find the minimum volume of a sphere that contains a right cylinder with volume 12p3πcubic centimeters.

Answer:

y = 16

Step-by-step explanation:

You are in putting 2, meaning that x = 2. Plug in the corresponding numbers to the corresponding variables:

[tex]y = 12x - 8\\x = 2\\\\y = 12(2) - 8[/tex]

Remember to follow the order of operations, PEMDAS. PEMDAS stands for:

Parenthesis

Exponents (& Roots)

Multiplications

Divisions

Additions

Subtractions

~

First, multiply 12 with 2, then subtract 8:

[tex]y = 12(2) - 8\\y = (12 * 2) - 8\\y = (24) - 8\\y = 16[/tex]

y = 16 is your answer.

~

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1. The inverse function, f^(-1)(x) = (x + 2)/5.

2. The derivative of the inverse function, (f^(-1))'(x) = 1/5.

3. (f^(-1))'(3) = 1/5.

We know that a function is invertible if and only if it is one-to-one and onto. In this case, we can easily see that f(x) is a one-to-one function because different inputs always give different outputs, and it is also onto because any real number can be obtained as an output. Therefore, f(x) is invertible.

To find (f^-1)'(3), we need to use the formula for the derivative of the inverse function:

(f^-1)'(3) = 1 / f'(f^-1(3))

First, we need to find f^-1(x). We can do this by solving the equation y = 5x - 2 for x in terms of y:

y = 5x - 2

y + 2 = 5x

x = (y + 2) / 5

Therefore, f^-1(x) = (x + 2) / 5.

Now we can find f'(x):

f(x) = 5x - 2

f'(x) = 5

Next, we need to find f^-1(3):

f^-1(3) = (3 + 2) / 5 = 1

Finally, we can use the formula to find (f^-1)'(3):

(f^-1)'(3) = 1 / f'(f^-1(3)) = 1 / f'(1) = 1 / 5

Therefore, the answer is b) 1/15.

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The hydrostatic pressure on the bottom of the aquarium is 4015 lb/ft2. The hydrostatic pressure on the bottom of the aquarium is 96360 lb. The hydrostatic pressure on one end of the aquarium is 97440 lb.

(a) The hydrostatic pressure on the bottom of the aquarium can be found using the formula P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the gravitational acceleration, and h is the depth. In this case, ρ = 62.5 lb/ft3, g = 32.2 ft/s2, and h = 2 ft. The pressure is:

P = ρgh = 62.5 lb/ft3 × 32.2 ft/s2 × 2 ft = 4015 lb/ft2

So the hydrostatic pressure on the bottom of the aquarium is 4015 lb/ft2.

(b) The hydrostatic force on the bottom of the aquarium can be found using the formula F = P A, where F is the force, P is the pressure, and A is the area. The area of the bottom of the aquarium is 6 ft × 4 ft = 24 ft2. The force is:

F = P A = 4015 lb/ft2 × 24 ft2 = 96360 lb

So the hydrostatic force on the bottom of the aquarium is 96360 lb.

(c) The hydrostatic force on one end of the aquarium can be found using the formula F = ρgAh, where A is the area of the end, which is 6 ft × 2 ft = 12 ft2. The depth of the end is 4 ft. So the force is:

F = ρgAh = 62.5 lb/ft3 × 32.2 ft/s2 × 12 ft2 × 4 ft = 97440 lb

So the hydrostatic force on one end of the aquarium is 97440 lb.

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True. This can be determined using the formula for calculating the z-score: Z = (X - M) / (S / sqrt(n)), where X is the score, M is the mean, S is the sample standard deviation, and n is the sample size. Substituting the given values, we get:

Z = (72 - 76) / (8 / sqrt(1)) = -0.5

Therefore, a score of X = 72 corresponds to Z = -0.50, given that the sample has a mean of M = 76 and a sample standard deviation of S = 8.
True. Given a sample with a mean (M) of 76 and a sample standard deviation (S) of 8, you can calculate the Z-score for a score of X = 72 using the formula:

Z = (X - M) / S

Z = (72 - 76) / 8

Z = (-4) / 8

Z = -0.50

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By following cosine law, The ladybug is 1 foot above the ground when the second hand points straight up, 0 feet above the ground after 30 seconds, approximately 0.29 feet above the ground after 45 seconds, and 2 feet above the ground after 60 seconds.

