A particle moves along a horizontal line according to the position equation x(t) = t^4 – 6t^3 + 12t^2 + 3 where t is greater than/equal to -2 and t is less than/equal to 4. When is the velocity increasing?

To eliminate the parameter in the given situation, we can solve for cos(t) and sin(t) using the equations x = h + r cos(t) and y = k + r sin(t). First, we can isolate cos(t) by subtracting h from both sides of the equation for x and dividing by r:

x - h = r cos(t)
cos(t) = (x - h) / r

Similarly, we can isolate sin(t) by subtracting k from both sides of the equation for y and dividing by r:

y - k = r sin(t)
sin(t) = (y - k) / r

Now we can substitute these expressions for cos(t) and sin(t) into the equation for a circle centered at (h, k) with radius r:

(x - h)^2 + (y - k)^2 = r^2
((x - h) / r)^2 + ((y - k) / r)^2 = 1

This is the equation of the circle in standard form, where the center is (h, k) and the radius is r.

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Potential difference between the two ends of the copper rod is 0.00126 V.

To determine the potential difference between the two ends of the copper rod, we need to use the formula:

V = BLVsinθ

where V is the potential difference, B is the magnetic field strength, L is the length of the copper rod, V is the velocity of the rod, and θ is the angle between the rod and the velocity vector.

Substituting the given values, we get:

V = (0.3T) * (0.12m) * (0.15m/s) * sin(50°)

V = 0.00126 V

Therefore, the potential difference between the two ends of the copper rod is 0.00126 V.

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For a equation r = 9 sin(θ), the tangent line(s) equation at the pole is equals to the y = 0. So, the option(a) is right answer for problem.

We have a equation, r = 9 sin(θ), we have to determine the equation of tangent line(s) at the pole. First we draw the graph present in above figure. It is a circle with pole. As we know in polar coordinates, x = rcos(θ), y= r sin(θ). The equation of tangent line is equals the dy/dx. Tangent is equals to slope of line. So, we determine the value of dy/dx.

Differentiating the equation x = rcos(θ),

[tex]\frac{dx}{d \theta} = \frac{ d(rcos(θ))}{d \theta}[/tex]

[tex]\frac{dx}{d \theta} = - r sin(\theta) + cos(\theta) \frac{ dr}{d\theta}[/tex]

Similarly, Differentiating the equation x = r sin(θ), with respect to x

[tex]\frac{dy}{d \theta} = \frac{ d(rsin(θ))}{d \theta}[/tex]

[tex]\frac{dy}{d \theta} = r cos(\theta) + sin(\theta) \frac{ dr}{d\theta}[/tex]

[tex]\frac{dy}{dx} = \frac{\frac{ dy}{d\theta}}{\frac{ dx}{d\theta}}[/tex]

[tex]\frac{dy}{dx} = \frac{ r cos (\theta) + sin( \theta)\frac{ dr}{ d\theta}}{r sin( \theta) + cos(\theta){\frac{ dr}{d\theta}}}[/tex]

[tex]\frac{dy}{dx} = \frac{ 9sin{\theta} cos (\theta) + 9 sin²(\theta)}{ 9 sin²{ \theta} + 9cos{\theta}sin(\theta)}[/tex]

Now, At pole, r = 0 => θ = 0°, so

=> [tex]\frac{dy}{dx} = 0 [/tex]

So, tangent line equation is y = 0.

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Complete question:

find the tangent line(s) at the pole (if any). (− < < . enter your answers as a comma-separated list.). equation r = 9 sin(θ)

a) y = 0

b) y = 1

c) x = 0

d) x = 1

To find v, we start at the end of u and add 2w to it. So, starting at the end of u (2, -1), we go 2 units to the right and 4 units up (since 2w = 2⟨1, 2⟩ = ⟨2, 4⟩). This takes us to point (4, 3), which is the end of the vector v.

