In an ellipse, the length of the semi -major axis a is 10 units and the length of the semi -minor axis b is 8 units, what is the distance c of the focus from the center?

(a) Aᵀ = [4 0]

            [2 2]

            [1 -1]

(b) AᵀA = [16 8 4]

               [8 8 0]

               [4 0 2]

(c) AAᵀ = [21 3]

               [3 5]

To find the required matrix operations, let's calculate them step by step:

Given matrix A:

A = [4 2 1]

[0 2 -1]

(a) Aᵀ (transpose of A):

To find the transpose of A, we simply interchange the rows and columns of the matrix. The resulting matrix will have dimensions 3x2.

Aᵀ = [4 0]

[2 2]

[1 -1]

(b) AᵀA:

To calculate AᵀA, we multiply the transpose of A by A. The resulting matrix will have dimensions 3x3.

AᵀA = Aᵀ * A

Aᵀ = [4 0]

[2 2]

[1 -1]

A = [4 2 1]

[0 2 -1]

To perform the matrix multiplication, we multiply the corresponding elements of the rows of Aᵀ with the columns of A and sum them up.

AᵀA = [44 + 00 42 + 02 41 + 0(-1)]

[24 + 20 22 + 22 21 + 2(-1)]

[14 + (-1)0 12 + (-1)2 11 + (-1)(-1)]

Simplifying the calculations:

AᵀA = [16 8 4]

[8 8 0]

[4 0 2]

(c) AAᵀ:

To calculate AAᵀ, we multiply A by the transpose of A. The resulting matrix will have dimensions 2x2.

AAᵀ = A * Aᵀ

A = [4 2 1]

[0 2 -1]

Aᵀ = [4 0]

[2 2]

[1 -1]

To perform the matrix multiplication, we multiply the corresponding elements of the rows of A with the columns of Aᵀ and sum them up.

AAᵀ = [44 + 22 + 11 40 + 22 + 1(-1)]

[04 + 22 + (-1)1 00 + 22 + (-1)(-1)]

Simplifying the calculations:

AAᵀ = [21 3]

[3 5]

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When two parallel lines are cut by a transversal, the sum of the interior angles on the same side of the transversal is always 180 degrees. This is known as the Angle Sum Property of Parallel Lines.

To understand why this is the case, let's consider an example.

Imagine you have two parallel lines, labeled line 1 and line 2. Now, draw a transversal line that intersects both parallel lines. This will create several pairs of corresponding angles, such as angle 1 and angle 2, angle 3 and angle 4, and so on.

The interior angles on the same side of the transversal are angle 1 and angle 4.

Now, if you measure the sum of angle 1 and angle 4, you will find that it always equals 180 degrees. This holds true for any pair of interior angles on the same side of the transversal.

Therefore, when given the expression (56x^(4)y)/(4),

it is not directly related to the Angle Sum Property of Parallel Lines.

It seems to be a separate mathematical expression or equation that requires evaluation or simplification.

To proceed, we need more information about what specifically needs to be done with this expression.

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154.3325 has two significant figures to maintain consistency with the original data.

To solve the given expression, we'll follow the order of operations (PEMDAS/BODMAS).

Multiply: 111 × 3.3447 = 370.398 (rounded to 3 significant figures).

Divide: 370.398 ÷ 2.4 = 154.3325 (rounded to 4 significant figures).

Significant figures are a way to express the precision of a measurement or calculation result. In this case, the original numbers (111, 3.3447, and 2.4) have varying significant figures.

To ensure the accuracy of the final answer, we round it to the same number of significant figures as the least precise value involved, which is 2.4 with two significant figures. Therefore, the answer, 154.3325 has two significant figures to maintain consistency with the original data.

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The rational zeros of the function q(x) = 3x⁴ + x³ - 11x² - 3x + 6 can be found using the Rational Zero Theorem, which states that if a polynomial function has integer coefficients, then any rational zero will have the form of p/q where p is a factor of the constant term and q is a factor of the leading coefficient.  

By applying the tests for 1 and −1, we find that neither of these are roots of the function. Hence, let's move to the next step of finding rational zeroes. Rational Zero Theorem states that all rational zeroes will be of the form p/q, where p is a factor of 6, and q is a factor of 3. These rational zeros can be positive or negative, so we need to consider all the possible combinations of the factors of 6 and 3.