The cosine law, commonly referred to as the law of cosines, is a rule that explains how a triangle's sides and angles relate to one another. According to this rule, the square of any side is equal to the difference between the squares of the other two sides added together, multiplied by two, and the cosine of the angle between the other two sides. It can be used to solve for missing information and is applicable to any triangles1. It makes the Pythagorean theorem more prevalent.

The second hand of the clock is rotating in a circle like the ladybug does. One foot, or the length of the second hand, makes up the circle's radius. The ladybug is 10 feet above the ground when the second hand is immediately to the right. With a radius of 10 feet, this indicates that the ladybug is travelling in a vertical circle.

The following formula can be used to determine the height above the ground:

radius is equal to (radius× cos(angle)) - distance.

where r is the circle's radius and is the angle formed by the second hand and vertical axis.

Angle = 0 degrees when the second hand is pointing up straight, so:

Distance is equal to 1 - (1× cos(0)) = **1 foot**.

Angle equals 90 degrees after 30 seconds, so:

Distance is equal to 1 - (1 × cos(90)) = 0 ft.

Angle = 135 degrees after 45 seconds, so:

Distance is equal to 1 - (1 ×cos(135)) **0.29 feet**.

Angle equals 180 degrees after 60 seconds, so:

Distance = 1 - (1×cos(180)), which is **2 feet**.

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5x² is the greatest common monomial factor (GCMF) of 5x²-10x³, and 5x²(1-2x) is the factored form.

We hunt for the greatest monomial that splits both terms evenly to obtain the GCMF of 5x²-10x³. In this situation, both words have an x² factor, hence the GCMF is 5x². Using the distributive property, we can factor this out:

5x² - 10x³ = 5x²(1 - 2x)

This is the factored version of the formula, which demonstrates that 5x2 is a common factor of both components and that (1-2x) is the remaining factor. We can verify this by multiplying 5x² by (1-2x) and getting 5x² - 10x³, which is the original formula.

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Area = πr²

= π × 8²

= 64π cm²

All positive integers n greater than 1. Therefore, we can conclude that n! < n^n for all n > 1

a) The statement P(2) is 2! < 2^2.

b) P(2) is true since 2! = 2 < 4 = 2^2.

c) The inductive hypothesis is to assume that P(k) is true for some positive integer k.

d) In the inductive step, we need to prove that P(k+1) is true, assuming that P(k) is true.

e) To complete the inductive step, we start with the assumption that P(k) is true, which means that k! < k^k. We then need to prove that (k+1)! < (k+1)^(k+1).

(k+1)! = (k+1) * k! < (k+1) * k^k (since k! < k^k by the inductive hypothesis)

< (k+1) * (k+1)^k

= (k+1)^(k+1)

Therefore, we have shown that (k+1)! < (k+1)^(k+1), and thus P(k+1) is true.

f) By completing the basis step and inductive step, we have shown that P(n) is true for all positive integers n greater than 1. Therefore, we can conclude that n! < n^n for all n > 1.

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To find the elasticity of the demand function 2p + 3q = 90 at the price p = 15, we need to first solve for q at that price level.

2(15) + 3q = 90

30 + 3q = 90

3q = 60

q = 20

So, at a price level of p = 15, the quantity demanded is q = 20.

Next, we need to find the derivative of the demand function with respect to price:

dQ/dp = -2/3

Then, we can use the formula for elasticity:

Elasticity = (dQ/dp) * (p/Q)

Elasticity = (-2/3) * (15/20)

Elasticity = -0.5

Therefore, the elasticity of the demand function 2p + 3q = 90 at the price p = 15 is -0.5.
To find the elasticity of the demand function 2p 3q = 90 at the price p = 15, we need to first find the corresponding quantity (q) and then calculate the price elasticity of demand.