To find the vector v, we use the given equation v = u + 2w. Substituting the values of u and w, we get:

v = ⟨2, -1⟩ + 2⟨1, 2⟩
v = ⟨2, -1⟩ + ⟨2, 4⟩
v = ⟨4, 3⟩

To illustrate the vector operations geometrically, we can draw a coordinate plane and plot the vectors u, w, and v. Starting at the origin (0, 0), we can draw the vector u which goes 2 units to the right (positive x direction) and 1 unit down (negative y direction). Similarly, we can draw the vector w which goes 1 unit to the right and 2 units up.

To find v, we start at the end of u and add 2w to it. So, starting at the end of u (2, -1), we go 2 units to the right and 4 units up (since 2w = 2⟨1, 2⟩ = ⟨2, 4⟩). This takes us to point (4, 3), which is the end of the vector v.

We can then draw the vector v from the origin to the point (4, 3) to complete the illustration. This gives us a visual representation of the vector operations.

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If Richard gets £12 more than Stephen after winning some money and sharing it in the ratio of 2:1, Stephen got £12.

The ratio refers to the relative size of one quantity compared to another.

Ratios show the fractional value of one quantity in relation to the whole.

Richard and Stephen's sharing ratio = 2:1

The sum of ratios = 3 (2+ 1)

The amount amount that Richard got more than Stephen = £12

The total amount that was shared = £36 (£12 x 3)

Thus, Stephen's share from the amount = £12 (£36 x 1/3)

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The probability of drawing a three given that we draw a nonface card is 1/9.

A "nonface" card refers to a card that is not a Jack, Queen, or King. There are 12 nonface cards of each suit: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, and Jack. Since we are drawing a card from a standard 52-card deck, there are 36 nonface cards in the deck.

Out of these 36 nonface cards, only four are threes: the three of clubs, the three of diamonds, the three of hearts, and the three of spades.

Therefore, the probability of drawing a three given that we draw a nonface card is:

P(three | nonface card) = number of favorable outcomes / number of possible outcomes

P(three | nonface card) = 4 / 36

P(three | nonface card) = 1 / 9

The probability of drawing a three given that we draw a nonface card is 1/9.

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The distance between the given parallel planes is 13.5/√30 units.

To find the distance between the given parallel planes, we need to find the perpendicular distance between them.

First, we need to find the normal vectors of the planes.

For the plane 5x – 2y + z = 15, the normal vector is <5, -2, 1>.
For the plane 10x – 4y + 2z = 3, the normal vector is <10, -4, 2>.

Next, we need to find the dot product of the normal vectors:

<5, -2, 1> · <10, -4, 2> = (5)(10) + (-2)(-4) + (1)(2) = 52

The dot product tells us that the normal vectors are not orthogonal, which means the planes are not perpendicular.

To find the distance between the planes, we need to use the formula:

distance = |(Ax + By + Cz - D)/√(A² + B² + C²)|

where A, B, and C are the coefficients of the variables in the equation of one of the planes, and D is the constant term.

Let's use the first plane:

distance = |(5x - 2y + z - 15)/√(5² + (-2)² + 1²)|

Distance = |(-13.5)| / √(25 + 4 + 1)
Distance = 13.5 / √30

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The northbound car has traveled approximately 8.4 kilometers distance (rounded to the tenth decimal place).

Distance is the amount of space between two points or objects. It can be measured in various units such as kilometers, miles, meters, feet, etc. In mathematics, distance is often calculated using the Pythagorean theorem or other distance formulas, depending on the context of the problem.

We can use the Pythagorean theorem to solve this problem. Let's call the distance traveled by the northbound car "d". According to the Pythagorean theorem:

d² + 3²=[tex]9^{2}[/tex]

Simplifying this equation, we get:

d² + 9 = 81

Subtracting 9 from both sides, we get:

d² = 72

Taking the square root of both sides, we get:

d = [tex]\sqrt{72}[/tex] = 6[tex]\sqrt{2}[/tex]

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In a case whereby Finn has 5 of the sweets are lemon and rest of the sweets are mint where  probability that Finn takes a mint flavoured sweet is 2/5, the nummber of mint flavoured sweets are in the bag is 8.