Here are all the possible rational zeros: ±1/1, ±2/1, ±3/1, ±6/1, ±1/3, ±2/3, ±1/−1, ±2/−1, ±3/−1, and ±6/−1.Thus, we can now use synthetic division to find which of these possible rational zeros are actual zeros of the function. Synthetic division for each of the possible rational zeros results in the following:

1: 3 4 -7 -10 -4 2: 3 10 -1 -14 -8 3: 3 13 18 57 180 6: 3 19 58 343 2262 −1: 3 0 -11 11 0 -2: 3 -2 -7 17 -40 -3: 3 -5 -8 49 -198 -6: 3 -12 1 151 -894 As we can see, the only rational zero of q(x) is x = 1.

Thus, using synthetic division, we can divide the function by (x - 1) to get a quadratic function.

The result of the division is: (x - 1)(3x³ + 4x² - 3x - 6) = 0We can use the quadratic formula or factoring to find the remaining zeroes of 3x³ + 4x² - 3x - 6. Factoring by grouping, we get: 3x³ + 4x² - 3x - 6 = (3x² - 2)(x + 3)Thus, the zeroes of the function q(x) = 3x⁴ + x³ - 11x² - 3x + 6 are:x = 1, x = -3, x = $\frac{2}{3}, and x = $\frac{-1}{3}

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Average score initially = 79, average score after 4 months is 57.47, after 24 months it is 35.09

Given: S(t)=79−14 ln(t+1),

(a) The average score when they initially took the test(i.e at t = 0)

S(0)= 79 − 14 ln (0+1) ⇒ S(0) = 79 - 14 ln(1) ⇒ S(0) = 79 - 14 (0) ⇒ S(0) = 79

Hence, the average score when they initially took the test was 79.

(b) The average score after 4 months (i.e at t = 4)

S(4) = 79 − 14 ln (4+1) ⇒ S(4) = 79 − 14 ln (5) ⇒ S(4) = 79 - 14(1.609) ⇒ S(4) = 57.47. Hence, the average score after 4 months was 57.47.

(c) The average score after 24 months (i.e at t = 24)

S(24) = 79 − 14 ln (24+1) ⇒ S(24) = 79 − 14 ln (25) ⇒ S(24) = 79 - 14(3.218) ⇒ S(24) = 35.09

Hence, the average score after 24 months was 35.09.

(d) Differentiating S(t) with respect to t, we get: S'(t) = dS(t)/dt= -14(1/(t+1))(1) ⇒ S'(t) = -14/(t+1)

Therefore, S'(t) = -14/(t+1).

(e) S'(4) = -14/(4+1) = -2.8 and S'(24) = -14/(24+1) = -0.5

The number S'(4) = -2.8 is interpreted as follows: If the students keep retaking the equivalent form of the test for every one more month after the first test (t=0), then on average, the score will drop by 2.8% after every one month. For example, if a student's score is initially 79 (in the first test) and keeps on taking the equivalent test every one month, then on average, their score after 1 month will be 76.2 (i.e 2.8% less than 79).

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The function  [tex]f (x)=x^ {1/2} - 5x - 4[/tex] is not a polynomial function because it contains a term with a fractional exponent (1/2). Polynomial functions must have integer exponents. The correct choice is C "It is not a polynomial".

The function  [tex]f (x)=x^ {1/2} - 5x - 4[/tex] is not a polynomial because it contains a term with a fractional exponent of 1/2. In polynomial functions, the exponents of variables must be non-negative integers. Since 1/2 is not a non-negative integer, the function does not meet the requirement for being a polynomial.

Therefore, the correct choice is C: It is not a polynomial.

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The equilibrium price p = 0.9887 and the equilibrium quantity q = 41.602.The consumer surplus at the equilibrium market is 2177.18.

