[tex]d^{2}=15^{2}+9^{2}+10^{2}[/tex]

Using the mean value theorem, we can find an upper bound for f(7) given the information provided. The mean value theorem states that for a differentiable function f(x) on the interval [a,b], there exists at least one point c in the interval such that:

f'(c) = (f(b) - f(a))/(b - a)

If we apply this theorem to the interval [4,7], we get:

f'(c) = (f(7) - f(4))/(7 - 4)

Since f '(x) ≥ 3 for 4 ≤ x ≤ 7, we know that f'(c) ≥ 3. We can use this inequality to find an upper bound for f(7):

3 ≤ (f(7) - 6)/3

9 ≤ f(7) - 6

f(7) ≥ 15

Therefore, the smallest possible value for f(7) is 15. This means that f(x) must be increasing at a rate of at least 3 between x=4 and x=7, and the smallest possible value of f(7) occurs when f(x) is increasing at a constant rate of 3.

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By algebra properties, the simplified form of the expressions are listed below in the following four cases:

Case 1: 6

Case 2: 1 / 5

Case 3: √3

Case 4: - 3

In this problem we must simplify expressions involving powers and roots by algebra properties, mainly power and root properties. Now we proceed to show how each expression is simplified:

Case 1

[tex](1^{3}+2^{3}+ 3^{3})^{\frac {1}{2}}[/tex]

[tex](1 + 8 + 27)^{\frac{1}{2}}[/tex]

[tex]36^{\frac{1}{2}}[/tex]

√36

6

Case 2

[tex]\left[\left(625^{-\frac{1}{2}}\right)^{-\frac{1}{4}}\right]^{2}[/tex]

[tex]\left[\left[\left(625^{\frac{1}{2}}\right)^{-1}\right]^{-\frac {1}{4}}\right]^{2}[/tex]

[tex]\left[\left(\frac{1}{25}\right)^{\frac{1}{4}}\right]^{2}[/tex]

[tex]\left(\frac{1}{25} \right)^{\frac{1}{2}}[/tex]

√(1 / 25)

1 / 5

Case 3

[tex]\frac{9^{\frac{1}{2}}\times 27^{- \frac {1}{3}}}{3^{\frac{1}{6}}\times 3^{-\frac{2}{3}}}[/tex]

[tex]\frac{(3^{2})^{\frac{1}{2}}\times (3^{3})^{-\frac{1}{3}}}{3^{\frac{1}{6}}\times 3^{-\frac{2}{3}}}[/tex]

[tex]\frac{3\times 3^{- 1}}{3^{\frac{1}{6}}\times 3^{- \frac{2}{3}}}\\[/tex]

[tex]\frac{1}{3^{-\frac{1}{2}}}[/tex]

[tex]3^{\frac{1}{2}}[/tex]

√3

Case 4

[tex]64^{-\frac{1}{3}}\cdot \left[64^{\frac{1}{3}}-64^{\frac{2}{3}}\right][/tex]

[tex]64^{-\frac{1}{3}}\cdot 64^{\frac{1}{3}}-64^{-\frac{1}{3}}\cdot 64^{\frac{2}{3}}[/tex]

[tex]1 - 64^{ \frac{1}{3}}[/tex]

1 - ∛64

1 - 4

- 3

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If money is invested at 6% compounded daily, the interest rate per day is 6%/365 = 0.01644%.

To find the number of days it takes to quadruple the money, we can use the formula A = P(1 + r/n)^(nt), where A is the final amount, P is the initial amount, r is the annual interest rate, n is the number of times compounded per year, and t is the time in years. In this case, we want A/P = 4, so we have:

4 = (1 + 0.0001644/1)^(1t)

ln(4) = tln(1 + 0.0001644/1)

t = ln(4)/ln(1 + 0.0001644/1) ≈ 123.73 days

Therefore, it will take about 123.73 days or approximately 4 months to quadruple the money at 6% compounded daily.

If money is invested at 6.9% compounded continuously, we can use the formula A = Pe^(rt) to find the time it takes to quadruple the money. Again, we want A/P = 4, so we have:

4 = e^(0.069t)

ln(4) = 0.069t

t = ln(4)/0.069 ≈ 10.04 years

Therefore, it will take about 10.04 years to quadruple the money at 6.9% compounded continuously.

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Mike saved his money for 7 years to earn $280 in interest at a simple interest rate of 2%.