Step 1: Solve for q in terms of p
2p 3q = 90
3q = 90 - 2p
q = (90 - 2p) / 3

Step 2: Substitute p = 15 into the equation
q = (90 - 2(15)) / 3
q = (90 - 30) / 3
q = 60 / 3
q = 20

Now we have the point (p, q) = (15, 20) on the demand curve.

Step 3: Differentiate the demand function with respect to p
dq/dp = -2/3

Step 4: Calculate the price elasticity of demand (E)
E = (dq/dp) * (p/q)
E = (-2/3) * (15/20)

E = -0.5

The elasticity of the demand function 2p 3q = 90 at the price p = 15 is -0.5.

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The component form of v, we need to determine its x and y components. We can use trigonometry to do this. Therefore, the component form of v is: v = (-7/4, (7/4)√3)

We know that the magnitude of v is 7/2, so we can use this information to find the length of the hypotenuse of the right triangle formed by the x and y components of v. Let h be the hypotenuse:

h = ||v|| = 7/2

Next, we can use the angle θ to determine the ratios of the sides of the right triangle:

cos(θ) = adj/h = x/7/2
sin(θ) = opp/h = y/7/2

where x is the x component of v and y is the y component of v.

Substituting in the given values, we have:

cos(150∘) = x/7/2
sin(150∘) = y/7/2

Simplifying these equations, we get:

x = -7/4
y = (7/4)√3

Therefore, the component form of v is:

v = (-7/4, (7/4)√3)

To sketch v, we can plot the point (-7/4, (7/4)√3) in the Cartesian plane. The x component is negative, so the point will be in the third quadrant. The y component is positive and greater than the x component, so the point will be above the x-axis and closer to the y-axis. The resulting vector should be pointing in the direction of 150∘ from the positive x-axis.

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Sure, here are the steps to solve this problem:

1. Since the scale of the blueprint is 1:40, it means that any 1 unit on the blueprint represents 40 units on the actual building.

2. The room on the building measures 3.4 m by 4.8 m.

3. So for the dimensions of the room on the blueprint, we divide the measurements by the scale ratio.

4. 1:40 scale means 1 unit = 40 units.

5. So,

3.4 m / 40 units = 0.085 units = 0.08 units (round to 0.08 units)

4.8 m / 40 units = 0.12 units

6. Therefore, the room on the blueprint measures 0.08 units by 0.12 units.

Let me know if this explanation helps or if you have any other questions! I'm happy to help further.

step-by-step:

Room dimensions on building: 3.4 m by 4.8 m

Scale of blueprint: 1 : 40

Step 1) 1 unit on blueprint = 40 units on building

Step 2) 3.4 m / 40 units = 0.085 units (round to 0.08 units)

Step 3) 4.8 m / 40 units = 0.12 units

Step 4) Room dimensions on blueprint = 0.08 units by 0.12 units

Does this help explain the steps? Let me know if any part is still confusing!

Okay, let's break this down step-by-step:

* The shipping crate holds up to 24 cubic feet of space.

* The box measures:

Width: 2ft 1 1/2in = 2.75ft

Length: 1 1/2ft = 1.5ft

Height: 0.8ft

* To convert to cubic feet:

Width x Length x Height = (2.75ft) x (1.5ft) x (0.8ft) = 4.2 cubic feet

* So each box takes up 4.2 cubic feet of space.

* To fill the 24 cubic feet in the crate:

24 cubic feet / 4.2 cubic feet per box = 5 boxes

Therefore, the shipping crate can hold up to 5 of those rectangular boxes.

Let me know if you have any other questions!

Answer:

The first one is the right one

Step-by-step explanation:

The recursive sequence that produces the sequence 4, -14, 58, ... is given by:

a₁ = 4

aₙ = -4aₙ₋₁ - 2, for n ≥ 2

a. 12 out of 20 is 60%

b 62 out of 80 is 77.5%

a. 12% of 125 is 15

b. 18.3% of 28 is 5.12.

The percentage can be found by dividing the value by the total value and then multiplying the result by 100.

Hence, let's find the percentage of the following:

a.

12 / 20 × 100 = 1200 / 20 = 60%

b.