To find the consumer surplus at the equilibrium market,  to determine the equilibrium price and quantity by setting the demand and supply functions equal to each other:

900²(-0.1) = 3²(0.9)

To solve this equation,  take the natural logarithm (ln) of both sides:

ln(900²(-0.1)) = ln(3²(0.9))

Using the logarithmic properties bring down the exponent:

-0.1 × ln(900) = 0.9 × ln(3)

Now calculate the values:

ln(900) ≈ 6.8024

ln(3) ≈ 1.0986

-0.1 × 6.8024 ≈ -0.6802

0.9 × 1.0986 ≈ 0.9887

Therefore, the equilibrium price (p) is approximately 0.9887, and the equilibrium quantity (q)  obtained by substituting this price into either the demand or supply function. Let's use the demand function to find q:

q = 900²(-0.1) ≈ 41.602

To calculate the consumer surplus, to integrate the area under the demand curve (which represents the willingness to pay) from 0 to the equilibrium quantity (q) and subtract the area under the supply curve (which represents the cost) from 0 to the equilibrium quantity.

Consumer Surplus = ∫[0 to q] Demand Function dx - ∫[0 to q] Supply Function dx

Let's calculate the consumer surplus:

Consumer Surplus = ∫[0 to 41.602] 900²(-0.1) dx - ∫[0 to 41.602] 3²(0.9) dx

Integrating the demand function:

∫[0 to 41.602] 900²(-0.1) dx = [10 × (900²(0.9) - 900²(0.9) × x)] [0 to 41.602]

Simplifying the expression:

= 10 × (900²(0.9) - 900²(0.9) × 41.602)

Integrating the supply function:

∫[0 to 41.602] 3²(0.9) dx = [10 ×(3²(0.9) × x)] [0 to 41.602]

Simplifying the expression:

= 10 ×(3²(0.9) × 41.602)

Now, calculate the consumer surplus:

Consumer Surplus = 10 × (900²(0.9) - 900²(0.9) × 41.602) - 10 ×(3²(0.9) ×41.602)

Evaluate the values using a calculator:

Consumer Surplus ≈ 2177.18

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The mathematical solution to the equation is x = 25/19.

The equation (3x/5) - ((x-3)/2) = (x+2)/3, we can start by clearing the fractions.

Multiplying every term by the least common multiple (LCM) of the denominators, which is 30, will help us eliminate the fractions:

30 * (3x/5) - 30 * ((x-3)/2) = 30 * ((x+2)/3)

This simplifies to:

6x - 15(x-3) = 10(x+2)

Now we can expand and simplify:

6x - 15x + 45 = 10x + 20

Combining like terms:

-9x + 45 = 10x + 20

Next, let's isolate the variable terms on one side and the constant terms on the other side:

-9x - 10x = 20 - 45

-19x = -25

To solve for x, divide both sides by -19:

x = -25/-19

Simplifying the fraction:

x = 25/19

Therefore, the mathematical solution to the equation is x = 25/19.

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The car will have a value of $0 after approximately 84 years of ownership( through finding the exponential function)

To find the exponential function that represents the value of the car after t years, we need to use the formula for continuous depreciation:

V(t) = V0 * e^(kt),
where V(t) represents the value of the car after t years, V0 is the initial value of the car (which is $25,000 in this case), e is the base of the natural logarithm (approximately 2.71828), k is the rate of depreciation expressed as a decimal, and t is the number of years of ownership.

In this case, the rate of depreciation is 12%, which can be written as 0.12 in decimal form. Therefore, the exponential function that represents the value of the car after t years is:
V(t) = 25000 * e^(0.12t).
To find when the car will have a value of $0, we can set V(t) equal to 0 and solve for t:
0 = 25000 * e^(0.12t).
To isolate the exponential term, we can divide both sides of the equation by 25000:
0.12t = -ln(0),
where ln represents the natural logarithm. The natural logarithm of 0 is undefined, so there is no value of t that makes the car's value exactly $0.

However, we can find the time when the car's value is very close to $0 by setting V(t) equal to a small positive value, such as $1:
1 = 25000 * e^(0.12t).
To solve for t, we divide both sides of the equation by 25000:
0.00004 = e^(0.12t).
To isolate t, we can take the natural logarithm of both sides:
ln(0.00004) = 0.12t.

Using a calculator, we find that ln(0.00004) is approximately -10.09. Dividing by 0.12, we get:
t = -10.09 / 0.12,
t ≈ -84.08.