The simple interest formula:

I = P × r × t

Where:

I is the interest earned

P is the principal (the initial amount of money saved)

r is the interest rate

t is the time (in years)

We know that Mike saves $2000 at a simple interest rate of 2% and he earns $280 in interest.

So we can plug in these values and solve for "t":

280 = 2000 × 0.02 × t

Dividing both sides by (2000 × 0.02):

280 / (2000 × 0.02) = t

t = 7

I = P r t is the formula for calculating interest.

P stands for principle, which is the original sum of money saved and r stands for interest rate.

The date is (in years).

We are aware that Mike gets $280 in interest on his savings of $2000 at a basic interest rate of 2%.

Thus, we may enter these numbers and find the value of "t":

280 = 2000 × 0.02 × t

by (2000 0.02), divide both sides:

280 / (2000 × 0.02)= 7

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The given series 4^n / 2^n 3^n is convergent.

To see why, we can use the ratio test, which states that if the limit of the ratio of consecutive terms is less than 1, then the series converges. Applying the ratio test to the given series, we get:

lim n→∞ |(4^n+1 / 2^n+1 3^n+1) / (4^n / 2^n 3^n)|

= lim n→∞ |4 / 3(1 + 1/2n+1)|

= 4/3

Since the limit is less than 1, the series converges. To find its sum, we can use the formula for the sum of a convergent geometric series:

S = a / (1 - r)

where a is the first term and r is the common ratio. In this case, a = 4/6 = 2/3 and r = 2/3, so we get:

S = (2/3) / (1 - 2/3) = 2

Therefore, the sum of the series is 2.

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the question is that the zeros of the polynomial f(x)=(x−5)5(x+1)7 are x=5 and x=-1, and their multiplicities are 5 and 7, respectively.

the zeros and their multiplicities is as follows:

To find the zeros of the polynomial, we set each factor equal to zero and solve for x.

For the factor (x−5)5, we get x=5 as the only zero.

For the factor (x+1)7, we get x=-1 as the only zero.

To determine the multiplicities of the zeros, we count the number of times each zero appears as a factor.

Since (x−5)5 is a factor of the polynomial, the zero x=5 has a multiplicity of 5.

Similarly, since (x+1)7 is a factor of the polynomial, the zero x=-1 has a multiplicity of 7.

the zeros of the polynomial f(x)=(x−5)5(x+1)7 are x=5 and x=-1, and their multiplicities are 5 and 7, respectively.

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Answer:

Angle PRQ = 28 degrees.

Step-by-step explanation:

Angles P and Q are the same! That's bc this is an isosceles triangle.

The total of all 3 angles = 180. So to find R, subtract the other 2 angles from 180.

So 180-76-76 = 28. That's angle R

To test the hypothesis that the mean salary for associate professors in research institutions is $2000 higher than for those in other institutions, we can perform a two-sample t-test.

The null hypothesis is that the difference in means is not significantly different from $2000, while the alternative hypothesis is that the difference is greater than $2000.

Using the given information, we can calculate the t-statistic as (70750 - 65200 - 2000) / sqrt((6000^2/200) + (5000^2/200)) = 5.39. With 398 degrees of freedom (n1 + n2 - 2), the p-value for this one-sided test is less than 0.0001.

Since this p-value is much smaller than any reasonable level of significance, we reject the null hypothesis and conclude that there is strong evidence that the mean salary for associate professors in research institutions is significantly higher than for those in other institutions by $2000.

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To test the hypothesis that the mean salary for associate professors in research institutions is $2000 higher than for those in other institutions, we can perform a two-sample t-test.

The null hypothesis is that the difference in means is not significantly different from $2000, while the alternative hypothesis is that the difference is greater than $2000.

Using the given information, we can calculate the t-statistic as (70750 - 65200 - 2000) / sqrt((6000^2/200) + (5000^2/200)) = 5.39. With 398 degrees of freedom (n1 + n2 - 2), the p-value for this one-sided test is less than 0.0001.

Since this p-value is much smaller than any reasonable level of significance, we reject the null hypothesis and conclude that there is strong evidence that the mean salary for associate professors in research institutions is significantly higher than for those in other institutions by $2000.

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Answer:

x = 26

Step-by-step explanation:

complementary angles sum to 90° , that is

∠ ABC + ∠ CBD = 90

2x + 38 = 90 ( subtract 38 from both sides )

2x = 52 ( divide both sides by 2 )

x = 26

Answer

D. 126 inches squared

Step-by-step explanation:

8 x 17 = 136

2 x 5 = 10

136-10= 126

The half-life of the goo is approximately 18.75 minutes. The formula for g(t), the amount of goo remaining at time t, is g(t) = 132 * (1/2)^(t/18.75).