Let's assume the number of mint flavoured sweets in the bag as "m". Then, the total number of sweets in the bag can be calculated as: Total number of sweets = number of lemon sweets + number of strawberry sweets + number of mint sweets

Total number of sweets = 5 + 7 + m

Total number of sweets = 12 + m

The probability of picking a mint flavoured sweet can be expressed as the ratio of the number of mint sweets to the total number of sweets:

Probability of picking a mint sweet = m / (12 + m)

According to the problem statement, the probability of picking a mint flavoured sweet is 2/5. We can use this information to set up an equation:

m / (12 + m) = 2/5

Solving for "m", we get:

5m = 2(12 + m)

5m = 24 + 2m

3m = 24

m = 8

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complete question:

Finn has some sweets in a bag. 5 of the sweets are lemon. 7 of the sweets are strawberry. The rest of the sweets are mint. The probability that Finn takes a mint flavoured sweet is 2/5, How many mint flavoured sweets are in the bag?​

(a)  The average number of orders in a 1-minute interval of time is 12. (b)  The probability that at most 8 orders are placed in a 1-minute interval 0.0659 (c)  The probability that no orders are placed in the next 12 seconds 0.8187. (d)  The expected value and variance for the number of orders arriving in the next 12 seconds is 0.2.

a) The total time period is 6 hours or 360 minutes. So the average number of orders placed in a 1-minute interval of time can be estimated as:

Average number of orders = Total number of orders / Total time in minutes

= 4320 / 360 = 12

Therefore, the estimated average number of orders in a 1-minute interval of time is 12.

b) The number of orders in a 1-minute interval of time follows a Poisson distribution with mean λ = 12. To find the probability that at most 8 orders are placed in a 1-minute interval, we can use the Poisson probability formula:

P(X ≤ 8) =[tex]\sum_{x=0}^{\infty} \frac{e^{-\lambda}\lambda^x}{x!}[/tex], for x = 0, 1, 2, ..., 8

P(X ≤ 8) = [tex]\sum_{x=0}^{\infty} \frac{e^{-\ 12}\ 12^x}{x!}[/tex], for x = 0, 1, 2, ..., 8

Using a calculator, we get:

P(X ≤ 8) = 0.0659

Therefore, the probability that at most 8 orders are placed in a 1-minute interval is approximately 0.0659.

c) The probability that no orders are placed in the next 12 seconds can be calculated using the Poisson probability formula again, but with λ = 12/60 = 0.2 (since there are 60 seconds in a minute):

P(X = 0) = ([tex]e^{0.2}[/tex] 0.2⁰) / 0!

= [tex]e^{0.2}[/tex]

≈ 0.8187

Therefore, the probability that no orders are placed in the next 12 seconds is approximately 0.8187.

d) The expected value and variance for the number of orders arriving in the next 12 seconds can be calculated using the Poisson distribution parameters:

Expected value = λ = 12/60 = 0.2

Variance = λ = 0.2

Therefore, the expected value and variance for the number of orders arriving in the next 12 seconds are both 0.2.

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Therefore, it will take approximately 0.34 years for the substance to decay to half of its original amount, rounded to the nearest hundredth year.

A function is a rule that assigns to each input value (or argument) from a set called the domain, a unique output value (or result) from a set called the range. The function is usually denoted by a symbol such as f(x), where "f" is the name of the function and "x" is the input value.

A function can be visualized as a mapping between two sets, where each element of the domain is paired with exactly one element of the range. This mapping can be represented graphically by a plot of the function, which shows how the output values change as the input values vary.