Since time cannot be negative in this context, we round up to the nearest whole number:
t ≈ -84.

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The surface area of the square base pyramid is 1425 inches².

The surface area of the square base pyramid can be found as follows:

surface area of square base pyramid = a² + 2al

where

Therefore,

a = 19 inches

l = 28 inches

Therefore,

surface area of square base pyramid = 19² + 2 × 19 × 28

surface area of square base pyramid = 361 + 1064

surface area of square base pyramid = 1425 inches²

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The value of B is 90° and the values of a, b, c, A, and B are 18.8 cm, 37.95 cm, 45.7 cm, 65.06°, and 90° respectively.

Given that, the right triangle ABC, where C=90° and a=18.8 cm, c=45.7 cm. We need to find the value of b, A, and B.Step 1:As we know that `a^2 + b^2 = c^2`Plugging the values in the above equation 18.8^2 + b^2 = 45.7^2b^2 = 45.7^2 - 18.8^2b^2 = 1794.89 - 353.44b^2 = 1441.45Taking square root on both the sidesb = 37.95 cmStep 2:Finding value of sin AUsing the formula `sin A = a/c`sin A = 18.8/45.7sin A = 0.411 = 24.94°Step 3:Finding value of AAs we know that, A + 90° + 24.94° = 180°A + 114.94° = 180°A = 180° - 114.94°A = 65.06°Therefore, the value of B is 90° and the values of a, b, c, A, and B are 18.8 cm, 37.95 cm, 45.7 cm, 65.06°, and 90° respectively.

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In similar triangles, the angle corresponding to adjacent sides are equal.

"Similar triangles are triangles that have the same shape, but their sizes may vary. In short, Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles.

If the corresponding angles of two triangles are equal, then the triangles are similar. They are called equiangular triangles. In similar triangles, angle corresponding to adjacent sides are equal.

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Compare the corresponding angles in ABC and DEF. In general, what do your observations suggest about the angle measures in two similar triangles? Use the definition of similarity transformations to explain the relationship between corresponding angle measures.

Let n be a positive integer not divisible by 3. Prove that n² − 1 is always divisible by 3.We have to use proof by contradiction to prove this statement. Proof by contradiction is a type of proof in which we first assume the opposite of the statement we want to prove and then show that it leads to a contradiction or absurdity. This will show that our original assumption must have been correct.

Let us assume that n² − 1 is not divisible by 3. Then, we have two possibilities:It is possible that n is itself divisible by 3, which we know is not true because we have assumed that n is not divisible by 3.

It is also possible that n is not divisible by 3 but (n² - 1) leaves a remainder of 1 when divided by 3, which is also not possible since (n² - 1) must be divisible by 3.

However, neither of these possibilities can be true since we have already assumed that n² − 1 is not divisible by 3. Therefore, our assumption must be incorrect and n² − 1 is always divisible by 3 when n is a positive integer not divisible by 3.

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Correct Option is. False

When two random variables, X and Y, are independent, the variance of their sum, X+Y, is equal to the sum of their individual variances. In this case, both X and Y have a variance of 20. Therefore, the variance of X+Y would be the sum of 20 and 20, which is 40, not 20.

To understand why this is the case, we can consider the definition of variance. Variance measures how spread out the values of a random variable are from its mean. When two variables are independent, their joint distribution is simply the product of their individual distributions. The variance of the sum of two independent variables is obtained by summing their variances.

In this scenario, each variable has a variance of 20. However, when we add them together, the variances do not add up. Instead, the variance of the sum is the sum of the individual variances, resulting in a variance of 40 for X+Y.

Therefore, the statement "var(X+Y)=20" is false. The correct answer is that the variance of X+Y is 40.

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The radian measure of 240 degrees is 4π/3.

To convert an angle from degrees to radians, we can use the fact that π radians is equivalent to 180 degrees. We can set up a proportion to find the radian measure.