To find the half-life of the goo, we can use the formula for exponential decay: A(t) = A0 * (1/2)^(t/h), where A(t) is the amount of radioactive substance at time t, A0 is the initial amount, h is the half-life, and t is time. We are given A0 = 132 grams, A(75) = 2.0625 grams, and we need to solve for h. Plugging in these values, we get:

2.0625 = 132 * (1/2)^(75/h)

Solving for h, we get h ≈ 18.75 minutes.

The formula for g(t) is g(t) = A0 * (1/2)^(t/h). Plugging in A0 = 132 and h = 18.75, we get g(t) = 132 * (1/2)^(t/18.75). This formula gives us the amount of goo remaining at time t, where t is measured in minutes.

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The volume of the square pyramid that has sides of length 15 cm and height of 14 cm is: D. 1050 cm³

To find the volume of a square pyramid, you can use the formula: Volume = (1/3) * Base Area * Height.

Since the base of the square pyramid has sides of length 15 cm, the base area can be calculated as:

Base area = 15 cm * 15 cm

= 225 cm².

Plugging the values into the formula, the volume of the pyramid:

= (1/3) * 225 * 14 cm

= 1050 cm³.

Therefore, the volume of the square pyramid is 1050 cm³.

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Answer:

should be 18 ft

Step-by-step explanation:

6÷4= 1.5

27÷1.5= 18

Answer:

8√3

Step-by-step explanation:

method 1

180°-(30°+90°)= 60°

8=sin 30° × chord

sin 30°=1/2

chord=16

x^2 + 8^2 = 16^2

x=√256 - 64

x= √192 = 8√3

method 2:

use arcsin & arccos

method 3:

...

The true statements regarding the measures of center and variability of these data sets are: statements A, C and E.

To analyze the statements regarding the measures of center and variability, let's calculate the required measures for each data set.

For Yard Works:

Tree heights: 7, 9, 7, 12, 5

Mean: (7 + 9 + 7 + 12 + 5) / 5 = 40 / 5 = 8

Median: 7, 7, 9, 12, 5 → Median = 7

Range: 12 - 5 = 7

Mean absolute deviation (MAD): Calculate the absolute difference of each value from the mean, then find the average of those differences.

|7 - 8| + |9 - 8| + |7 - 8| + |12 - 8| + |5 - 8| = 1 + 1 + 1 + 4 + 3 = 10 / 5 = 2

For The Grow Station:

Tree heights: 9, 11, 6, 12, 7

Mean: (9 + 11 + 6 + 12 + 7) / 5 = 45 / 5 = 9

Median: 6, 7, 9, 11, 12 → Median = 9

Range: 12 - 6 = 6

Mean absolute deviation (MAD):

|9 - 9| + |11 - 9| + |6 - 9| + |12 - 9| + |7 - 9| = 0 + 2 + 3 + 3 + 2 = 10 / 5 = 2

Analyzing the statements:

A. The mean of the tree heights at Yard Works is less than the mean of the tree heights at The Grow Station.

True. Mean Yard Works < Mean The Grow Station (8 < 9)

B. The median of the tree heights at Yard Works is greater than the median of the tree heights at The Grow Station.

False. Median Yard Works = Median The Grow Station (7 = 9)

C. The range of the tree heights at Yard Works is greater than the range of the tree heights at The Grow Station.

True. Range Yard Works > Range The Grow Station (7 > 6)

D. The mean absolute deviation of the tree heights at Yard Works is greater than the mean absolute deviation of the tree heights at The Grow Station.

False. MAD Yard Works = MAD The Grow Station (2 = 2)

E. The mean absolute deviation of the tree heights at Yard Works is equal to the mean absolute deviation of the tree heights at The Grow Station.

True. MAD Yard Works = MAD The Grow Station (2 = 2)

In summary, the true statements are: A, C, and E.

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The owner of the bookstore sells the used books for $6 each. J.

The price of a used book in the bookstore we need to calculate how much the owner is selling the books for.

The owner of the bookstore buys the used books from customers for $1.50 each.

The owner resells the used books for we need to multiply the cost price by 400%:

$1.50 x 400% = $1.50 x 4

= $6

The markup percentage for the used books is very high.

The owner is reselling the used books for four times the amount he paid for them.

