrued = Principal amount * Interest rate * (Number of days / 365)

Interest accrued = $13,500 * 11% * (50 / 365)

Interest accrued ≈ $201.37

Next, let's calculate the remaining principal balance after the partial payment:

Remaining principal balance = Principal amount - Partial payment

Remaining principal balance = $13,500 - $1,000

Remaining principal balance = $12,500

Now, let's calculate the interest accrued on the remaining principal balance for the remaining loan period (180 - 50 days):

Interest accrued = Remaining principal balance * Interest rate * (Number of days / 365)

Interest accrued = $12,500 * 11% * (130 / 365)

Interest accrued ≈ $570.41

Finally, we can calculate the final amount due on the loan by adding the remaining principal balance and the interest accrued:

Final amount due = Remaining principal balance + Interest accrued

Final amount due = $12,500 + $570.41

Final amount due ≈ $13,070.41

Therefore, the final amount due on the loan after the partial payment is approximately $13,070.41 (rounded to the nearest cent).

Learn more about loan here

https://brainly.com/question/30130621

#SPJ11

The indefinite integral becomes after the evaluation using reduction formulas is ∫(x+4)√(8x+[tex]x^{2}[/tex]) dx = 64[(1/2)sec(θ)[tex]tan^{2}[/tex](θ) + (1/2)ln|sec(θ) + tan(θ)|] + C.

To evaluate the indefinite integral ∫(x+4)[tex]\sqrt{8x+x^{2} }[/tex] dx, we can use a combination of algebraic manipulation and integration techniques. Let's go step by step:

First, let's rewrite the expression under the square root as a perfect square. We complete the square for the quadratic term:

8x + [tex]x^{2}[/tex] = ([tex]x^{2}[/tex] + 8x + 16) - 16 =[tex]{ (x + 4)^{2} - 16.}[/tex]

∫(x + 4)[tex]\sqrt{ (x + 4)^{2} - 16.}[/tex] dx.

Next, we can apply a substitution to simplify the integral. Let's substitute u = x + 4. Then, du = dx.

The integral becomes:

∫u√([tex]u^{2}[/tex] - 16) du.

Now, we can use a trigonometric substitution to further simplify the integral. Let's substitute u = 4sec(θ), which implies du = 4sec(θ)tan(θ) dθ.

Using the identity [tex]sec^{2}[/tex](θ) = 1 + [tex]tan^{2}[/tex](θ),

u^2 - 16 = 16 [tex]sec^{2}[/tex](θ) - 16 = 16( [tex]sec^{2}[/tex](θ) - 1) = 16[tex]tan^{2}[/tex](θ)

The integral now becomes:

∫(4sec(θ))(4tan(θ))(4sec(θ)tan(θ)) dθ

= 64∫[tex]sec^{3}[/tex](θ)[tex]tan^{2}[/tex](θ) dθ.

To integrate [tex]sec^{3}[/tex](θ)[tex]tan^{2}[/tex](θ) we can use a reduction formula. Let's rewrite the integral as:

64∫sec(θ)[tex]tan^{2}[/tex](θ)[tex]sec^{2}[/tex](θ) dθ.

Let I(n) represent the integral of [tex]sec^{n}[/tex](θ) dθ. The reduction formula states:

I(n) = (1/(n-1))[tex]sec^{n-2}[/tex](θ)tan(θ) + (n-2)/(n-1)I(n-2),

where n > 2.

Using the reduction formula, we have:

∫sec(θ)[tex]tan^{2}[/tex](θ)[tex]sec^{2}[/tex](θ)dθ = (1/2)sec(θ)[tex]tan^{2}[/tex](θ) + (1/2)∫sec(θ)dθ.

The integral of sec(θ) can be found using a common integral result:

∫sec(θ)dθ = ln|sec(θ) + tan(θ)| + C.