We know that 180 degrees is equal to π radians. Therefore, we can set up the following proportion:

180 degrees / π radians = 240 degrees / x radians

To solve for x, we can cross-multiply and divide:

180x = 240π

x = (240π) / 180

x = 4π/3

In terms of explanation, when converting from degrees to radians, we use the fact that one complete revolution around a circle is equal to 2π radians or 360 degrees. Therefore, to convert a given angle from degrees to radians, we divide the angle by 360 and multiply by 2π. In this case, since 240 degrees is two-thirds of a full revolution, it corresponds to 4/3 times the value of π, which gives us 4π/3 as the radian measure.

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The exact value of `cos(θ)` is `cos(θ) = sqrt(63)/8`.

Given that `sin(θ) = 1/8`, we can use the Pythagorean identity: `sin^2(θ) + cos^2(θ) = 1`.

Squaring both sides of `sin(θ) = 1/8`, we have `sin^2(θ) = (1/8)^2 = 1/64`.

Substituting `sin^2(θ)` in the Pythagorean identity, we get `cos^2(θ) = 1 - sin^2(θ)`.

Plugging in the values, we have `cos^2(θ) = 1 - 1/64`.

Simplifying further, `cos^2(θ) = 63/64`.

To find the value of `cos(θ)`, we take the square root of both sides.

Since θ is in Quadrant I, where `cos(θ)` is positive, we take the positive square root.

Therefore, the exact value of `cos(θ)` is `cos(θ) = sqrt(63)/8`.

This represents the positive value of `cos(θ)` when `sin(θ) = 1/8` and θ is in Quadrant I.

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(a) To calculate f(6), we have n = 2 as there are only two prime numbers less than 6, which are 2 and 3.Thus, f(6) = 2n + 1 = 2(2) + 1 = 5 To calculate f(7), we have n = 3 as there are only three prime numbers less than 7, which are 2, 3 and 5.Thus, f(7) = 2n + 1 = 2(3) + 1 = 7 To calculate f(8), we have n = 4 as there are only four prime numbers less than 8, which are 2, 3, 5 and 7.Thus, f(8) = 2n + 1 = 2(4) + 1 = 9

(b) To calculate g({2, 3, 5}), the size of the set is 3.Thus, g({2, 3, 5}) = 3To calculate g({3, 4, 5}), the size of the set is 3.Thus, g({3, 4, 5}) = 3

(c) f∘g means f(g(x)). To find f∘g, we need to find g(x) first and then use this value of g(x) to find f(g(x)). Let’s use the set {1, 3, 5, 7, 9} to find f∘g.g({1,3,5,7,9}) = |{1,3,5,7,9}| = 5n = 4 as there are only four prime numbers less than 10, which are 2, 3, 5, and 7f(g({1,3,5,7,9})) = f(5) = 2n + 1 = 2(4) + 1 = 9 Therefore, f∘g({1,3,5,7,9}) = 9(d) g∘f means g(f(x)).

To find g∘f, we need to find f(x) first and then use this value of f(x) to find g(f(x)). Let’s use the value 10 to find g∘f.f(10) = 2n + 1 = 2(4) + 1 = 9g(f(10)) = g(9) = 1 As we have found the value of g(f(10)), g∘f(10) exists and equals to 1.

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Answer:

-37.5

Step-by-step explanation:

[tex]69÷(7-9)-6^{2} ÷12 = \\ 69 \div ( - 2) - 36 \div 12 = \\ - 34.5 - 3 = - 37.5[/tex]

The value of the perpetuity today, with an annual interest rate of 8%, is $4,125.25.

To calculate the present value of a perpetuity, we can use the formula: Present Value = Cash Flow / Interest Rate.

In this case, the cash flow is $500 per year, and the interest rate is 8% (or 0.08 in decimal form). Plugging these values into the formula, we get: Present Value = $500 / 0.08 = $6,250.

However, this calculation gives us the present value of the perpetuity starting from today. Since the payments start 5 years from today, we need to discount the value by the present value of $1 received 5 years from today.

Using the formula for the present value of a single amount, we find that the present value of $1 received 5 years from today, with an 8% interest rate, is approximately 0.6806.

To calculate the present value of the perpetuity starting 5 years from today, we multiply the present value of $6,250 by the discount factor of 0.6806: Present Value = $6,250 * 0.6806 ≈ $4,250.25.

