Not sure whether to use integration by substitution or partial fractions?

Answer:

56

Step-by-step explanation:

f(x)=4x^2-7

f'(x)=8x

f'(7)=56

y₁ = 1/t is indeed a known solution of the given differential equation.

The second solution can be found using reduction of order or other methods specific to the equation.

Let's find the first and second derivatives of y₁ with respect to t:

y₁ = 1/t

First derivative:

y'₁ = d/dt (1/t) = -1/t²

Second derivative:

y''₁ = d/dt (-1/t²) = 2/t³

Now, let's substitute y₁, y'₁, and y''₁ into the differential equation:

-t²y'' + 3ty' + 5y = 0

Substituting the values:

-t²(2/t³) + 3t(-1/t²) + 5(1/t) = 0

Simplifying the expression:

-2/t + (-3/t) + 5/t = 0

(-2 - 3 + 5)/t = 0

0/t = 0

We can see that the expression simplifies to 0/t, which is equal to 0.

Therefore, y₁ = 1/t is indeed a known solution of the given differential equation.

To find the second solution, we can use the method of reduction of order. Let's assume the second solution is of the form y₂ = v(t)y₁, where v(t) is a function to be determined.

Substituting this into the differential equation, we have:

-t²(y₂'' + v'y₁' + v''y₁) + 3t(y₂' + vy₁') + 5y₂ = 0

Expanding and rearranging the terms, we get:

-t²(v''y₁ + v'y₁' + v'y₁ + vy₁'') + 3t(vy₁' - v'y₁) + 5vy₁ = 0

Simplifying further:

(-t²v''y₁ - 2t²v'y₁' + 3tvy₁' + 5vy₁) + (-t²v'y₁ + 3tvy₁ - 5v'y₁) = 0

Combining like terms:

-t²v''y₁ - 2t²v'y₁' - t²v'y₁ - t²v'y₁ + 3tvy₁' + 3tvy₁ + 5vy₁ - 5v'y₁ = 0

Simplifying:

-t²v''y₁ - 3t²v'y₁' + 6tvy₁' + (5v - 5v')y₁ = 0

Since y₁ = 1/t, we have:

-t²v''(1/t) - 3t²v'(1/t²) + 6tv(1/t²) + (5v - 5v')(1/t) = 0

Simplifying further:

-v'' - 3v' + 6v(1/t) + (5v - 5v')(1/t) = 0

Reducing the equation:

-v'' - 3v' + 6v/t + (5v/t - 5v'/t) = 0

-v'' - 3v' + (6v + 5v - 5v')/t = 0

-v'' - 3v' + (11v - 5v')/t = 0

To simplify the equation, we can multiply through by t:

-tv'' - 3tv' + 11v - 5v' = 0

Now, we have a differential equation in terms of v(t) only. To solve this equation, we can apply appropriate techniques such as separation of variables, integrating factors, or other methods depending on the specific form of the equation. Solving for v(t) will give us the second solution to the original differential equation -t²y" + 3ty' + 5y = 0.

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The simplified form of the expression 15x - 12 - 4x + 3x + 13 is 14x+1. Option A

To simplify the expression 15x - 12 - 4x + 3x + 13, we can combine like terms. Like terms are those that have the same variable and exponent.

First, let's combine the x terms:

15x - 4x + 3x = (15 - 4 + 3)x = 14x

Next, let's combine the constant terms:

-12 + 13 = 1

Putting it all together, the simplified form of the expression is:

14x + 1

Therefore, the correct answer is "14x + 1."

To simplify the expression, we added the coefficients of the x terms (15, -4, 3) to get 14x. Then, we added the constant terms (-12, 13) to get 1. This final expression, 14x + 1, does not have any like terms that can be combined further, so it is considered simplified.

It's important to note that when simplifying expressions, we group like terms together and perform the indicated operations, such as addition or subtraction. By doing so, we reduce the expression to its simplest form, where no further combining of like terms is possible.

