The opening of a tunnel that travels through a mountainside can be modeled by \(y = - \frac {2}{15} (x - 15)(x + 15)\), where x and y are measured in feet. The x-axis represents the ground.
​a. Find the width of the tunnel at ground level.
b. How tall is the tunnel?

To find the marginal revenue equations and determine the production levels that will maximize revenue, we need to find the partial derivatives of the revenue function R(x, y) with respect to x and y. Then, we set these partial derivatives equal to zero and solve the resulting system of equations.

The revenue function is given by R(x, y) = 130x + 160y - 3x^2 - 4y^2 - xy.

To find the marginal revenue equations, we take the partial derivatives of R(x, y) with respect to x and y:

∂R/∂x = 130 - 6x - y

∂R/∂y = 160 - 8y - x

Next, we set these partial derivatives equal to zero and solve the resulting system of equations:

130 - 6x - y = 0   ...(1)

160 - 8y - x = 0   ...(2)

Solving equations (1) and (2) simultaneously will give us the production levels that will maximize revenue. This can be done by substitution or elimination methods.

Once the values of x and y are determined, we can plug them back into the revenue function R(x, y) to find the maximum revenue achieved.

Note: The given revenue function is quadratic, so it is important to confirm that the obtained solution corresponds to a maximum and not a minimum or saddle point by checking the second partial derivatives or using other optimization techniques.

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Answer:

4 : 2 = 10 : 5.

Step-by-step explanation:

Let's put this in a fraction way. 4/2 = 2, and 10/5 = 2. Likewise, we can keep going and putting things that equal to 2. (e.g. 42 : 21)

Answer:

$2.16 / gallon

Step-by-step explanation:

4 gallons were bought for $8.64

4 gallons = $8.64

1 gallon = ?

cross multiply and divide

$8.64 / 4 gallons = $2.16 per gallon

check:

$2.16 * 4 = $8.64

Answer:

6

Step-by-step explanation:

To solve this problem, you are supposed to take the GCF of 60 and 42.

Factors of 42: 1,2,3,6,7,14,21

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30

The common numbers are: 1,2,3,6

The greatest common factor is 6.

So, the greatest number of boxes she can make are 6

Rounding to the nearest tenth, the area of the sector is approximately 3.4 square meters.

The area of a sector of a circle can be found using the formula:
Area = (θ/2π) * πr²
where θ is the central angle and r is the radius of the circle.
In this case, the radius is given as 3.2 m and the central angle is 2π/3.
Substituting these values into the formula, we get:
Area = (2π/3 * 1/2π) * π(3.2)²
Simplifying further, we have:
Area = (1/3) * 3.2²
Calculating the value, we get:
Area ≈ 3.4133 square meters
Rounding to the nearest tenth, the area of the sector is approximately 3.4 square meters.

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Answer:

\(p > 80\)

Step-by-step explanation:

Given

Points = More than 80

Required [Missing from the question]

Represent as an inequality

Let p represents the number of points.

More than means greater than i.e. >

Hence, more than 80 is represented as;

\(p > 80\)

Below is an example implementation of the modified Student class with the requested features:

public class Student {
   private String name;
   private int age;
   private int[] testScores;
   
   public Student(String name, int age, int score1, int score2, int score3) {
       this.name = name;
       this.age = age;
       this.testScores = new int[]{score1, score2, score3};
   }
   
   public Student(String name, int age) {
       this.name = name;
       this.age = age;
       this.testScores = new int[3];
   }
   
   public void setTestScore(int testNumber, int score) {
       if (testNumber >= 1 && testNumber <= 3) {
           testScores[testNumber - 1] = score;
       } else {
           System.out.println("Invalid test number.");
       }
   }
   
   public int getTestScore(int testNumber) {
       if (testNumber >= 1 && testNumber <= 3) {
           return testScores[testNumber - 1];
       } else {
           System.out.println("Invalid test number.");
           return 0;
       }
   }
   
   public int average() {
       int sum = 0;
       for (int score : testScores) {
           sum += score;
       }
       return Math.round(sum / 3.0f);
   }
   
   Override
   public String toString() {
       String studentString = "Name: " + name + "\nAge: " + age;
       
       String testScoresString = "";
       for (int i = 0; i < 3; i++) {
           testScoresString += "Test " + (i + 1) + ": " + testScores[i] + "\n";
       }
       
       int avg = average();
       String averageString = "Average: " + avg + " with tests: " + testScores[0] + ", " + testScores[1] + ", " + testScores[2];
       
       return studentString + "\n" + testScoresString + averageString;
   }
}

With this implementation, you can create Student objects, set test scores using setTestScore(), retrieve test scores using getTestScore(), calculate the average using average(), and display all the information including test scores and average using toString().