This is a common practice in the used book industry as it allows the owner to make a profit on the books they sell.

It is important for customers to be aware of the markup and shop around for the best prices.

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The bearing form D from A, according to the figure is

205 degrees

Bearings are measured form the North and in the clockwise direction

Examining the figure and applying the clockwise direction to measure the angles, we have the bearing as

bearing form D from A = angle N to B + angle B to C + angle C to D

angle C to D is not given and solved using sum of angles in a point

angle C to D = 360 - 155 - 15 - 166

angle C to D = 24 degrees

plugging in the values

bearing form D from A = 15 + 166 + 24

bearing form D from A = 205 degrees

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A tile is drawn and replaced, and then a second tile is drawn. Therefore, option A and B are correct answers.

The first two events would be considered independent because the drawing and replacing/removing of one tile does not affect the outcome of the next tile. The third event would not be considered independent because how the first tile is drawn will affect the second one being drawn (since only one of each color is available). The fourth event would also not be considered independent because the outcome of the first tile drawn will affect the second one.

Therefore, option A and B are correct answers.

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To find the equations of the tangent and normal lines to the hyperbola x^2/4 − y^2/1 = 1 at the point where x = 4, we need to first find the y-coordinate of the point of tangency. We can do this by substituting x = 4 into the equation of the hyperbola and solving for y:

x^2/4 - y^2/1 = 1

(4)^2/4 - y^2/1 = 1

16/4 - y^2/1 = 1

4 - y^2 = 1

y^2 = 3

y = ±√3

So, the point of tangency is (4, √3).

Now, to find the equation of the tangent line at this point, we need to take the derivative of the equation of the hyperbola implicitly with respect to x:

x^2/4 - y^2/1 = 1

Differentiating both sides with respect to x:

x/2 - 2y(dy/dx) = 0

dy/dx = x/(4y)

At the point (4, √3), we have:

dy/dx = 4/(4√3) = √3/3

So the slope of the tangent line at this point is √3/3. Using the point-slope form of the equation of a line, we can write the equation of the tangent line as:

y - √3 = (√3/3)(x - 4)

Simplifying, we get:

y = (√3/3)x - (√3/3)∙4 + √3

y = (√3/3)x - (√3/3) + √3

y = (√3/3)x + 2√3/3

To find the equation of the normal line, we first need to find its slope, which is the negative reciprocal of the slope of the tangent line. So:

m(normal) = -1/m(tangent) = -1/(√3/3) = -√3

Using the point-slope form again, the equation of the normal line is:

y - √3 = (-√3)(x - 4)

Simplifying, we get:

y = -√3x + 4√3 + √3

y = -√3x + 5√3

So the equations of the tangent and normal lines to the hyperbola x^2/4 − y^2/1 = 1 at the point where x = 4 are:

Tangent line: y = (√3/3)x + 2√3/3

Normal line: y = -√3x + 5√3

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The solution of the two system of equations using elimination method is x = 8, and y = 3.

The solution of the two system of equations using elimination method is calculated as follows;

The given equations;

2x - 5y = 1   -------- (1)

-3x + 2y = -18 ----- (2)

To eliminate x, multiply equation (1)  by 3 and equation (2) by 2, and add the to equations together;

3:   6x  -  15y = 3

2:   -6x  + 4y  = -36

-----------------------------------

             -11y = -33

               y = 33/11  =  3

Now, solve for the value of x by substituting the value of y back into any of the equations.

2x - 5y = 1

2x - 5(3) = 1

2x - 15 = 1

2x = 16

x = 8

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Answer:

129*

Step-by-step explanation:

First notice the 2 parallel lines. Since the 2 parallel lines are intersecting the same line, the angle measures would be the same for both. Since the first angle is 51*, the corresponding angle for the next line is also 51*. Since it intersects a straight line, the angles need to add up to 180*. 51 + x = 180 so x = 129*

Answer:

1.5.1 : constant difference is 8

1.5.2: When there are 10 chairs stacked, it's 127 cm tall.

Step-by-step explanation:

There is a linear relationship between the height of the stack and the number of chairs.

1 chair = 55 cm + 0 extra cm = 55cm

2 chairs = 55cm + 8cm = 63 cm

3 chairs = 55cm + 8(2)cm = 71 cm

4 chairs = 55cm + 8(3)cm = 79 cm

1.5.1 the constant difference between all the underlined numbers above is 8.

1.5.2 You could just keep calculating above until you get 127 cm. (Your teacher might not like that, but it's an option!)