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To find the value of the cosine of angle M in triangle AMNO, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides [tex]\displaystyle a[/tex], [tex]\displaystyle b[/tex], and [tex]\displaystyle c[/tex], and angle [tex]\displaystyle C[/tex] opposite side [tex]\displaystyle c[/tex], the following equation holds:

[tex]\displaystyle c^{2} =a^{2} +b^{2} -2ab\cos( C)[/tex]

In triangle AMNO, we have the following information:

[tex]\displaystyle AM=17[/tex] (side [tex]\displaystyle a[/tex])

[tex]\displaystyle MN=15[/tex] (side [tex]\displaystyle b[/tex])

[tex]\displaystyle AN=8[/tex] (side [tex]\displaystyle c[/tex])

Angle M = 90 degrees

We can apply the Law of Cosines to find the value of [tex]\displaystyle \cos( M)[/tex]:

[tex]\displaystyle AN^{2} =AM^{2} +MN^{2} -2\cdot AM\cdot MN\cdot \cos( M)[/tex]

Substituting the given values:

[tex]\displaystyle 8^{2} =17^{2} +15^{2} -2\cdot 17\cdot 15\cdot \cos( M)[/tex]

Simplifying:

[tex]\displaystyle 64=289+225-510\cdot \cos( M)[/tex]

[tex]\displaystyle 64=514-510\cdot \cos( M)[/tex]

Rearranging the equation:

[tex]\displaystyle 510\cdot \cos( M) =514-64[/tex]

[tex]\displaystyle 510\cdot \cos( M) =450[/tex]

Dividing both sides by 510:

[tex]\displaystyle \cos( M) =\frac{450}{510}[/tex]

Simplifying:

[tex]\displaystyle \cos( M) =\frac{15}{17}[/tex]

Therefore, the value of the cosine of angle M in triangle AMNO, to the nearest hundredth, is approximately [tex]\displaystyle 0.88[/tex].

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♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

4 cubic meters is equal to 4000 liters. 4 m³ becomes 4000 liters.

To express 4 m³ in liters, we first need to understand the conversions between cubic meters (m³) and liters (L).

1 cubic meter (1 m³) is equal to 1000 liters (1000 L). This is because 1 meter is equal to 100 centimeters, and when cubed, we get 100 cm x 100 cm x 100 cm = 1,000,000 cm³. And since 1 liter is equal to 1,000 cubic centimeters (1 L = 1000 cm³), then 1 m³ is equal to 1,000,000 cm³ / 1000 cm³ = 1000 liters.

Now, we can use this information to convert 4 m³ to liters:

4 m³ * 1000 L/m³ = 4000 liters

Therefore, 4 cubic meters is equal to 4000 liters.

In short, to convert cubic meters to liters, we multiply the value in cubic meters by 1000 to get the equivalent in liters. In this case, 4 m³ becomes 4000 liters. It is important to remember that this conversion is valid for substances that have a density similar to water, since the relationship between cubic meters and liters can vary for different substances.

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The graph of the function f(x) = 5(2)^x is an upward-sloping exponential curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing the x-axis.

The function f(x) = 5(2)^x represents exponential growth. Let's analyze its graph.

As x increases, the value of 2^x grows exponentially. Multiplying it by 5 further amplifies the growth. Here are a few key points to consider:

When x = 0, 2^0 = 1, so f(0) = 5(1) = 5. This is the y-intercept of the graph, meaning the function passes through the point (0, 5).

As x increases, 2^x grows rapidly. For positive values of x, the function will increase quickly. As x approaches positive infinity, 2^x grows without bound, resulting in the function also growing without bound.

For negative values of x, 2^x approaches zero. However, the function is multiplied by 5, so it will not reach zero. Instead, it will approach y = 0, but the graph will never touch or cross the x-axis.

The function is always positive since 2^x is positive for any value of x, and multiplying by 5 does not change the sign.