62 / 80 × 100 = 6200 / 80 = 77.5%

Therefore,

12% of 125 = 12 / 100 × 125 = 1500 / 100 = 15

18.3% of 28 = 18.3 / 100 × 28 = 512.4 / 100 = 5.12

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The coefficient of x¹⁰ in (1 × x × x² × x³ × …)ⁿ is 1 for n=5 and n=10. For other values of n, the coefficient of x^10 will be 0, as there are no other possible combinations to achieve x¹⁰.

To find the coefficient of x¹⁰ in (1 × x × x² × x³ × …)ⁿ, you need to determine the possible ways to select terms from the sequence (1 × x × x² × x³ × …) such that their product is x¹⁰ and there are n terms.

Let's consider the following possible combinations of terms that can result in x^10:

1. x × x² × x² × x² × x³ (Here, n=5)
2. x² × x² × x² × x² × x² (Here, n=10)

These are the only two combinations that result in x¹⁰, assuming all powers of x are positive. For the first combination, there is only one way to select the terms, so the coefficient is 1. For the second combination, since all terms are the same, there is also only one way to select the terms, so the coefficient is 1.

Therefore, the coefficient of x¹⁰ in (1 × x × x² × x³ × …)ⁿ is 1 for n=5 and n=10. For other values of n, the coefficient of x^10 will be 0, as there are no other possible combinations to achieve x¹⁰.

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The coefficient of x¹⁰ in (1 × x × x² × x³ × …)ⁿ is 1 for n=5 and n=10. For other values of n, the coefficient of x^10 will be 0, as there are no other possible combinations to achieve x¹⁰.

To find the coefficient of x¹⁰ in (1 × x × x² × x³ × …)ⁿ, you need to determine the possible ways to select terms from the sequence (1 × x × x² × x³ × …) such that their product is x¹⁰ and there are n terms.

Let's consider the following possible combinations of terms that can result in x^10:

1. x × x² × x² × x² × x³ (Here, n=5)
2. x² × x² × x² × x² × x² (Here, n=10)

These are the only two combinations that result in x¹⁰, assuming all powers of x are positive. For the first combination, there is only one way to select the terms, so the coefficient is 1. For the second combination, since all terms are the same, there is also only one way to select the terms, so the coefficient is 1.

Therefore, the coefficient of x¹⁰ in (1 × x × x² × x³ × …)ⁿ is 1 for n=5 and n=10. For other values of n, the coefficient of x^10 will be 0, as there are no other possible combinations to achieve x¹⁰.

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There are approximately 10.764 square feet in an area of 1.00 square metre. This conversion is a mathematical relation and is applicable in the physical universe.

In order to convert square meters to square feet, you can use the following conversion factor: 1 square meter is equal to 10.764 square feet. So, in an area of 1.00 square meters, there are approximately 10.764 square feet. This conversion is applicable in the physical universe.

The use of a unit depends on the context. For instance, the area of a room is measured in meters, but a pencil's length and thickness are measured in centimetres and millimeters, respectively.

As a result, we must convert from one unit to another. We must comprehend the relationship between units before we can comprehend the idea of unit conversion.

We need to convert between units in order to ensure accuracy and prevent measurement confusion. For example, we do not measure a pencil's length in kilometres. In this scenario, it is necessary to convert from kilometres (km) to centimetres (cm). In most cases, multiplicative conversion factors are used to convert one unit to another of the same quantity.

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The 95% confident that the true population mean is between 220.26 and 247.74 when standard deviation σ= 55.

If the statistical model used to construct the interval is reliable, a 95% confidence interval is a range of values that is calculated from a sample of data and is anticipated to contain the real population parameter with a probability of 0.95. To put it another way, we would anticipate that 95% of the confidence intervals calculated for each sample taken from the same population will contain the true population value. A broader interval will come from a greater confidence level (such as 99%), whereas a narrower gap will result from a lower confidence level (such as 90%).

The 95% confidence interval is determined by the formula:

CI = X ± z(α/2) * (σ/√n)

Now, given α/2 (α/2 = 0.025 for a 95% confidence interval).

Thus,

CI = 234.00 ± 1.96 * (55/√65)

CI = 234.00 ± 13.74

CI = (220.26, 247.74)

Hence, the 95% confident that the true population mean is between 220.26 and 247.74.