You can find the equation & either solve for the number of chairs OR graph it.

So if we let C = the number of stacked chairs, our equation for H (height) would be:

H = 55 + 8(C-1)

If we substitute H = 127, solve for C.

127 = 55+ 8(c-1)

127 = 55+ 8c-8

127 = 47 + 8c

127 -47 = 8c

80 = 8c

10=c

When there are 10 chairs stacked, it's 127 cm tall.

Check that the answer works:

55 cm (1st chair) + 8*9 (8cm for each additional chair) = 55+ 72 = 127 cm

The area of the surface obtained by rotating the curve y = (x³/6) + (1/2x) from 1/2 to 1 about the x-axis is given by the above expression is 2π (1/6 x √(1+ 9x² - 3x⁴/4)).

Calculate the arc length of the curve

We first need to calculate the arc length of the curve, which can be done using the formula:

L = ∫aᵇ √(1+ (dy/dx)²) dx

where,

dy/dx = (3x² - 1/2x²)/6

Therefore, the arc length of the curve is given by:

L = ∫1/2¹√(1+ (3x² - 1/2x² )/6)dx

Calculate the area of the surface

Once we have the arc length of the curve, we can calculate the area of the surface obtained by rotating the curve about the x-axis. This can be done using the formula:

A = 2π × L

Substituting the arc length of the curve in the formula, we get:

A = 2π × ∫1/2¹√(1+ (3x² - 1/2x²)/6)dx

Evaluate the integral

Finally, we need to evaluate the integral in order to calculate the area of the surface. We can do this using integration by parts, which gives us:

A = 2π × ∫1/2¹√(1+ (3x² - 1/2x²)/6)dx

= 2π (1/6 x √(1+ 9x² - 3x⁴/4) - (1/6) ∫1/2¹ (9x² - 3x⁴/4)/√(1+ 9x² - 3x⁴4) dx)

Therefore, the area of  the surface is 2π (1/6 x √(1+ 9x² - 3x⁴/4)).

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The x and y intercept of the equation is (8,12)

A linear equation is an algebraic equation of the form y=mx+b. It involves only a constant and a first-order term, where m is the slope and b is the y-intercept.

For 3x +2y = 24

we need to put it to the standard form

2y = 24 - 3x

divide both sides by 2

y = 12 - 3/2x

Here b is 12 and m is -3/2

therefore the y intercept is 12

when y = 0

0 = 12 -3/2 x

3/2 x = 12

3x = 24

divide both sides by 3

x = 24/3 = 8

therefore the x intercept is 8

The x and y intercept of the equation is ( 8,12)

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The length of line CD is 15 cm.

To calculate the length of line CD, we can use the property of similar triangles.

In triangle ABC, we can see that triangle ABE is similar to triangle ACD.

Using the property of similar triangles, we can set up the following proportion:

AB/AC = BE/CD

Substituting the given values:

12/18 = 10/CD

To solve for CD, we can cross-multiply and solve the resulting equation:

12 × CD = 18 × 10

CD = (18 × 10) / 12

CD = 180 / 12

CD = 15

Therefore, the length of line CD is 15 cm.

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(a) False
(b) True
(c) True
(d) True
(e) True
(f) True
(a) False: ∇f(a,b,c) is not parallel to the tangent plane to S at (a,b,c).

(b) True: ∇f(a,b,c) is perpendicular to the tangent plane to S at (a,b,c).

(c) True: If ⟨m,n,q⟩ is a nonzero vector on the tangent plane to S at (a,b,c), then ⟨m,n,q⟩×∇f(a,b,c) must be ⟨0,0,0⟩.

(d) True: If ⟨m,n,q⟩ is a nonzero derivative vector on the tangent plane to S at (a,b,c), then ⟨m,n,q⟩.∇f(a,b,c) must be 0.

(e) True: ∣∇f(a,b,c)∣=∣−∇f(a,b,c)∣

(f) True: Let u be a unit vector in R3. Then, −∣∇f(a,b,c)∣≤Duf(a,b,c)≤∣∇f(a,b,c)∣

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The norm of u is 276.46 and the distance between u and v is 259.98 in M22 with the standard inner product.

To find the norm ||u|| of the vector u=[39 276] in M22 with the standard inner product, we use the formula:

||u|| = sqrt(<u,u>)

where <u,u> is the dot product of u with itself.