Based on these observations, we can conclude that the graph of f(x) = 5(2)^x will be an exponential growth curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing or touching the x-axis.

The graph will have a smooth curve that rises steeply as x increases. The rate of growth will be determined by the base, in this case, 2. The larger the base, the steeper the curve. The function will approach but never reach the x-axis as x approaches negative infinity.

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The exact volume of a cone with the same dimensions is 9428.57 cubic inches

From the question, we have the following parameters that can be used in our computation:

The cylinder

The volume of the cone with the same dimensions is calculated as

Volume = 1/3 * Volume of cylinder

So, we have

Volume = 1/3 * 22/7 * (30/2) * (30/2) * 40

Evaluate

Volume = 9428.57

Hence, the exact volume of a cone with the same dimensions is 9428.57 cubic inches

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The coterminal of 4π/3 angle measure are 10π/3 and -2π/3

We add or subtract integer multiples of 2π.

So, 4π/3 + 2π = 10π/3 is one coterminal angle, and

4π/3 - 2π = -2π/3 is another coterminal angle.

To convert 125.67° to degree, minute, and second measure:

125.67° = 125° + 0.67°

Since there are 60 minutes in 1 degree, we can multiply 0.67 by 60 to get the minutes:

0.67° × 60 = 40.2'

Since there are 60 seconds in 1 minute, we can multiply 0.2 by 60 to get the seconds:

0.2' × 60 = 12"

So, 125.67° is equivalent to 125° 40.2' 12".

To find the degree measure equivalent to 10π/13 rounded to the nearest hundredth of a degree, we can divide 10π/13 by π and then multiply by 180° to convert from radians to degrees:

(10π/13) / π * 180° ≈ 138.46° (rounded to the nearest hundredth).

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The correct time(s) when you are at the same height as the top of the tower are approximately -1.57 hours, 1.57 hours, 4.71 hours, 7.85 hours, 11.00 hours, and so on.

To find the time or times during the ride on the London Eye when you are at the same height as the top of the Elizabeth Tower, we can solve the given equation for t.

320 = -197cos(π/15(t)) + 246

First, let's isolate the cosine term:

-197cos(π/15(t)) = 320 - 246

-197cos(π/15(t)) = 74

Next, divide both sides by -197:

cos(π/15(t)) = 74 / -197

Now, we can take the inverse cosine (arccos) of both sides to solve for t:

π/15(t) = arccos(74 / -197)

To isolate t, multiply both sides by 15/π:

t = (15/π) * arccos(74 / -197)

Using a calculator to evaluate the arccosine term and performing the calculation, we find the value(s) of t:

t ≈ -1.57, 1.57, 4.71, 7.85, 11.00, ...

These values represent the time(s) during the ride on the London Eye when you are at the same height as the top of the Elizabeth Tower. Note that time is typically measured in hours, so these values can be converted accordingly.

In light of this, the appropriate time(s) when you are at the same altitude as the tower's peak are roughly -1.57 hours, 1.57 hours, 4.71 hours, 7.85 hours, 11.00 hours, and so forth.

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The slope of 0.75 corresponds to option 3. So, the correct answer is 3) 0.75.

To find the slope, we can use the formula for the slope of a line:

slope (m) = (change in y) / (change in x)

In this case, x represents the number of miles to the destination and y represents the total cost of the load.

Given that the charge to deliver a load 40 miles is $280 and the charge to deliver a load 56 miles is $292, we can set up two points on the line: (40, 280) and (56, 292).

Now let's calculate the change in y and change in x:

Change in y = 292 - 280 = 12

Change in x = 56 - 40 = 16

Plugging these values into the slope formula:

slope (m) = (change in y) / (change in x) = 12 / 16 = 0.75

Therefore, the slope of the line representing the relationship between the number of miles (x) and the total cost of the load (y) is 0.75.

Option 3 is correct.

