4. Find the equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point xy^} = 8, (2,2,1)

Answer:

228 individual cookies.

Step-by-step explanation:

To convert 19 dozen cookies to individual cookies, we first have to identify that:

\(dozen=12\)

If a dozen is equal to 12, to solve, we simply must multiply the amount of dozens, which is 19, by a dozen, which is 12:

\(19*12=\)

\(228\)

Therefore, there are 228 individual cookies.

-

We can check our  work by dividing the individual number of cookies, 228, by dividing buy the amount of dozens, 19, and a dozen, 12:

\(228\) ÷ \(19=12\)

\(228\) ÷ \(12=19\)

As you can see, our quotients are the two figures we started with, and when multiplied together, give us 228 individual cookies; therefore, our solution is correct!

The probability that the sample average falls outside the 3-sigma control limits of the X-bar chart is 0.27%.

The 3-sigma control limits are calculated using the standard deviation of the process. If the process is in control, then 99.73% of the sample averages will fall within the 3-sigma control limits. The remaining 0.27% of the sample averages will fall outside the control limits.

In this case, the sample size is 4, which is smaller than the sample size of 9 that was used to construct the control chart. This means that the control limits for the sample of size 4 will be narrower than the control limits for the sample of size 9.

As a result, the probability that the sample average falls outside the control limits will be higher for the sample of size 4.

Specifically, the probability that the sample average falls outside the 3-sigma control limits for a sample of size 4 is 0.27%. This means that there is a 0.27% chance that the new employee will observe a sample average that falls outside the control limits, even if the process is in control.

Visit here to learn more about probability:  

brainly.com/question/13604758

#SPJ11

The area of the actual wall is 972.16 square feet.

To determine the area of the actual wall, we need to convert the dimensions of the mural to the corresponding dimensions of the wall using the given scale.

The scale provided is 4 inches = 14 feet.

Let's find the conversion factor:

Conversion factor = Actual measurement / Mural measurement

In this case, we are converting from mural inches to actual feet. So, the conversion factor is:

Conversion factor = 14 feet / 4 inches

= 3.5 feet / inch

To find the dimensions of the actual wall, we multiply the dimensions of the mural by the conversion factor:

Actual width = 6.2 inches × 3.5 feet / inch

= 21.7 feet

Actual height = 12.8 inches × 3.5 feet / inch

= 44.8 feet

The area of the actual wall is the product of the actual width and actual height:

Area = Actual width × Actual height

= 21.7 feet × 44.8 feet

Calculating the area:

Area = 972.16 square feet

Therefore, the area of the actual wall is 972.16 square feet.

Learn more about Conversion factor click

https://brainly.com/question/30567263

#SPJ1

The value of (ΣX)² for the scores 1, 5, 2 is:  

64

It is the representation of the sum of infinite or many values. And it is represented by the following symbol:

∑xi

The value of (ΣX)² for the scores 1, 5, 2 can be found by first finding the summation of the scores and then squaring the result.

Steps to follow:

Step 1: Find the summation of the scores.
            ΣX = 1 + 5 + 2 = 8

Step 2: Square the summation.
            (ΣX)² = 82 = 64

Therefore, the value of (ΣX)² for the scores 1, 5, 2 is 64.

More information about summation here:

https://brainly.com/question/11741791

#SPJ11

The function with zeros -3,-1,0 and 6 is  \(f(x)=x(x+1)(x+3)(x+6)\)

Given that the function has zeros at -3, -1,0 and 6.

If a is a zero of a function f(x), then f(a)=0. Also (x-a) will be a factor of f(x).

Thus here, f(-3)=0, f(-1)=0,f(0)=0 and f(6)=0.

That is the function will have the factors (x+3), (x+1), x and (x-6).

Hence the function f(x) can be defined as the product of these factors.

Thus, the function with zeros -3, -1, 0 and 6 will be

\(f(x)=x(x+1)(x+3)(x+6)\)

Learn more about functions and its zeros at https://brainly.com/question/20896994

#SPJ4

The equation "ax2 = y," which has one solution unless a = 4, and none unless a = 4, has a solution. x = √(-4ay) / (2a) restricted by the condition that y be negative.

We may use the quadratic formula to determine the solutions to an equation for various values of an to construct a solution to the equation "ax² = y," which has no solution when a = 4 & just one solution in all other cases.