The amount of a radioactive substance at a given time t is given by the function [tex]A(t) = A_{0} e^{(-kt)}[/tex], where A₀ is the initial amount and k is the decay constant. In this case, we are given A₀ = 500 and k = 2.04.

To find the time it takes for the substance to decay to half of its original amount, we need to solve the equation:

[tex]A(t) = 0.5A_{0}[/tex]

Substituting the given values, we get:

[tex]0.5A_{0} = 500e^{(-2.04t)}[/tex]

Dividing both sides by 500, we get:

[tex]e^{(-2.04t)} = 0.5[/tex]

Taking the natural logarithm of both sides, we get:

[tex]-2.04t = ln(0.5)[/tex]

Solving for t, we get:

[tex]t = -ln(0.5)/2.04[/tex]

Using a calculator, we get:

t ≈ 0.34

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Answer:

Step-by-step explanation:

28.3

Answer: I think it's 6m^2

Step-by-step explanation:

Using the formula ∫ t 0 f(τ) dτ = ℒ−1 {F(s)/s}, we have:

∫ t 0 (1/τ^3) (1/(s-1)) ds

= ∫ t 0 (1/τ^3) (1/(s-1)) ds

= [(-1/2) (1/τ^3) e^(s-1)]_0^t

= (-1/2) [(1/t^3) e^(t-1) - (1/0^3) e^(0-1)]

= (-1/2) [(1/t^3) e^(t-1) - e^(-1)]

Therefore, the inverse Laplace transform of 1/s^3(s-1) is:

ℒ−1 {1/s^3(s-1)} = (-1/2) [(1/t^3) e^(t-1) - e^(-1)]
To evaluate the given inverse Laplace transform, ℒ^−1{1/s^3(s − 1)}, we can use the property (8), which states that ∫ t 0 f(τ) dτ = ℒ^−1{F(s)/s}. In this case, F(s) = 1/s^2(s - 1).

First, perform partial fraction decomposition on F(s):

1/s^2(s - 1) = A/s + B/s^2 + C/(s - 1)

Multiplying both sides by s^2(s - 1) to eliminate the denominators:

1 = A(s^2)(s - 1) + B(s)(s - 1) + Cs^2

Now, we will find the values of A, B, and C:

1. Setting s = 0: 1 = -A => A = -1
2. Setting s = 1: 1 = C => C = 1
3. Differentiating the equation with respect to s and setting s = 0:
  0 = 2As + Bs - B + 2Cs
  0 = -B => B = 0

Now we can rewrite F(s) using the values of A, B, and C:

F(s) = -1/s + 0/s^2 + 1/(s - 1)

Next, we can find the inverse Laplace transform of each term separately:

ℒ^−1{-1/s} = -1
ℒ^−1{0/s^2} = 0
ℒ^−1{1/(s - 1)} = e^t

Finally, combine these results and multiply by the unit step function u(t) to obtain the final answer:

f(t) = (-1 + 0 + e^t)u(t) = (e^t - 1)u(t)

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The equation that represents the table is y = 4x.

A term with x as the highest power appears in a linear equation, which is a polynomial equation of the first degree. Y = mx + b, where m is the slope (the rate of change of y with respect to x) and b is the y-intercept (the value of y when x = 0), is the typical form of a linear equation. Straight lines can be used to express linear equations on a coordinate plane, with the slope denoting the line's steepness and the y-intercept designating the line's point of intersection with the y-axis. Linear equations are important to the study of algebra and calculus and have numerous applications in math, science, engineering, economics, and other disciplines.

The slope of the line is given by the formula:

m = change in y / change in x

Using the coordinates from the table we have:

m = 24 - 12 / 6 - 3

m = 12 / 3 = 4

The equation of the line is given as y = mx + b.

Substituting the value of slope we have:

y = 4x + b

The equation that has the slope 4 is y = 4x.

Hence, the equation that represents the table is y = 4x.