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Answer:

30

Step-by-step explanation:

22/33=20/x

cross multiply

22x=33x20

22x=660

x=660/22

x=30

The proof to show that FA = RN should be completed with the following step and reasons;

Step                                                        Reason_______

FR = AN                                                   Given

RA = RA                                   Reflexive property of Equality

FR + RA = AN + RA                 Addition Property of Equality

FR + RA = FA                         Segment Addition Postulate

AN + RA = RN                        Segment Addition Postulate

FA = RN                                 Transitive Property of Equality

In Geometry, the Segment Addition Postulate states that when there are two end points on a line segment (F) and (N), a third point (A) would lie on the line segment (RN), if and only if the magnitude of the distances between the end points satisfy the requirements of these equations;

FR + RA = FA.

AN + RA = RN.

This ultimately implies that, the Segment Addition Postulate is only applicable on a line segment that contains three collinear points.

By applying the Segment Addition Postulate to the given end points, we can logically deduce that line segment FA is equal to line segment RN based on the steps and reasons stated in the two-column proof shown above.

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Answer:

35.7 ft

Step-by-step explanation:

Given

Hypotenuse (length of the ladder) = 50 ft

Base (distance from the ladder to wall) = 35 ft

Height (of the wall) = [tex]\sqrt{50^{2}-35^{2} }[/tex] = [tex]\sqrt{1275}[/tex] = 35.7 ft

The z-score for Brazil is given as follows:

Z = 0.87.

The z-score formula is defined as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

In which:

The parameters for this problem are given as follows:

[tex]X = 6.24, \mu = 4.8, \sigma = 1.66[/tex]

Hence the z-score for Brazil is given as follows:

Z = (6.24 - 4.8)/1.66

Z = 0.87.

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Answer:

Step-by-step explanation:

To evaluate the expression {p - q, p - q}, which represents the inner product of the polynomial (p - q) with itself, you can follow these steps:

Given p(x) = 5x^2 - x - 3 and q(x) = 2x^2 + 4x - 3.

Subtract q(x) from p(x) to get (p - q):

(p - q)(x) = (5x^2 - x - 3) - (2x^2 + 4x - 3)

= 5x^2 - x - 3 - 2x^2 - 4x + 3

= (5x^2 - 2x^2) + (-x - 4x) + (-3 + 3)

= 3x^2 - 5x

Now, calculate the inner product of (p - q) with itself using the given inner product formula:

{p - q, p - q} = 5(a1)(a2) + 4(b1)(b2) + 3(c1)(c2)

= 5(3)(3) + 4(-5)(-5) + 3(0)(0)

= 45 + 100 + 0

= 145

Therefore, the value of {p - q, p - q} is 145.

Answer:

D The base is 7 units and the height is 3 units.

Step-by-step explanation:

The answer is d I counted the width/base then the height/length and found answer.

Answer:

Step-by-step explanation:

Let's represent the number of people that Ms. Hernandez can bring to the zoo as "x". Each person will require an admission ticket, which costs $10.40 per person. Additionally, there is a parking fee of $3.

The total cost of admission tickets and parking is given as $150. We can set up the equation as follows:

10.40x + 3 = 150

To solve for x, we need to isolate the variable:

10.40x = 150 - 3

10.40x = 147

Now, divide both sides of the equation by 10.40 to solve for x:

x = 147 / 10.40

Using a calculator, we find:

x ≈ 14.13

Since we can't have a fractional number of people, we need to round down to the nearest whole number since we can't bring a fraction of a person. Therefore, Ms. Hernandez can bring 14 people to the zoo, including herself.

The average speed of a car is calculated by dividing the total distance traveled by the total time taken. In this case, the car travels 50+ miles in "of an hour".

To calculate the average speed in miles per hour, we need to determine the value of "of an hour". If the value is given as a fraction, we need to convert it to a decimal.

Assuming "of an hour" is 1/2 (0.5), the average speed can be calculated as:

Average speed = Total distance / Total time

Average speed = 50+ miles / (1/2) hour

To divide by a fraction, we can multiply by its reciprocal:

Average speed = 50+ miles * (2/1) hour

Average speed = 100+ miles per hour

Therefore, the average speed of the car is 100+ miles per hour.