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Answer:

X = 21

Step-by-step explanation:

Following the distributive property, on the left side we get 3x-12 = 9 + 2x.

Combine like terms, from 3x, remove 2x and add 12 to 9. This gives us X = 21.

To find the standard matrix of a linear transformation, we need to apply the transformation to the standard basis vectors of the domain and express the results in terms of the standard basis vectors of the codomain.

In this case, the linear transformation is the projection onto the line y=6x, which means that any vector in ℝ2 will be projected onto the closest point on the line.

The standard basis vectors of ℝ2 are (1,0) and (0,1), so let's apply the transformation to each of these vectors:

- (1,0) will be projected onto the point (x, 6x) that lies on the line y=6x. The closest point on the line to (1,0) is when x=0, so the projection of (1,0) onto the line is (0,0). Therefore, the first column of the standard matrix will be (0,0).
- (0,1) will be projected onto the point (x, 6x) that lies on the line y=6x. The closest point on the line to (0,1) is when x=1/6, so the projection of (0,1) onto the line is (1/6,1). Therefore, the second column of the standard matrix will be (1/6,1).

Putting these columns together, we get the standard matrix of the projection onto the line y=6x:

[0  1/6]
[0   1 ]

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The given set is {(x, y) | x ≠ −3}, open and connected. Option a and b are correct.

The set is open because for any point (x, y) in the set, we can find a small neighborhood around it (an open ball) that is entirely contained within the set. Specifically, we can choose a radius smaller than the distance from x to -3 to get an open ball around x that does not intersect -3.

The set is connected because any two points in the set can be connected by a continuous path within the set. This follows from the fact that the set is an open interval in the x-axis, which is a connected space.

The set is not simply connected because it has a "hole" at x = -3. Specifically, any closed curve in the set that encircles x = -3 cannot be continuously shrunk to a point within the set. This means that the set fails to satisfy the more stringent condition of simply connectedness, which requires that every closed curve in the set can be continuously shrunk to a point within the set. Option a and b are correct.

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Answer:

VU and TU

Step-by-step explanation:

the marked angle between the lines VU and TU is ∠ VUT or ∠ TUV

that is the 2 lines forming the angle between them

Answer:

VU and TU

Step-by-step explanation:

i did this and the rest of it to

In general, when A is not invertible, we cannot guarantee that B = C. Since we can not apply the inverse of A, we cannot cancel out the A matrix on both sides, and thus cannot prove that B = C in such cases.

We are given that AB = AC, where B and C are nxp matrices and A is invertible. We need to show that B = C and discuss whether this is true when A is not invertible.

Step 1: Since A is invertible, we can apply the inverse of A to both sides of the equation AB = AC. We will multiply both sides on the left by A⁻¹.

Step 2: Applying A⁻¹ to both sides, we get A⁻¹(AB) = A⁻¹(AC).

Step 3: Using the associative property of matrix multiplication, we can rearrange the parentheses as follows: (A⁻¹A)B = (A⁻¹A)C.

Step 4: According to the property of the inverse matrix, A⁻¹A = I (the identity matrix). Therefore, we have IB = IC.

Step 5: Since the identity matrix does not change the matrix it is multiplied with, we get B = C.

So, in general, when A is not invertible, we cannot guarantee that B = C. Without the ability to apply the inverse of A, we cannot cancel out the A matrix on both sides, and thus cannot prove that B = C in such cases.

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To determine the margin of error for a 98% confidence interval, we need to use the formula: Margin of Error = Z* * Standard Error.

Where Z* is the z-value from the standard normal distribution that corresponds to a 98% confidence level, and Standard Error is the standard deviation of the sampling distribution of proportions.

Using the given table, we can find that the z-value for a 98% confidence level is 2.33, To find the standard error, we use the formula: Standard Error = √((p(1-p))/n).