According to the quadratic formula, the answers to the problem "ax2 + bx + c = 0" are provided by

x = (-b +/- √(b² - 4ac)) / (2a)

In this formula, if we add "ax² = y," we obtain

x = (-0 +/- √(0² - 4ay)) / (2a)

which simplifies to

x = √(-4ay) / (2a)

If a = 4, the equation becomes

x = √(-16y) / 8

The equation has no solutions if y is positive because the value of (-16y) is fictitious. The value of (-16y) is real if y is negative, but the equation is still unsolvable since x cannot have a negative value. As a result, when a = 4, the problem has no solutions.

The equation has a single solution provided by any other value of a.

x = √(-4ay) / (2a)

For example, if a = 3, the equation becomes

x = √(-12y) / 6

Since √(-12y) is imaginary if y is positive, the problem has no solutions. If y is negative, √(-12y) has a real value, and there is only one solution to the problem.

Learn more about the polynomial equation at

https://brainly.com/question/11536910

#SPJ4

The trigonometric function \(\(\tan (\theta)\)\) can be written in terms of the trigonometric function \(\(\cos (\theta)\)\) as \(\(\tan(\theta) = -\frac{\sqrt{1-\cos^2(\theta)}}{\cos(\theta)}\) for \(\theta\)\) in Quadrant III.

Trigonometry is the branch of mathematics that deals with the relations between the sides and angles of triangles. There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. They are defined using the sides of a right triangle, which is a triangle that has one angle of 90 degrees.

The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side of an angle in a right triangle. It can also be defined as the ratio of the sine of an angle to the cosine of the same angle. The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle. It can also be defined as the x-coordinate of a point on the unit circle that is located at a certain angle.

The trigonometric functions can be related to each other using trigonometric identities.

For example, the Pythagorean identity states that sin²(θ) + cos²(θ) = 1.

This means that if you know the value of one trigonometric function, you can find the value of another using this identity.

In Quadrant III, the cosine function is negative and the tangent function is positive. To write the tangent function in terms of the cosine function, we can use the identity

tan(θ) = sin(θ)/cos(θ).

Since sin(θ) is negative in Quadrant III, we need to use the negative square root to ensure that the value of the tangent function is positive. This gives us the expression

\(\(\tan(\theta) = -\frac{\sqrt{1-\cos^2(\theta)}}{\cos(\theta)}\)\)

To conclude, we have seen that the tangent function can be written in terms of the cosine function using the identity tan(θ) = sin(θ)/cos(θ). In Quadrant III, the cosine function is negative and the tangent function is positive, so we need to use the negative square root to ensure that the value of the tangent function is positive.

The resulting expression is

\(\(\tan(\theta) = -\frac{\sqrt{1-\cos^2(\theta)}}{\cos(\theta)}\)\)

Learn more about trigonometric function visit:

brainly.com/question/25618616

#SPJ11

Answer:

C

Step-by-step explanation:

5 times a number is the 5n and the four less is the - 4 so that makes 5n-4

Answer:

Step-by-step explanation:

The Cartesian equation of the line that contains the points A and B, where A is the intersection point of the given plane with the x-axis and B is the intersection point with the z-axis, is x = 1 and z = 3.

To find the Cartesian equation of the line, we need to determine the values of x and z while allowing y to vary freely. Since point A lies on the x-axis, its y-coordinate is 0, so we have x = 1 and y = 0 for point A. Similarly, since point B lies on the z-axis, its y-coordinate is also 0, so we have z = 3 and y = 0 for point B.

Thus, the equation x = 1 represents the line that contains point A, and the equation z = 3 represents the line that contains point B. Since y can vary freely, we do not include it in the equations. Therefore, the Cartesian equation of the line that contains points A and B is x = 1 and z = 3.

to learn more about Cartesian equation click here:

brainly.com/question/27927590

#SPJ11

It is gvien that y varies inversely as x.

The relation between x and y in mathematical expressin is given as.

The value of y is 11 when x is 8 , we have to determine the value of y when x is 10.

Apply the mathematical expression and substitute the value of x and y ,

Now , substitute the value of k in the mathematical expression and value of x which is 10, to find the value of y.

Thus the value of y is 8.8 when x is 10.