<u,u> = (39 * 39) + (276 * 276) = 76461

Therefore, ||u|| = sqrt(76461) = 276.46 (rounded to two decimal places).

To find the distance d(u,v) between vectors u=[39 276] and v=[-64 19] in M22 with the standard inner product, we use the formula:

d(u,v) = sqrt(<u-v,u-v>)

where <u-v,u-v> is the dot product of the difference between u and v with itself.

<u-v,u-v> = (39 - (-64))^2 + (276 - 19)^2 = 12769 + 54756 = 67525

Therefore, d(u,v) = sqrt(67525) = 259.98 (rounded to two decimal places).

Therefore, the norm of u is 276.46 and the distance between u and v is 259.98 in M22 with the standard inner product.

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The volume of the solid is (11π/3) cubic units.

We have,

To find the volume of the solid obtained by rotating the region bounded by y = x and y = x^2 about the line x = -3, we can use the method of cylindrical shells.

The formula for the volume using cylindrical shells is given by:

V = 2π ∫ [a, b] x h(x) dx,

where [a, b] is the interval of integration, x represents the variable of integration, and h(x) represents the height of the shell at each value of x.

In this case, we want to rotate the region bounded by y = x and y = x² about the line x = -3.

Since we are rotating about a vertical line, the height of the shell at each value of x will be given by the difference between the x-coordinate of the curve and the line of rotation:

h(x) = (x - (-3)) = x + 3.

To find the interval of integration, we need to determine the x-values where the two curves intersect.

Setting x = x², we have:

x = x²,

x² - x = 0,

x (x - 1) = 0.

This gives us two intersection points: x = 0 and x = 1.

Therefore, the interval of integration is [0, 1].

Now we can set up the integral to find the volume:

V = 2π ∫ [0, 1] x (x + 3) dx.

Evaluating this integral, we have:

V = 2π ∫ [0, 1] (x² + 3x) dx

= 2π [x³/3 + (3/2)x²] evaluated from 0 to 1

= 2π [(1/3 + 3/2) - (0/3 + 0/2)]

= 2π [(2/6 + 9/6) - 0]

= 2π (11/6)

= (22π/6)

= (11π/3).

Therefore,

The volume of the solid is (11π/3) cubic units.

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Thus, the general solution is y(x) = -x + 2x^3 + c₁x + c₂x^3.

To find the general solution of the differential equation x^2y'' + 3xy' + 3xy = 4x^7, we can use the method of variation of parameters.

Given that {x, x^3} is a fundamental set of solutions of the homogeneous equation x^2y'' - 3xy' + 3y = 0, we can use these solutions to find the particular solution.

Let's assume the particular solution has the form y_p = u(x)x + v(x)x^3, where u(x) and v(x) are unknown functions.

Differentiating y_p:

y_p' = u'x + u + v'x^3 + 3v(x)x^2

Differentiating again:

y_p'' = u''x + 2u' + v''x^3 + 6v'x^2 + 6v(x)x

Substituting these derivatives into the original differential equation, we have:

x^2(u''x + 2u' + v''x^3 + 6v'x^2 + 6v(x)x) + 3x(u'x + u + v'x^3 + 3v(x)x^2) + 3x(u(x)x + v(x)x^3) = 4x^7

Simplifying and grouping like terms:

x^3(u'' + 3v') + x^2(2u' + 3v'' + 3v) + x(u + 3v' + 3v) + (2u + v) = 4x^5

Setting the coefficients of each power of x to zero, we get the following system of equations:

x^3: u'' + 3v' = 0

x^2: 2u' + 3v'' + 3v = 0

x^1: u + 3v' + 3v = 0

x^0: 2u + v = 4

Solving this system of equations, we find:

u = -1

v = 2

Therefore, the particular solution is y_p = -x + 2x^3.

The general solution of the differential equation x^2y'' + 3xy' + 3xy = 4x^7 is given by the sum of the particular solution and the homogeneous solutions:

y(x) = y_p + c₁x + c₂x^3

where c₁ and c₂ are arbitrary constants.

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The bottom right graph shows the line of best fit for the data.

For a data-set, the definition of a residual is that it is the difference of the actual output value by the predicted output value, that is:

Residual = Observed - Predicted.

Hence the graph of the line of best fit should have the smallest possible residual values, meaning that the points on the scatter plot are the closest possible to the line.

For this problem, we have that the bottom right graph has the smaller residuals, hence it shows the line of best fit for the data.

More can be learned about residuals at brainly.com/question/1447173

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