Where p is the sample proportion and n is the sample size, For part (a), where n=100 and p=0.70, the standard error is: √((0.70(1-0.70))/100) = 0.0463,Therefore, the margin of error is: 2.33 * 0.0463 = 0.1077,

So the margin of error for a 98% confidence interval to estimate the population proportion with a sample proportion equal to 0.70 and sample size n=100 is 0.108 (rounded to three decimal places). For part (b), where n=200 and p=0.70, the standard error is: √((0.70(1-0.70))/200) = 0.0327, Therefore, the margin of error is: 2.33 * 0.0327 = 0.0762

So the margin of error for a 98% confidence interval to estimate the population proportion with a sample proportion equal to 0.70 and sample size n=200 is 0.076 (rounded to three decimal places). For part (c), where n=250 and p=0.70, the standard error is: √((0.70(1-0.70))/250) = 0.0293,

Therefore, the margin of error is: 2.33 * 0.0293 = 0.0681, So the margin of error for a 98% confidence interval to estimate the population proportion with a sample proportion equal to 0.70 and sample size n=250 is 0.068 (rounded to three decimal places).

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The relation xOy is not reflexive and not transitive, it is not an equivalence relation. There are no equivalence classes to describe.

To determine whether the relation xoy on the set of all integers, where xoy if x+y is odd, is an equivalence relation, we need to check if it satisfies the three properties of reflexivity, symmetry, and transitivity.

1. Reflexivity:

For any integer x, x+x=2x, which is even.

Therefore, x0x is false, and the relation is not reflexive.

2. Symmetry:

If xOy, then x+y is odd. But y+x is also odd since addition is commutative.

Therefore, yOx, and the relation is symmetric.

3. Transitivity:

If xOy and yOz, then x+y is odd and y+z is odd. Adding these equations together,

we get x+y+y+z=x+z+2y, which is even.

Therefore, x+z is even, and xOz is false. Thus, the relation is not transitive.

Since the relation xOy is not reflexive and not transitive, it is not an equivalence relation. There are no equivalence classes to describe.

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The relation xOy is not reflexive and not transitive, it is not an equivalence relation. There are no equivalence classes to describe.

To determine whether the relation xoy on the set of all integers, where xoy if x+y is odd, is an equivalence relation, we need to check if it satisfies the three properties of reflexivity, symmetry, and transitivity.

1. Reflexivity:

For any integer x, x+x=2x, which is even.

Therefore, x0x is false, and the relation is not reflexive.

2. Symmetry:

If xOy, then x+y is odd. But y+x is also odd since addition is commutative.

Therefore, yOx, and the relation is symmetric.

3. Transitivity:

If xOy and yOz, then x+y is odd and y+z is odd. Adding these equations together,

we get x+y+y+z=x+z+2y, which is even.

Therefore, x+z is even, and xOz is false. Thus, the relation is not transitive.

Since the relation xOy is not reflexive and not transitive, it is not an equivalence relation. There are no equivalence classes to describe.

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The trapezoid ABCD have adjacent angles to be supplementary and values of the variable x = 4 while the measure of m∠D = 78°.

The adjacent angles of the the trapezium are supplementary, so their sum is equal to 180°.

m∠A and m∠D are supplementary so;

14x + 46 + 7x + 50 = 180°

21x + 96° = 180°

21x = 180° - 96° {subtract 96° from both sides}

x = 84°/21

x = 4

m∠D = 7(4) + 50

m∠D = 78°

Therefore, the trapezoid ABCD have adjacent angles to be supplementary and values of the variable x = 4 while the measure of m∠D = 78°.

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The length of side x is given as follows:

[tex]x = 2\sqrt{7}[/tex]

The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:

In the context of this problem, we have that the parameters are given as follows:

Hence we apply the sine of 30º to obtain the length x, as follows:

sin(30º) = sqrt(7)/x

[tex]\frac{1}{2} = \frac{\sqrt{7}}{x}[/tex]

[tex]x = 2\sqrt{7}[/tex]

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a) x1 and x2 cannot span R3 because we would need a third vector in order to do so and it would also have to be linearly independent.

b) In order for X = (x1,x2,x3) we would need all three vectors to be linearly independent such that

ax1+bx2+cx3 = 0 only when a=b=c=0

c) let x3 = (0,0,-1)