Therefore, the value of the perpetuity today, with an 8% annual interest rate and payments starting 5 years from today, is approximately $4,125.25.

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The numerical length of line segment RS is 10 units, obtained by substituting x=3 into the expression RS = 4x - 2.

To determine the numerical length of the line segment RS, we need to find the value of x and substitute it into the expression RS = 4x - 2.

Given that R is on the line segment QS, we can set up the equation QR + RS = QS:

(3x - 6) + (4x - 2) = 5x - 2.

Simplifying the equation, we have:

7x - 8 = 5x - 2.

Subtracting 5x from both sides, we get:

2x - 8 = -2.

Adding 8 to both sides, we have:

2x = 6.

Dividing both sides by 2, we find:

x = 3.

Now, we can substitute the value of x into the expression RS = 4x - 2:

RS = 4(3) - 2 = 12 - 2 = 10.

Therefore, the numerical length of line segment RS is 10 units.

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Rounding two decimal places gives a period of rotation of 0.25 seconds only.

When a car's engine makes less than 240 revolutions per minute, it stalls.

To find the period of the rotation of the engine when it is about to stall, we can use the formula,T = 60/n

Where T is the time in seconds for one revolution and n is the number of revolutions per minute. To find the period when the engine is about to stall, we substitute 240 into n.

T = 60/n

  = 60/240

  = 0.25 seconds.

Rounding this to two decimal places gives a period of rotation of 0.25 seconds only.

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Let us assume that the first angle is 7x, then the second angle would be 2x (as the ratio of their measures is 7:2).

We know that the sum of complementary angles is 90 degrees.

Therefore, we can write an equation as: 7x + 2x = 90

(Since the first angle is 7x and the second angle is 2x, their sum is 7x + 2x)

Simplify: 9x = 90

Divide both sides by 9:x = 10

Calculating angle we get;

So, the first angle would be: 7x = 7 × 10 = 70

The second angle would be: 2x = 2 × 10 = 20

Therefore, the two angles are 70 degrees and 20 degrees.

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Answer:

Surface Area = 2 * (length * width + length * height + width * height)

Step-by-step explanation:

To find the surface area of a rectangular solid (also known as a rectangular prism), you need to calculate the areas of its six faces and then add them together. The formula for the surface area of a rectangular solid is:

Surface Area = 2 * (length * width + length * height + width * height)

Here are the steps to find the surface area:

Identify the length, width, and height of the rectangular solid.

Multiply the length by the width to find the area of the top and bottom faces.

Multiply the length by the height to find the area of the front and back faces.

Multiply the width by the height to find the area of the left and right faces.

Add up all the areas calculated in steps 2, 3, and 4.

Multiply the sum by 2 to account for the two identical sets of faces.

The result will be the surface area of the rectangular solid.

It's important to note that all measurements should be in the same unit (e.g., centimeters, inches) for accurate results.

Answer:

Step-by-step explanation:

The area of ​​the rectangular solid (parallelepiped) is calculated by adding the lateral area and twice the base area: Stot=Slat+2Sb; the area of ​​the rectangular parallelepiped is given by the sum of the areas of the six rectangles that make up its surface, ie Stot=2(ab+ah+bh).

The main answers to the given derivatives are:

a)  42x⁶ - 44x - 2x⁻³

b)  4x(x² + 2)

c) [tex]\(-\frac{9}{2}x^{-3/2} - 3x^{-5/2}\)[/tex]

To find the derivative of [tex]\(6x^7 - 22x^2 + \frac{1}{x^2}\)[/tex] with respect to x, we can differentiate each term separately using the power rule and the rule for differentiating a constant:

[tex]\(\frac{d}{dx}\left[6x^7 - 22x^2 + \frac{1}{x^2}\right] = 6 \cdot \frac{d}{dx}(x^7) - 22 \cdot \frac{d}{dx}(x^2) + \frac{d}{dx}\left(\frac{1}{x^2}\right)\)[/tex]

Applying the power rule, we have:

[tex]\(= 6 \cdot 7x^{7-1} - 22 \cdot 2x^{2-1} + \frac{d}{dx}\left(\frac{1}{x^2}\right)\)[/tex]

Simplifying:

[tex]\(= 42x^6 - 44x + \frac{d}{dx}\left(\frac{1}{x^2}\right)\)[/tex]

To find the derivative of \(\frac{1}{x^2}\), we can use the power rule again:

[tex]\(\frac{d}{dx}\left(\frac{1}{x^2}\right) = \frac{d}{dx}(x^{-2}) = -2x^{-2-1} = -2x^{-3}\)[/tex]

Substituting this result back into the previous equation:

[tex]\(= 42x^6 - 44x - 2x^{-3}\)[/tex]

Therefore, the derivative of [tex]\(6x^7 - 22x^2 + \frac{1}{x^2}\)[/tex] with respect to[tex]\(x\) is \(42x^6 - 44x - 2x^{-3}\).[/tex]

b) To differentiate[tex]\(\left(x^2+2\right)^2\)[/tex] with respect to x, we can use the chain rule. Let's define u = x² + 2. Now, the function becomes u². Applying the chain rule:

[tex]\(D_x\left[\left(x^2+2\right)^2\right] = \frac{d}{du}(u^2) \cdot \frac{du}{dx}\)[/tex]

Differentiating u² with respect to u:

= 2u

Now, finding [tex]\(\frac{du}{dx}\)[/tex]  using the power rule:

[tex]\(\frac{du}{dx} = \frac{d}{dx}(x^2 + 2) = \frac{d}{dx}(x^2) + \frac{d}{dx}(2) = 2x\)[/tex]

Substituting the values back into the equation:

[tex]\(D_x\left[\left(x^2+2\right)^2\right] = 2u \cdot 2x = 4ux\)[/tex]

Since we defined u = x² + 2, the final result is:

[tex]\(D_x\left[\left(x^2+2\right)^2\right] = 4(x^2 + 2)x = 4x(x^2 + 2)\)[/tex]

Therefore, the derivative of (x²+2t)²with respect to x is 4x(x² + 2).

To differentiate [tex]\(9x^{-1/2} + \frac{2}{x^{3/2}}\)[/tex]  with respect to x, we can differentiate each term using the power rule and the rule for differentiating a constant:

[tex]\(\frac{d}{dx}\left[9x^{-1/2} + \frac{2}{x^{3/2}}\right] = 9 \cdot \frac{d}{dx}(x^{-1/2}) + 2 \cdot \frac{d}{dx}\left(\frac{1}{x^{3/2}}\right)\)[/tex]

Applying the power rule:

[tex]\(= 9 \cdot \left(-\frac{1}{2}\right)x^{-1/2-1} + 2 \cdot \frac{d}{dx}\left(\frac{1}{x^{3/2}}\right)\)[/tex]

Simplifying:

[tex]\(= -\frac{9}{2}x^{-3/2} + 2 \cdot \frac{d}{dx}\left(\frac{1}{x^{3/2}}\right)\)[/tex]

To find the derivative of [tex]\(\frac{1}{x^{3/2}}\)[/tex], we can use the power rule:

[tex]\(\frac{d}{dx}\left(\frac{1}{x^{3/2}}\right) = \frac{d}{dx}(x^{-3/2}) = -\frac{3}{2}x^{-3/2-1} = -\frac{3}{2}x^{-5/2}\)[/tex]

Substituting this result back into the previous equation:

[tex]\(= -\frac{9}{2}x^{-3/2} + 2 \cdot \left(-\frac{3}{2}x^{-5/2}\right)\)[/tex]

Simplifying further:

[tex]\(= -\frac{9}{2}x^{-3/2} - 3x^{-5/2}\)[/tex]

Therefore, the derivative of [tex]\(9x^{-1/2} + \frac{2}{x^{3/2}}\)[/tex] with respect to x is[tex]\(-\frac{9}{2}x^{-3/2} - 3x^{-5/2}\).[/tex]

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The simplified expression of the trig function is  4.

Substitute the exact value of each trig function and then simplify the result of tan²45°+tan²60°.

We have the following information; tan 45° = 1 and tan 60° = √3 / 1.

Substituting these values, we get; tan²45°+tan²60°= 1² + (√3 / 1)²= 1 + 3= 4.

Therefore, the expression tan²45°+tan²60°  is simplified is 4.