The tangent line at that point is:

y = (-3/25)*x + 6/5

so m = -3/25, and b = 6/5

To find the slope at that point, we need to evaluate the derivative at that point.

y = 3/x

The derivative is:

y' = -3/x²

When x = 5, we have:

y' = -3/5² = -3/25

So that is the slope, m.

Now let's find the line.

The line must pass trhough the point (5, 3/5), then:

3/5 = (-3/25)*5 + b

3/5 = -3/5 + b

3/5 + 3/5 = b

6/5 = b

The equation of the line is:

y = (-3/25)*x + 6/5

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Answer: D. 145°

Step-by-step explanation:

Since it is a parallelogram given by the symbol, then angle B is equal to angle D which is 145°.

Answer:

Step-by-step explanation:

To evaluate the expression (8y)², where y = 5, we substitute the value of y into the expression and perform the operations.

First, substitute y = 5:

(8y)² = (8 * 5)²

Next, perform the operation inside the parentheses:

(8 * 5)² = 40²

Now, calculate the square of 40:

40² = 1600

Therefore, when y = 5, (8y)² is equal to 1600.

1. the linear equation 5(x + 7) - 3(x - 4) = 7x + 2, x = 9

2. In linear equation 4(3x + 5) - 3 = 9x - 7, x = -8

3. the linear equation 1/3(5x - 9) = 2(1/3x + 6), x = 15

A linear equation is an equation in only on variable.

1. To solve the linear equation 5(x + 7) - 3(x - 4) = 7x + 2, we proceed as follows

5(x + 7) - 3(x - 4) = 7x + 2

Expanding the brackets, we have

5x + 35 - 3x + 12 = 7x + 2

Collecting like terms in the expression, we have

5x + 35 - 3x + 12 = 7x + 2

5x - 3x - 7x = 2 - 35 - 12

-5x = -45

x = -45/-5

x = 9

2. To solve linear equation 4(3x + 5) - 3 = 9x - 7, we proceed as follows

4(3x + 5) - 3 = 9x - 7

Expanding the brackets, we have

12x + 20 - 3 = 9x - 7

Collecting like terms in the expression, we have

12x - 9x = - 7 + 3 - 20

3x = -24

x = -24/3

x = -8

3. To solve  the linear equation 1/3(5x - 9) = 2(1/3x + 6), we proceed as follows

1/3(5x - 9) = 2(1/3x + 6)

Expanding the brackets, we have

5x/3 - 3 = 2/3x + 12

Collecting like terms, we have

5x/3 - 2x/3 = 12 + 3

3x/3 = 15

x = 15

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Answer:

y = -3

Step-by-step explanation:

The midline of a sinusoidal function is the horizontal center line about which the function oscillates periodically.

The midline is positioned halfway between the maximum (peaks) and minimum (troughs) values of the graph. It serves as a baseline that helps visualize the oscillations of the function.

To find the equation of the midline, we need to determine the average y-value between the maximum and minimum y-values.

In this case, the maximum y-value is -1, and the minimum y-value is -5. To find the equation of the midline, sum the maximum and minimum y-values, and divide by 2:

[tex]y=\dfrac{-1 + (-5)}{2} = \dfrac{-6}{2}=-3[/tex]

Therefore, the equation of the midline for the graphed sinusoidal function is y = -3.

Answer:

Given the information you provided, we can model cellular phone usage over time with an exponential growth model. An exponential growth model follows the equation:

`y = a * b^(x - h) + k`

where:

- `y` is the quantity you're interested in (cell phone usage),

- `a` is the initial quantity (34 million in 1995),

- `b` is the growth factor (1.22, representing 22% growth per year),

- `x` is the time (the year),

- `h` is the time at which the initial quantity `a` is given (1995), and

- `k` is the vertical shift of the graph (0 in this case, as we're assuming growth starts from the initial quantity).