Now we place the three vectors into a 3x3 matrix and perform row reductions

1 3  0

1 -1 0

1 4 -1

Add (-1 * row1) to row2

1     3     0

0     -4     0

1     4     -1

Add (-1 * row1) to row3

1     3     0

0     -4     0

0     1     -1

Divide row2 by -4

1     3     0

0     1     0

0     1     -1

Add (-1 * row2) to row3

1     3     0

0     1     0

0     0     -1

Divide row3 by -1

1     3     0

0     1     0

0     0     1

Add (-3 * row2) to row1

1     0     0

0     1     0

0     0     1

So, indeed x3=(0 0 -1) does work and lets X be basis for R3.

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A third vector x3 is [0, 0, 1]

We need to locate a third vector that is linearly independent of the first two in order to extend the set "x1, x2" to a basis of R3. The cross product is one method for accomplishing this.

The following is how we can locate the third vector, x3, assuming that x1 and x2 are not zeros in R3:

Take the cross result of x1 and x2: x1 × x2.

Verify that the final vector is not zero. x1  x2 can be used as x3 if it is linearly independent of x1 and x2. We must locate another vector if it is zero.

Therefore, if x1 = [1, 0, 0] and x2 = [0, 1, 0], we can find x3 as follows:

x1 × x2 = [0, 0, 1]

[0, 0, 1] can be used as x3 because it is linearly independent of x1 and x2 and has a non-zero cross product with x2. In this manner, the set {x1, x2, x3} is a reason for R3.

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The coefficient of x7 in the expansion of (3x + 4)10 is 53,248,000.

To find the coefficient of x^7 in the expansion of (3x + 4)^10 using the binomial theorem, we need to identify the term that has x^7.

The binomial theorem states that (a + b)^n = Σ (nCk) * a^(n-k) * b^k, where k goes from 0 to n and nCk denotes the binomial coefficient, which is the combination of choosing k items from n.

In our case, a = 3x, b = 4, and n = 10. We need to find the term with x^7, so the power of a (3x) should be 3 (since 3x raised to the power of 3 is x^7). This means the term will have the form:

10C3 * (3x)^3 * 4^(10-3)

Now we calculate the coefficients:

10C3 = 10! / (3! * (10 - 3)!) = 120
(3x)^3 = 27x^{7}
4^7 = 16384

Now, we multiply the coefficients together:

120 * 27 * 16384 = 53,248,000

Therefore, the coefficient of x^7 in the expansion of (3x + 4)^10 is 53,248,000.

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As the marginal cost and average cost are both equal to $8 at x = 5, we can conclude that the marginal cost and average cost are equal at this production level.

To find the production level at which the average cost is a minimum, we need to first find the average cost function. The average cost function is given by:

[tex]AC(x) = C(x)/x[/tex]

Substituting C(x) from the given equation, we get:

[tex]AC(x) = (x^3 - 10x^2 + 33x)/x[/tex]

Simplifying this, we get:

[tex]AC(x) = x^2 - 10x + 33[/tex]

To find the production level at which the average cost is a minimum, we need to find the value of x that minimizes the average cost function. We can do this by taking the derivative of the average cost function and setting it equal to zero:

[tex]d/dx (x^2 - 10x + 33) = 2x - 10 = 0[/tex]

Solving for x, we get:

x = 5

Therefore, the production level at which the average cost is a minimum is x = 5.

To show that the marginal cost and average cost are equal at this production level, we need to first find the marginal cost function. The marginal cost function is given by the derivative of the cost function:

[tex]MC(x) = d/dx (x^3 - 10x^2 + 33x) = 3x^2 - 20x + 33[/tex]

Substituting x = 5, we get:

[tex]MC(5) = 3(5)^2 - 20(5) + 33 = 8[/tex]

Therefore, the marginal cost at x = 5 is $8.

To find the average cost at x = 5, we can substitute x = 5 into the average cost function:

[tex]AC(5) = 5^2 - 10(5) + 33 = 8[/tex]

Therefore, the average cost at x = 5 is also $8.

Since the marginal cost and average cost are both equal to $8 at x = 5, we can conclude that the marginal cost and average cost are equal at this production level.

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