Answer:

11+15√5

Step-by-step explanation:

mat.hway

Answer: x can equal 9 or -1

Step-by-step explanation:

There are 7000 ways to select 2 men, 4 women, 3 boys, and 3 girls with the given condition. To solve this problem, we can first choose the particular man and woman that must be selected. There are 6 choices for the man and 8 choices for the woman, so there are 6 x 8 = 48 ways to choose them.

After we have chosen the particular man and woman, we need to select 1 more man, 3 more women, 3 boys, and 3 girls from the remaining people.

There are 5 men left to choose from, so we have 5 choices for the second man. Then, there are 7 women left to choose from (since we've already selected one), so we have 7C3 = 35 ways to choose 3 more women.

For the boys, there are 4 boys left to choose from, so we have 4C3 = 4 ways to choose 3 boys. Similarly, there are 5 girls left to choose from, so we have 5C3 = 10 ways to choose 3 girls.

Therefore, the total number of ways to select 2 men, 4 women, 3 boys, and 3 girls if a particular man and woman must be selected is:

48 x 5 x 35 x 4 x 10 = 2,688,000

So there are 2,688,000 ways to make this selection.

To solve this problem, we'll first select the particular man and woman that must be chosen, and then choose the remaining members from the remaining groups.

Since the particular man and woman are already selected, we now need to choose 1 more man from the remaining 5 men, 3 more women from the remaining 7 women, 3 boys from the available 4 boys, and 3 girls from the available 5 girls.

Number of ways to choose 1 man from 5: C(5,1) = 5
Number of ways to choose 3 women from 7: C(7,3) = 35
Number of ways to choose 3 boys from 4: C(4,3) = 4
Number of ways to choose 3 girls from 5: C(5,3) = 10

Multiply these numbers together to find the total number of ways to form the group:

5 (men) x 35 (women) x 4 (boys) x 10 (girls) = 7000

So, there are 7000 ways to select 2 men, 4 women, 3 boys, and 3 girls with the given condition.

Visit here to learn more about Multiply brainly.com/question/30875464

#SPJ11

The slope is:

Work/explanation:

To find the slope, I use the slope formula

\(\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}\)

where m = slope.

Plug in the data

\(\sf{m=\dfrac{-2-4}{14-(-1)}\)

Simplify

\(\sf{m=\dfrac{-6}{14+1}}\)

Simplify the denominator

\(\sf{m=-\dfrac{6}{15}}\)

Finally, reduce the fraction to its lowest terms.

\(\sf{m=-\dfrac{2}{5}}\)

We can proved that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Let's denote the vertices of the parallelogram as A, B, C, and D, with O as the midpoint of AC and BD.

First, we can see that triangles ABO and CDO are congruent by the Side-Side-Side (SSS) postulate, since they share side BD, and both have sides AB and CO of equal length due to O being the midpoint of BD. Therefore, angles AOB and COD are congruent, and we can denote their measure as θ.

Using the Law of Cosines in triangles ABO and CDO, we can express the squares of the diagonals AC and BD in terms of the sides of the parallelogram:

AC² = AB² + BC² - 2(AB)(BC)cosθ

BD² = AB² + BC² + 2(AB)(BC)cosθ

Adding these two equations together, we get:

AC² + BD² = 2(AB² + BC²)

which is the desired result. Therefore, we have shown that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Learn more about parallelogram at https://brainly.com/question/11645193

#SPJ11

If I were to hypothesize that communication students will have a higher average score on the oral communication measures,

I would have a research hypothesis. A research hypothesis is a statement that is used to explain a relationship between two or more variables,

in this case, the relationship between being a communication student and having a higher score on oral communication measures.

The hypothesis can then be tested through research and analysis of data to determine if there is a significant correlation between the two variables. In order to fully test this hypothesis,

it would be necessary to gather data on both communication students and non-communication students and compare their scores on oral communication measures.

learn more about communication here:brainly.com/question/22558440

#SPJ11

The series converges for x in the interval [-1, 1), with the left end included and the right end not included.

To find all the values of x for which the given series converges, we will use the Ratio Test. The series is given by:

\(\sum_{n=1}^\infty \frac{(4 x)^n}{n^{8}}\)

Let's apply the Ratio Test:

\(\lim_{n \to \infty} \frac{\frac{(4 x)^{n+1}}{(n+1)^{8}}}{\frac{(4 x)^n}{n^{8}}} = \lim_{n \to \infty} \frac{n^8 (4x)^{n+1}}{(n+1)^8 (4x)^n}\)

Simplify the expression:

\(\lim_{n \to \infty} \frac{4x n^8}{(n+1)^8}\)

Now, we need to find the values of x for which this limit is less than 1, as the Ratio Test states that the series converges if this limit is less than 1.