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Answer:

angle KML= 50 degrees

Step-by-step explanation:

because triangle JHK is congruent to triangle LMK

so y = 50 then angle KML = 50 degrees

The renowned French jeweler Cartier sells engagement rings that cost more than $100,000. This is an illustration of premium pricing.

A premium pricing strategy is one in which a corporation sets high prices for its products or services in order to target high-end or luxury markets. The company's high costs are intended to give an appearance of exclusivity, quality, and status, as well as to appeal to clients ready to pay a premium for the brand and the related lifestyle.

Cartier, for example, targets wealthy customers searching for high-quality, luxury jewelry to commemorate major events in their lives by producing engagement rings that cost more than $100,000. The premium pricing strategy assists Cartier in positioning itself as a high-end luxury brand and distinguishing itself from competitors.

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In cylindrical coordinates, the vector dot del operator (also known as the dot product of the gradient operator).

Here, A_r, A_θ, and A_z are the radial, azimuthal, and axial components of the vector field A, respectively. The operator ∇ is the gradient operator, which in cylindrical coordinates.

The dot product of these two operators gives the divergence of the vector field A. This formula can be used to calculate the divergence of a vector field in cylindrical coordinates.
Hi! A vector dot del in cylindrical coordinates, also known as the scalar product of the gradient operator and a vector field, represents the directional derivative of a scalar function along a vector field. In cylindrical coordinates.
The vector field in cylindrical coordinates, and ∇ • A denotes the vector dot del operation.

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The equation of the circle is:  x² + y² = 123

The distance  through the center of the circle is called the diameter. The distance from the center of the circle to any point on the border is called the radius. The radius is half  the diameter; 2r = d 2 r = d.

A straight line connecting two points of a circle is a chord.  The equation of a circle with center (0,0) and radius r is given by:

x² + y² = r²

To find the value of R, we can use the fact that the point (5.7√2) is on the circumference of the circle. Substituting x=5 and y=7√2 in the circular equation, we get:

5² + (7√2)² = r²

Simplifying this equation, we get:

25 + 98 = r²

123 = r²

Taking the square root of both sides gives us:

r = √123

So the equation of the circle is:

x² + y² = 123

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1. The most appropriate graphical representation for the data would be a stem-and-leaf plot.

2. The correct answer is Bus 18, with a mean of 12. This is because the mean, or average, is a better measure of center when there are outliers present.

A stem-and-leaf plot is a type of display that shows numerical data in a form that allows individual data points to be seen. It consists of a stem, which is the first digit(s) of each number, and a leaf, which is the last digit of each number.

1. The most appropriate graphical representation for the data would be a stem-and-leaf plot.

The stem-and-leaf plot is used when there is a large set of data, which is the case with the 10 different tortilla sandwich wraps sold in the grocery store.

2. The correct answer is Bus 18, with a mean of 12. This is because the mean, or average, is a better measure of center when there are outliers present.

In this case, the outliers are the dots at 18 and 20 for Bus 14. The mean takes into account all the data points, including these outliers, so it gives a better representation of the center.

The median, on the other hand, is the midpoint of the data set and ignores any outliers. Therefore, it is not an accurate measure of the center in this situation.

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The horizontal distance between the base of the ladder and the wall is approximately 16.06 feet.

To find the flat distance between the foundation of the  stool and the wall, we utilize geometry. For this situation, we can utilize the sine capability, which relates the contrary side of a right triangle to the hypotenuse and the point inverse the contrary side.

Let x be the flat distance between the foundation of the stepping stool and the wall. Then, utilizing the given point of 64 degrees, we can compose:

sin(64) = inverse/hypotenuse

where the hypotenuse is the length of the stepping stool, which is 18 feet.

Addressing for the contrary side, which is the even distance we need to find, we get:

inverse = sin(64) x hypotenuse

inverse = sin(64) x 18

inverse = 16.06 feet

Accordingly, the even distance between the foundation of the stepping stool and the wall is roughly 16.06 feet.