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If he rolls higher than 10 , you pay him $11., the expected value of the bet to you is -$0.5.

The formula for the expected value is:

Expected value = (Probability of a winning outcome x Value of winning outcome) - (Probability of a losing outcome x Value of losing outcome)

From the given information, if your friend rolls a number between 1 and 10 (inclusive), he will pay you $11. The probability of rolling a number between 1 and 10 is 10/20 or 0.5.

The value of this winning outcome is $11.On the other hand, if he rolls a number between 11 and 20 (inclusive), you have to pay him $11.

The probability of rolling a number between 11 and 20 is also 0.5. The value of this losing outcome is -$11.

Therefore, using the formula for the expected value, we have:

Expected value = (0.5 x $11) - (0.5 x $11)

Expected value = $5.5 - $5.5

Expected value = -$0.5

So the expected value of the bet to you is -$0.5.

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To find the equation of a parabola given its vertex and focus, determine the direction, calculate the value of p, and use the appropriate equation form based on the orientation of the parabola.

To find the equation of a parabola given its vertex and focus, you can follow these steps:

Step 1: Identify the coordinates of the vertex and focus.

Let's assume the vertex is given as (h, k) and the focus is given as (a, b).

Step 2: Determine the direction of the parabola.

If the parabola opens upwards or downwards, it is a vertical parabola. If it opens sideways (left or right), it is a horizontal parabola. This will help you determine the form of the equation.

Step 3: Determine the value of p.

The distance between the vertex and focus is denoted by p. Calculate the value of p using the distance formula: p = sqrt((a-h)^2 + (b-k)^2).

Step 4: Write the equation.

a) For a vertical parabola:

If the parabola opens upwards: (x-h)^2 = 4p(y-k)

If the parabola opens downwards: (x-h)^2 = -4p(y-k)

b) For a horizontal parabola:

If the parabola opens to the right: (y-k)^2 = 4p(x-h)

If the parabola opens to the left: (y-k)^2 = -4p(x-h)

Substitute the values of h, k, and p into the appropriate equation based on the direction of the parabola to obtain the final equation.

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To find the freshman enrollment of 2018, we need to determine the original enrollment figure before the 13.2% increase.

Let's assume the freshman enrollment of 2018 is represented by "x."

According to the information given, the freshman enrollment in 2019 was a 13.2% increase over the previous record in 2018. This can be expressed as:

x + 0.132x = 1060

Combining like terms:

1.132x = 1060

Dividing both sides by 1.132:

x = 1060 / 1.132

x ≈ 937.26

Rounded to the nearest whole number, the freshman enrollment of 2018 at Howard University was approximately 937.

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The equation of the line in slope-intercept form which passes through the point (-1,-2), and is perpendicular to the line with the equation y = -(3)/(4)x + (9)/(4) is y = (4)/(3)x - (2)/(3).

Given that, the point on the line is (-1,-2) and the line is perpendicular to y = -(3)/(4)x + (9)/(4). Now we will convert the given equation into slope-intercept form y = mx + c, where m is the slope and c is the y-intercept, to identify the slope of the line: y = -(3)/(4)x+(9)/(4) ⇒ y = mx + c, where m = -(3)/(4)

So the slope of the line perpendicular to the above line is given by:

Slope of the perpendicular line = negative reciprocal of the slope of the above line

Therefore, the slope of the perpendicular line is (4)/(3)

We use the point-slope form of a line y - y1 = m(x - x1), where m = slope and (x1, y1) = point on the line. Substituting the values, we get the equation of the line which passes through the point (-1,-2) and is perpendicular to the line with the equation y = -(3)/(4)x + (9)/(4) as:

y - (-2) = (4)/(3)(x - (-1))

y + 2 = (4)/(3)x + (4)/(3)

y = (4)/(3)x - 2(2)/(3)

y = (4)/(3)x - (2)/(3)

Thus, the required equation of the line is y = (4)/(3)x - (2)/(3) in the slope-intercept form.

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Answer:

100

Step-by-step explanation:

2x2=4 4 squared is 16

16+3=21

in the brackets is 2+2 which is 4

21x4=84

4x4=16

84+16=100

im so sorry if i get this wrong