\(\frac{4x}{(1+\frac{1}{n})^8} < 1\)

Divide both sides by 4:

\(x < \frac{1}{(1+\frac{1}{n})^8}\)

As n approaches infinity, the term \(\frac{1}{n}\) approaches 0:

\(x < \frac{1}{(1+0)^8} = \frac{1}{1}\)

So, for the series to converge, x must be less than 1.

The series is convergent from x = -1, left end included (Y) to x = 1, right end not included (N).

Your answer: The series converges for x in the interval [-1, 1), with the left end included and the right end not included.

Learn more about Convergence at:
https://brainly.com/question/30114464

#SPJ11

There are different models that can be used to add fractions, but one common model is the area model, which represents fractions as parts of a whole.

To use the area model to add fractions, you can follow these steps:

1. Draw a rectangle to represent the whole. The size of the rectangle can be arbitrary, but it should be divided into equal parts to match the denominators of the fractions you are adding.

2. Divide the rectangle into the appropriate number of equal parts to represent the denominators of the fractions. For example, if you are adding 1/3 and 1/4, divide the rectangle into 12 equal parts (3 x 4 = 12).

3. Shade the appropriate number of parts in each fraction. For example, shade 4 out of 12 parts for 1/3 and 3 out of 12 parts for 1/4.

4. Count the total number of shaded parts in the rectangle. This represents the numerator of the sum of the fractions.

5. Write the sum of the fractions as the total number of shaded parts over the total number of parts in the rectangle.

For example, to add 1/3 and 1/4 using the area model:

1. Draw a rectangle to represent the whole.

```

+---+---+---+---+

| | | | |

+---+---+---+---+

| | | | |

+---+---+---+---+

| | | | |

+---+---+---+---+

```

2. Divide the rectangle into 12 equal parts.

```

+---+---+---+---+

| | | | |

+---+---+---+---+

| | | | |

+---+---+---+---+

| | | | |

+---+---+---+---+

1 2 3 4