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To find the partial fraction decomposition of x^2 + 36/ x^3 + x^2, we first need to factor the denominator using the sum of cubes formula: x^3 + x^2 = x^2(x+1) + x^2 = x^2(x+1) + 1(x+1) = (x+1)(x^2+1).

Now we can write our expression as: (x^2 + 36)/[(x+1)(x^2+1)]. Next, we can write our partial fraction decomposition in the form: f(x)/(x+1) + g(x)/(x^2+1). To find f(x), we multiply both sides of the equation by (x+1): (x^2 + 36)/(x^2+1) = f(x) + g(x)(x+1)/(x^2+1), Then, we can substitute x = -1 to solve for f(-1): 37 = f(-1) To find g(x), we can multiply both sides of the equation by (x^2+1): (x^2 + 36)/(x+1) = f(x)(x^2+1)/(x+1) + g(x).

Simplifying this equation, we get: x^2 + 36 = f(x)(x^2+1) + g(x)(x+1), Now we can substitute x = 0 and x = i to get two equations with two unknowns (f(0) and g(0)), and solve for both variables: f(0) = 36/g(1), g(i) = -i^2f(i) Using these equations, we can solve for f(x) and g(x): f(x) = 37(x^2+1)/(x+1)(x^2+1), g(x) = -36x/(x^2+1), Finally, we can rewrite our partial fraction decomposition in the form: 37/(x+1) - 36x/(x^2+1).

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2xy dA, D is the triangular region with vertices (0,0), (1,2), and (0,3) total value of the integral is: 4/3.

To calculate the value of 2xy dA for the triangular region D with vertices (0,0), (1,2), and (0,3), we first need to set up the integral:

∫∫D 2xy dA

Since D is a triangular region, we can express it as the union of two right triangles with legs along the x and y axes:

D = D1 ∪ D2

where D1 is the right triangle with vertices (0,0), (1,0), and (0,3), and D2 is the right triangle with vertices (1,0), (1,2), and (0,3). We can then write the integral as the sum of integrals over D1 and D2:

∫∫D 2xy dA = ∫∫D1 2xy dA + ∫∫D2 2xy dA

To evaluate each integral, we need to express x and y in terms of the coordinates of the region. For D1, we have x ranging from 0 to 1 and y ranging from 0 to 3-x, so we can write:

∫∫D1 2xy dA = ∫0¹ ∫0³⁻ˣ 2xy dy dx

Integrating with respect to y first, we get:

∫∫D1 2xy dA = ∫0¹ 2x/2 (3-x)² dx = ∫0¹(3x² - 2x³) dx = 3/2 - 1/2 = 1

For D2, we have x ranging from 0 to 1 and y ranging from 0 to 2x, so we can write:

∫∫D2 2xy dA = ∫0^1 ∫0^(2x) 2xy dy dx

Integrating with respect to y first, we get:

∫∫D2 2xy dA = ∫0¹ x² dx = 1/3

Therefore, the total value of the integral is:

∫∫D 2xy dA = ∫∫D1 2xy dA + ∫∫D2 2xy dA = 1 + 1/3 = 4/3

So the answer to the question is 4/3.

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The total number of strings of length less than equal to 6 is are found to be 126, the formula is based on combination formula.

The length of the string is 6 or less. Now, at one position in the string, there should be either 1 or 0.

The binary digits combinations has to be found, the length of which has to be less than equal to 6.

The formula to calculate the string will be,

= ⁿCₐ, where n and a means that we have to make a number of combinations from n elements.

So, finally the total number of strings.

= 2 + (2 x 2) + (2 x 2 x 2) + (2 x 2 x 2 x 2) + (2 x 2 x 2 x 2 x 2) + (2 x 2 x 2 x 2 x 2 x2)

= 2 + 4 + 8 + 16 + 32 + 64

= 126.

So, the total number of strings are 126.