So, our specific model becomes:

`y = 34 * 1.22^(x - 1995)`

To find the cellular usage in 2003, we plug 2003 in for x:

`y = 34 * 1.22^(2003 - 1995)`

Calculating this out will give us the cellular usage in 2003.

Let's calculate this:

`y = 34 * 1.22^(2003 - 1995)`

So,

`y = 34 * 1.22^8`

Calculating the above expression gives us:

`y ≈ 97.97` million.

So, the cellular phone usage in 2003, according to this model, is approximately 98 million.

Approximately 76.5% of the student body at Walton High School loves math.

To determine the percentage of the student body that loves math, we need to consider the proportions of males and females in the Walton High School student body and their respective percentages of loving math.

Given that 45% of the student body are males, we can deduce that 55% are females (since the total percentage must add up to 100%). Now let's calculate the percentage of the student body that loves math:

For the females:

55% of the student body are females.

90% of the females love math.

So, the percentage of females who love math is 55% * 90% = 49.5% of the student body.

For the males:

45% of the student body are males.

60% of the males love math.

So, the percentage of males who love math is 45% * 60% = 27% of the student body.

To find the total percentage of the student body that loves math, we add the percentages of females who love math and males who love math:

49.5% + 27% = 76.5%

As a result, 76.5% of Walton High School's student body enjoys maths.

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The area of the shaded part is 640.56 m²

The area of a shape is the space occupied by the boundary of a plane figures like circles, rectangles, and triangles.

The figure is a concentric circle, i.e a circle Ina circle. Therefore to calculate the area of the shaded part,

Area of shaded part = area of big circle - area of small circle

area of big circle = 3.14 × 20²

= 1256

area of small circle = 3.14 × 14²

= 615.44

Area of shaded part = 1256 - 615.44

= 640.56m²

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Answer:

would this not be a converse statement?

Step-by-step explanation:

triangle ABC is the hypothesis and it's conclusion is triangle DEF

Answer:

Step-by-step explanation:

To determine the length of AB, we need to find the value of x.

We are given that AC = 2x + 7, BC = x, and AB = 5x + 9.

Since B is on the line segment AC, the sum of lengths AB and BC should equal the length of AC. Therefore, we can set up the equation:

AB + BC = AC

Substituting the given values, we have:

(5x + 9) + x = 2x + 7

Simplifying the equation:

6x + 9 = 2x + 7

Bringing like terms to one side:

6x - 2x = 7 - 9

4x = -2

Dividing both sides by 4:

x = -2/4

Simplifying:

x = -1/2

Now that we have the value of x, we can substitute it back into the expression for AB to find its numerical length:

AB = 5x + 9 = 5(-1/2) + 9 = -5/2 + 9 = (18 - 5)/2 = 13/2 = 6.5

Therefore, the numerical length of AB is 6.5.

Sampling based upon equal probability is called d. Simple random sampling. The correct answer is d. Simple random sampling.

Simple random sampling is a sampling technique where each individual in the population has an equal probability of being selected for the sample. It is based on the principle of equal probability, ensuring that every element has the same chance of being chosen. This method involves randomly selecting samples without any specific grouping or stratification.

Cluster sampling involves dividing the population into clusters or groups and randomly selecting entire clusters for inclusion in the sample. It does not guarantee equal probability for individual units within each cluster.

Probability sampling is a general term that encompasses different sampling methods, including simple random sampling, stratified random sampling, and cluster sampling. It refers to sampling techniques that rely on random selection and allow for the calculation of probabilities associated with sample estimates.

Stratified random sampling involves dividing the population into distinct strata based on certain characteristics and then selecting samples from each stratum in proportion to their representation in the population. It does not guarantee equal probability of selection for all individuals.

Systematic sampling involves selecting every kth individual from a population list after randomly selecting a starting point. It does not guarantee equal probability of selection for all individuals.

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