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Answer:

A relation is defined as the collection of inputs and outputs which are related to each other in some way. In case, if each input in relation has accurately one output, then the relation is called a function.

Based on the given relation, we found that it is not a function because it has repeating x-values. Remember, for a relation to be a function, each input (x-value) must correspond to exactly one output (y-value).

To determine if a relation is a function, you need to check if each input (x-value) corresponds to exactly one output (y-value). You can use the following steps:

1. Identify the given relation as a set of ordered pairs, where each ordered pair represents an input-output pair.

2. Check if there are any repeating x-values in the relation. If there are no repeating x-values, move to the next step. If there are repeating x-values, the relation is not a function.

3. For each unique x-value, check if there is only one corresponding y-value. If there is exactly one y-value for each x-value, then the relation is a function. If there is more than one y-value for any x-value, then the relation is not a function.

Let's consider an example relation: {(1, 2), (2, 3), (3, 4), (2, 5)}.

Step 1: Identify the relation as a set of ordered pairs: {(1, 2), (2, 3), (3, 4), (2, 5)}.

Step 2: Check for repeating x-values. In our example, we have a repeating x-value of 2. Therefore, the relation is not a function.

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The dimension of a rectangle is 17m and 17m.

Here we have to find the area of a rectangle as large as possible.

Data given:

Perimeter = 68m

The formula for the perimeter of the rectangle:

perimeter = 2(length + breadth)

68 = 2( l + b)

l + b = 34

We have to find the largest area, so when we will take the length and breadth same then we get the largest area.

So length = breadth

2 length = 34

length = 17m

length = breadth = 17m

Therefore the dimensions are 17m and 17m.

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\(\huge \bf༆ Answer ༄\)

For the given triangles to be congruent by SSS criterion, the sides HJ and LN should be equal ~

therefore correct choice is ~ C

\( \large \boxed{ \sf \: HJ \cong LN}\)

The additional information needed to prove that both triangles are congruent by the SSS Congruence Theorem would be: C. HJ ≅ LN

Recall:

Thus, in the two triangles given, the two triangles has:

Therefore, an additional information needed to prove that both triangles are congruent by the SSS Congruence Theorem would be: C. HJ ≅ LN

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The correct option regarding the sampling distribution of p, considering the Central Limit Theorem, is given as follows:

D. The shape of the sampling distrbution of p is approximately normal because n <= 0.05N and np(1 -p) > 10.

The mean of the sampling distribution of p is of:

0.388.

The standard deviation of the sampling distribution of p is of:

0.0184.

The Central Limit Theorem defines the distribution of the sampling distribution of sample proportions of a proportion p in a sample of size n, in which:

As long as these two conditions are respected:

The values of the parameters in this context are given as follows:

N = 30000, n = 700, p = 0.388.

Hence the conditions are:

Then the shape is approximately normal and option D is correct.

The mean of the distribution is of:

\(\mu = p = 0.388\)

The standard error of the distribution is of:

\(s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.388(0.612)}{700}} = 0.0184\)

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In general, equipotential lines do not cross each other. However, there are certain circumstances where they can cross.

Equipotential lines represent regions of equal potential in a physical system, such as electric or gravitational fields. These lines are perpendicular to the field lines and indicate points with the same potential value. Under normal conditions, equipotential lines do not intersect because each line corresponds to a unique potential value, and no two points in a system can have the same potential value.

However, there are situations where equipotential lines can cross. This can occur when there are multiple sources of potential in the system or when the potential varies in a complex manner. In such cases, the equipotential lines may intersect each other, indicating regions with different potential values coming into close proximity.