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value of angle θ in the triangle is 94.04°

The cosine law, also known as the law of cosines, is a mathematical formula used to calculate the length of a side or measure of an angle in a non-right triangle. The formula relates the lengths of the sides of the triangle to the cosine of one of its angles.

Specifically, the cosine law states that for any non-right triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

b²= a² + c² – 2bc cos θ

where cos( θ) is the cosine of angle θ.

In the given triangle,

Sides are a=6,b=13 and c=12.

To find the value of angle θ

Using cosine law:

b²= a² + c² – 2bc cos θ

13²=12²+6²+2×13×6cosθ

Cosθ=-11/156

θ=Cos⁻¹-11/156

θ=94.04°

hence, value of angleθ in the triangle is 94.04°

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Product of polynomials: [tex]x^2y^4(4 - 9y^2).[/tex]

A polynomial is a mathematical expression consisting of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and exponentiation. The variables in a polynomial can only have non-negative integer exponents.

For example, [tex]3x^2 - 5x + 2[/tex] is a polynomial in the variable x, where 3, -5, and 2 are the coefficients, and [tex]x^2[/tex], x, and 1 are the variables with exponents 2, 1, and 0, respectively. The degree of a polynomial is the highest exponent of the variable in the expression.

Now to factorize the polynomial [tex]4x^2y^4 - 9x^2y^6[/tex], we can factor out the greatest common factor (GCF) of the two terms.

The GCF is [tex]x^2y^4[/tex], so we can write:

[tex]4x^2y^4- 9x^2y^6 = x^2y^4(4 - 9y^2)[/tex]

Therefore, [tex]4x^2y^4- 9x^2y^6[/tex] can be written as

a product of polynomials: [tex]x^2y^4(4 - 9y^2).[/tex]

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The nullspace of B is P, and B is the desired matrix.

To find a basis for the orthogonal complement of the plane P in R4 satisfying x1 + x2 + x3 + x4 = 0, we need to find a set of vectors that are orthogonal to all vectors in P.

Any vector in P can be written as (x1, x2, x3, x4) where x1 + x2 + x3 + x4 = 0. Thus, a vector (a, b, c, d) is orthogonal to P if and only if a + b + c + d = 0.

So, one basis for P⊥ is {(1, -1, 0, 0), (1, 0, -1, 0), (1, 0, 0, -1), (0, 1, -1, 0), (0, 1, 0, -1), (0, 0, 1, -1)}.

To construct a matrix that has P as its nullspace, we can use the basis for P⊥ and write it as a row matrix A:

A = [1, -1, 0, 0, 1, 0, 0, -1, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, 0, -1]

Then, we can construct a matrix B by taking the transpose of A and stacking it as columns:

B = [1, 1, 1, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 0, 1, 1, 0, -1, 0, -1]

Note that any vector x in P satisfies the equation Ax = 0, since x is orthogonal to all vectors in P⊥. Therefore, the nullspace of B is P, and B is the desired matrix.

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For the height of tallest and taller buildings in the city are 456. 4 meters and 431.6 meters respectively. The tallest building is total 15.8 meters taller.

Substraction is one of four basic arithmetic operation in mathematics. Subtraction is an operation that represents removal of objects from a collection. It is used to calculate the difference between two numbers or values. For example substraction of 10 from 20 results 10. It is denoted be '-' symbol. The height of tallest building in the city = 456.4 meters

The height of second tallest building in the city = 431.6 meters

We have to determine the how much taller is the tallest building. Now, to calculate the difference between the height of both buldings we use the sybstraction method. So, the difference between the height of tallest and taller buldings = 456.4 meters - 431.6 meters

= 15.8 meters

Hence, required value is 15.8 meters.

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The given set S is linearly independent.

To test for linear independence in P3 for Exercise 21, we can use Exercise 14 and property 2 of Theorem 5.