∫(x+4)√(8x+[tex]x^{2}[/tex]) dx = 64[(1/2)sec(θ)[tex]tan^{2}[/tex](θ) + (1/2)ln|sec(θ) + tan(θ)|] + C

Learn more about  indefinite integral ;

https://brainly.com/question/30404875

#SPJ4

Answer:

a. To find the equation h(t) after t years, we need to integrate the given growth rate dh/dt = 1.5t + 5 with respect to t. This gives us:

h(t) = ∫(1.5t + 5) dt = (1.5/2)t^2 + 5t + C = 0.75t^2 + 5t + C

where C is the constant of integration. We can find the value of C using the initial condition that the seedlings are 12 cm tall when planted (i.e., when t = 0). Substituting these values into the equation above, we get:

h(0) = 0.75(0)^2 + 5(0) + C = 12 C = 12

So, the equation for the height of the shrub after t years is:

h(t) = 0.75t^2 + 5t + 12

b. To find out how tall the shrubs are when they are sold, we need to evaluate h(t) at t = 6, since the shrubs are sold after 6 years of growth and shaping:

h(6) = 0.75(6)^2 + 5(6) + 12 = 27 + 30 + 12 = 69

So, the shrubs are 69 cm tall when they are sold.

Step-by-step explanation:

The correct answer is:

C. One plane can be drawn so it contains all three points.

The amount of glass needed to cover the entire pyramid is approximately 144.431 square feet. Since the answer choices are rounded, the closest option is 144.53 ft2.

To determine the amount of glass needed to cover the entire pyramid, we need to calculate the surface area of all its faces and add them together.

The rectangular pyramid has a base with dimensions of 7 feet by 6 feet. The two large triangular faces have a height of 7.79 feet, and the two small triangular faces have a height of 8 feet.

To calculate the surface area of the rectangular base, we use the formula for the area of a rectangle: Area = length × width. In this case, the area of the base is 7 feet × 6 feet = 42 square feet.

The two large triangular faces each have a base equal to the length of the rectangle, which is 7 feet, and a height of 7.79 feet. To calculate the area of each large triangular face, we use the formula for the area of a triangle: Area = 1/2 × base × height. Therefore, the area of each large triangular face is (1/2) × 7 feet × 7.79 feet = 27.2155 square feet.

The two small triangular faces each have a base equal to the width of the rectangle, which is 6 feet, and a height of 8 feet. Using the same formula for the area of a triangle, the area of each small triangular face is (1/2) × 6 feet × 8 feet = 24 square feet.

Now, to find the total surface area of the pyramid, we add up the areas of all the faces: 42 square feet (base) + 27.2155 square feet × 2 (large faces) + 24 square feet × 2 (small faces).

Calculating the total surface area, we get:

42 square feet + 27.2155 square feet × 2 + 24 square feet × 2 = 42 square feet + 54.431 square feet + 48 square feet = 144.431 square feet.

Therefore, the amount of glass needed to cover the entire pyramid is approximately 144.431 square feet. Since the answer choices are rounded, the closest option is 144.53 ft2.

for more such question on pyramid visit

https://brainly.com/question/30615121

#SPJ8

To find the limit using L'Hôpital's Rule, we differentiate the numerator and denominator separately until we obtain an indeterminate form.

a) limx→0​ xln(x^2)/ex

Taking the derivative of the numerator and denominator, we have:

limx→0​ (ln(x^2) + 2x/x) / ex

As x approaches 0, ln(x^2) and 2x/x both tend to 0, so we have:

limx→0​ (0 + 0) / ex

This simplifies to:

limx→0​ 0 / ex = 0

Therefore, the limit is 0.

b) limx→∞​ xln(x^2)/ex

Taking the derivative of the numerator and denominator, we have:

limx→∞​ (ln(x^2) + 2x/x) / ex

As x approaches infinity, ln(x^2) and 2x/x both tend to infinity, so we have an indeterminate form of ∞/∞.

Applying L'Hôpital's Rule again, we differentiate the numerator and denominator:

limx→∞​ (2/x) / ex

Simplifying further, we have:

limx→∞​ 2/(xex)

As x approaches infinity, the denominator grows much faster than the numerator, so the limit tends to 0:

limx→∞​ 2/(xex) = 0

Therefore, the limit is 0.

L'Hôpital's Rule is a powerful tool in calculus for evaluating limits involving indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions of x is of an indeterminate form, then the limit of the ratio of their derivatives will give the same result. In both cases, we applied L'Hôpital's Rule to evaluate the limits by taking the derivatives of the numerator and denominator. The first limit, as x approaches 0, resulted in a simple calculation where the denominator's exponential term dominates the numerator, leading to a limit of 0. The second limit, as x approaches infinity, required multiple applications of L'Hôpital's Rule to simplify the expression and determine that the limit is also 0. L'Hôpital's Rule is a useful technique for resolving indeterminate forms and finding precise limits in calculus.

Learn more about derivative here:

brainly.com/question/25324584

#SPJ11