```

3. Shade 4 out of 12 parts for 1/3 and 3 out of 12 parts for 1/4.

```

+---+---+---+---+

|###|###|###| |

+---+---+---+---+

|###|###|###| |

+---+---+---+---+

|###|###| | |

+---+---+---+---+

f is a linear transformation that is both injective and surjective, it is an isomorphism between X and F^n. Therefore, we have X ≅ F^n as desired.

To prove that X is isomorphic to F^n, where X is a finite-dimensional linear space over the field F of dimension n, we need to show that there exists an isomorphism between X and F^n.

Let (x_1, x_2, ..., x_n) be a basis of X. Any element x in X can be uniquely expressed as a linear combination of the basis vectors:

x = ∑(i=1 to n) μ_i * x_i, where μ_i ∈ F for all i.

Now, we define a function f: X -> F^n as follows:

f(x) = (μ_1, μ_2, ..., μ_n)

We claim that f is an isomorphism.

First, we need to show that f is well-defined, meaning that the mapping is independent of the choice of representation for x. Suppose x can be represented as a different linear combination of the basis vectors:

x = ∑(i=1 to n) ν_i * x_i, where ν_i ∈ F for all i.

Since both representations are linear combinations of the same basis vectors, we have:

∑(i=1 to n) μ_i * x_i = ∑(i=1 to n) ν_i * x_i

By the uniqueness of the representation, it follows that μ_i = ν_i for all i. Therefore, the function f(x) = (μ_1, μ_2, ..., μ_n) is well-defined.

Next, we need to show that f is a linear transformation. Let x, y ∈ X and α ∈ F. We have:

f(x + αy) = (μ_1 + αν_1, μ_2 + αν_2, ..., μ_n + αν_n)

         = (μ_1, μ_2, ..., μ_n) + α(ν_1, ν_2, ..., ν_n)

         = f(x) + αf(y)

This shows that f preserves vector addition and scalar multiplication, making it a linear transformation.

To prove that f is an isomorphism, we need to show that it is both injective and surjective.

Injectivity: Suppose f(x) = f(y), where x, y ∈ X. This implies (μ_1, μ_2, ..., μ_n) = (ν_1, ν_2, ..., ν_n), which further implies μ_i = ν_i for all i. Thus, x and y have the same unique representation in terms of the basis vectors, leading to x = y. Hence, f is injective.

Surjectivity: Let (a_1, a_2, ..., a_n) be an arbitrary element of F^n. We can construct an element x ∈ X such that f(x) = (a_1, a_2, ..., a_n) by choosing x = ∑(i=1 to n) a_i * x_i. This guarantees that f is surjective.

Since f is a linear transformation that is both injective and surjective, it is an isomorphism between X and F^n. Therefore, we have X ≅ F^n as desired.

Learn more about isomorphism here

https://brainly.com/question/30580081

#SPJ11

A stochastic process X(t) is called a martingale if the expected value of X(t) given all information available up to and including time s is equal to the value of X(s).

Thus, to show that the process X(t):=e^(t/2)cos(W(t)), 0 ≤ t ≤ T is a martingale w.r.t. any filtration for Brownian motion, we need to prove that E(X(t)|F_s) = X(s), where F_s is the sigma-algebra of all events up to time s.

As X(t) is of the form e^(t/2)cos(W(t)), we can use Itô's lemma to obtain the differential form:dX = e^(t/2)cos(W(t))dW - 1/2 e^(t/2)sin(W(t))dt

Taking the expectation on both sides of this equation gives:E(dX) = E(e^(t/2)cos(W(t))dW) - 1/2 E(e^(t/2)sin(W(t))dt)Now, as E(dW) = 0 and E(dW^2) = dt, the first term of the right-hand side vanishes.

For the second term, we can use the fact that sin(W(t)) is independent of F_s and therefore can be taken outside the conditional expectation:

E(dX) = - 1/2 E(e^(t/2)sin(W(t)))dt = 0Since dX is zero-mean, it follows that X(t) is a martingale w.r.t. any filtration for Brownian motion.

Now, let's represent X(t) as an Itô process on the interval [0,T]. Applying Itô's lemma to X(t) gives:

dX = e^(t/2)cos(W(t))dW - 1/2 e^(t/2)sin(W(t))dt= dM + 1/2 e^(t/2)sin(W(t))dt

where M is a martingale with M(0) = 0.

Thus, X(t) can be represented as an Itô process on [0,T] of the form:

X(t) = M(t) + ∫₀ᵗ 1/2 e^(s/2)sin(W(s))ds

Hence, we have shown that X(t) is a martingale w.r.t. any filtration for Brownian motion and represented it as an Itô process on any time interval [0,T], T > 0.

To know more about martingale visit:
brainly.com/question/32735198

#SPJ11

The partial derivative diffusivity equation for spherical flow is ∂²p/∂r² + (1/r) ∂p/∂r = ϕμC/k ∂p/∂t, and for hemispherical flow, it is the same equation.

1. The partial derivative diffusivity equation for spherical flow is derived from the spherical coordinate system and applies to radial flow in a spherical geometry. It can be expressed as ∂²p/∂r² + (1/r) ∂p/∂r = ϕμC/k ∂p/∂t.

2. The partial derivative diffusivity equation for hemispherical flow is derived from the hemispherical coordinate system and applies to radial flow in a hemispherical geometry. It can be expressed as ∂²p/∂r² + (1/r) ∂p/∂r = ϕμC/k ∂p/∂t.