First, we need to suppose that a linear combination of the given set equals zero. Let c1(x+1) + c2(x^2+1) + c3(x+1) + c4(1) = 0, where c1, c2, c3, and c4 are constants. Then, we can simplify this equation to (c1+c3)x^2 + (c1+c2)x + (c1+c3+c4) = 0. This means that the coefficients of the polynomial are all zero, so we can set up a system of equations to solve for the constants:

c1 + c3 = 0

c1 + c2 = 0

c1 + c3 + c4 = 0

Solving this system of equations, we get c1 = c2 = c3 = c4 = 0. Therefore, the set {x+1, x^2+1, x+1, 1} is linearly independent in P3.

For Exercise 14, we need to prove that (1, x, x^2, ..., x^n) is a linearly independent set in Pn. We can do this by supposing that p(x) = a0 + a1x + a2x^2 + ... + anx^n, where a0, a1, a2, ..., an are constants. Then, we can take the successive derivatives of p(x) as follows:

p(x) = a0 + a1x + a2x^2 + ... + anx^n

p'(x) = a1 + 2a2x + ... + nanx^(n-1)

p''(x) = 2a2 + 6a3x + ... + n(n-1)anx^(n-2)

...

p^(n)(x) = n!an

If we suppose that p(x) = (x), then p^(n)(x) = n!. But we know that (1, x, x^2, ..., x^n) is a basis for Pn, so any polynomial in Pn can be written as a linear combination of this basis. Therefore, we can write p(x) as a linear combination of (1, x, x^2, ..., x^n):

p(x) = c0(1) + c1(x) + c2(x^2) + ... + cn(x^n)

Substituting this into p^(n)(x) = n!, we get:

n! = n!cn

cn = 1

Substituting this back into p(x), we get:

p(x) = c0(1) + c1(x) + c2(x^2) + ... + cn(x^n) = c0(1) + c1(x) + c2(x^2) + ... + 1(x^n)

Since we know that (1, x, x^2, ..., x^n) is a basis for Pn, this linear combination can only equal zero if all the constants c0, c1, c2, ..., cn are zero. Therefore, (1, x, x^2, ..., x^n) is a linearly independent set in Pn.

Property 2 of Theorem 5 states that if the set S is linearly independent in V, then the set T obtained by adding a vector not in S to S is also linearly independent in V. In this case, V is RP (the set of real-valued polynomials) and S is the set (1, x, x^2, ..., x^n). So, if we add a vector not in S (for example, x^(n+1)), the resulting set T is also linearly independent in RP.

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To find the Laplace transform of a function, we use the formula:

L{f(t)} = ∫[0,∞) e^(-st) f(t) dt
where s is a complex number.

a. f(t) = t

Using the Laplace transform formula, we get:
L{t} = ∫[0,∞) e^(-st) t dt

Integrating by parts, we get:
L{t} = [-t e^(-st) / s]∞₀ + ∫[0,∞) e^(-st) / s dt

Evaluating the limits, we get:
L{t} = 0 + [1 / s^2] ∫[0,∞) s e^(-st) dt

Using the fact that ∫[0,∞) s e^(-st) dt = 1 / s^2, we get:
L{t} = 1 / s^2

Therefore, the Laplace transform of f(t) = t is 1 / s^2.

b. f(t) = t^2

Using the Laplace transform formula, we get:
L{t^2} = ∫[0,∞) e^(-st) t^2 dt

Integrating by parts twice, we get:
L{t^2} = [-t^2 e^(-st) / s]∞₀ + [2t e^(-st) / s^2]∞₀ + ∫[0,∞) 2e^(-st) / s^3 dt

Evaluating the limits, we get:
L{t^2} = 0 + 0 + [2 / s^3] ∫[0,∞) e^(-st) dt

Using the fact that ∫[0,∞) e^(-st) dt = 1 / s, we get:
L{t^2} = 2 / s^3

Therefore, the Laplace transform of f(t) = t^2 is 2 / s^3.

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