1. For the derivation of the partial derivative diffusivity equation for spherical flow, we consider a spherical coordinate system with the radial direction (r), the azimuthal angle (θ), and the polar angle (φ). By assuming steady-state flow and neglecting the other coordinate directions, we focus on radial flow. Applying the Laplace operator (∇²) in spherical coordinates, we obtain ∇²p = (1/r²) (∂/∂r) (r² ∂p/∂r). Simplifying this expression, we arrive at ∂²p/∂r² + (1/r) ∂p/∂r.

2. Similarly, for the derivation of the partial derivative diffusivity equation for hemispherical flow, we consider a hemispherical coordinate system with the radial direction (r), the azimuthal angle (θ), and the elevation angle (ε). Again, assuming steady-state flow and neglecting the other coordinate directions, we focus on radial flow. Applying the Laplace operator (∇²) in hemispherical coordinates, we obtain ∇²p = (1/r²) (∂/∂r) (r² ∂p/∂r). Simplifying this expression, we arrive at ∂²p/∂r² + (1/r) ∂p/∂r.

In both cases, the term ϕμC/k ∂p/∂t represents the source or sink term, where ϕ is the porosity, μ is the fluid viscosity, C is the compressibility, k is the permeability, and ∂p/∂t is the change in pressure over time.

These equations are commonly used in fluid mechanics and petroleum engineering to describe radial flow behavior in spherical and hemispherical geometries, respectively.

To learn more about partial derivative, click here: brainly.com/question/2293382

#SPJ11

Answer

1. f - 3/4

2. 62 + h

Step-by-step explanation:

1. 1/4 of the fish are angelfish, all fish are one whole, one whole - 3/4's = 1/4

2. we dont have the value of H so we in turn have to figure out the equation and 62 + h is the only answer we can come to

Answer:

x = 35

exterior angle measure = 82°

Step-by-step explanation:

The angles denoted by (2x + 12) and (3x - 7) are a linear pair of angles formed at the vertex created by extending one side of the quadrilateral

Therefore they are supplementary angles whose sum = 180°

Therefore
2x + 12 + 3x - 7 = 180
2x + 3x + 12 - 7 = 180
5x + 5 = 180

Dividing by 5:

5x/5 + 5/5 = 180/5
x + 1 = 36

x = 36 - 1 = 35

The exterior angle is the one which has the expression 2x + 12

Plug in x = 35 to get
exterior angle measure = 2(35) + 12 = 70 + 12 = 82°

To do this, we'll use the formula for the surface area of a rectangular prism, which is:

Surface Area = 2lw + 2lh + 2wh

where l is the length, w is the width, and h is the height.

Given the dimensions of the aquarium:
l = 50 cm
w = 40 cm
h = 30 cm

Now, plug these values into the formula:

Surface Area = 2(50)(40) + 2(50)(30) + 2(40)(30)

Next, perform the multiplications:

Surface Area = 2(2000) + 2(1500) + 2(1200)

Then, multiply by 2 for each term:

Surface Area = 4000 + 3000 + 2400

Finally, add the values together:

Surface Area = 9400 cm²

So, the surface area of the aquarium is 9400 cm².

To learn more about surface area : brainly.com/question/29101132

#SPJ11

Ms. Logrieco's door: 35 students per 10 minutes

Mr. Riley's door: 22 students per 8 minutes

Time Frame: 24 minutes

35 x 2 = 70

35 x 2/5 = 14

70 + 14 = 84

22 x 3 = 66

84 + 66 = 150

Thus, we can expect for 150 students to be in the cafeteria after 24 minutes.

Answer:

3.6

Step-by-step explanation:

To find the distance of two points you use the formula \(\sqrt{x2-x1)^{2} +(y2-y1)^{2} }\)

so all we have to do is plug in the x's and y's

the points given are (-8,6) and (-5,8) so  we would do

\(\sqrt{(-5-(-8))^{2} +(8-6)^{2} }\)

if you plug that into a calculator you get that the distance between the two points are 3.605551275

but you would just want to round to the nearest tenth so the distance between the two points are 3.6

hope this helps and if you have nay questions lmk :D

9514 1404 393

Answer:

Step-by-step explanation:

A and the angle marked 48° are vertical angles, hence congruent.

  A = 48°

B is part of a linear pair with the angle marked 65°, so it is supplementary to that angle.

  B = 115°

C is an alternate interior angle with angle B, so is congruent.

  C = 115°

D is a vertical angle with the one marked 67°, so is congruent.

  D = 67°

E is an alternate exterior angle with the one marked 48°, so is congruent. It is also a corresponding angle with A, so congruent to that as well.

  E = 48°

Rolle's Theorem can be applied on \(-x^2 + 9x\) on [0, 9]

What is Rolle's Theorem?

Rolle's Theorem states that

1) If f is continuous on [a, b]

2) f is differentiable on (a, b)

3) f(a) = f(b)

Then there exist a point c in (a, b) such that \(f^{'}(c) = 0\)

Here, f(x) = \(-x^2 + 9x\)

f is continuous on [0, 9] as it is a polynomial function.

f is differentiable on (0, 9)

f(0) = \(-0^2+9\times 0 = 0\)

f(9) = \(-9^2+9 \times 9 = 0\)

f(0) = f(9)

So Rolle's Theorem has been applied here

To learn more about Rolle's Theorem, refer to the link-

https://brainly.com/question/29438393

#SPJ4