It is important to note that the crossing of equipotential lines does not violate the basic principles of potential theory. Instead, it reflects the intricate and complex nature of the underlying physical system, where multiple influences or varying potentials can lead to the crossing of equipotential lines.

Therefore, while it is uncommon for equipotential lines to cross, certain circumstances can give rise to such crossings in systems with multiple sources of potential or complex potential variations.

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The linear function rule to model the distance d in feet of the car traveled after t seconds is d = 83t.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

The distance traveled after 9 seconds = 747 ft.

The function rule to model the distance d after t seconds.

d = 83t

When t = 9,

d = 83 x 9 = 747 ft

Thus,

The linear function is d = 83t.

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Answer:

$3.23

Step-by-step explanation:

1.71 ÷ 18 = 0.095 = 9.5%

9.5% × 34 = 3.23

The probability that a student lives within 3 miles of the school is 75%, and the probability that a student who lives within 3 miles walks to school is 20%. We can find the probability that a student both lives within 3 miles and walks to school by multiplying these probabilities:

0.75 x 0.20 = 0.15

Therefore, the probability that a student lives within 3 miles and walks to school is 0.15 or 15%.

Answer:

0.15  or   15%

Step-by-step explanation:

we can conclude that only about 2.28% of vehicles in Santa Clara are more than 6 years old, based on the given average age and standard deviation of the vehicle age distribution.

To answer this question, we can use the standard normal distribution table or a calculator to find the area under the standard normal distribution curve for the given probability. To do this, we need to standardize the variable of interest, which in this case is the age of the vehicle, into a standard normal variable with a mean of 0 and a standard deviation of 1. To standardize the variable of interest, we can use the formula

z = (x - mu) / sigma

where z is the standardized variable, x is the age of the vehicle, mu is the mean age of the vehicle (9 years), and sigma is the standard deviation of the vehicle age (18 months, or 1.5 years).

To find the percentage of vehicles that are more than 6 years old, we need to find the area under the standard normal distribution curve to the left of the standardized variable z = (6 - 9) / 1.5 = -2.

Using a standard normal distribution table or calculator, we can find that the area to the left of z = -2 is approximately 0.0228. This means that the percentage of vehicles that are more than 6 years old is approximately:

0.0228 * 100% = 2.28%

Therefore, we can conclude that only about 2.28% of vehicles in Santa Clara are more than 6 years old, based on the given average age and standard deviation of the vehicle age distribution.

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Answer:

m∠ADC = 132°

Step-by-step explanation:

Use sine rule to find m<ADB

\( \frac{b}{sin(B)} = \frac{d}{sin(D)} \)

b = AD = 35

B = m∠ABD = 120º

d = AB = 30

D = m∠ADB = ?

Plug in the values

\( \frac{35}{sin(120)} = \frac{30}{sin(D)} \)

\( \frac{35}{sin(120)} = \frac{30}{sin(D)} \)

Cross multiply

\( 35 \times sin(D) = 30 \times sin(120) \)

Divide both sides by 35

\( \frac{35 \times sin(D)}{35} = \frac{30 \times sin(120)}{35} \)

\( sin(D) = \frac{30 \times sin(120)}{35} \)

\( sin(D) = 0.7423 \)

\( D = sin^{-1}(0.7423) \)

\( D = 48 \) (nearest integer)

D = m∠ADB = 48°

m∠ADC = 180 - m∠ADB (angles on a straight line)

m∠ADC = 180 - 48° (substitution)

m∠ADC = 132°

Answer:

m∠ADC =  132

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

This shape is a trapezoid. We can divide into two parts: a triangle and a rectangle.

■■■■■■■■■■■■■■■■■■■■■■■■■■

Let A' be the area of the triangle.

● A'= (b*h)/2

b is the base and h is teh heigth.

b= 26-20 = 6 mm

● A'= (6*14)/2 = 42 mm^2

●●●●●●●●●●●●●●●●●●●●●●●●

Let A" be the area of the rectangle.

A"= L*w

L is the length and w is the width.

A"= 14*20

A"= 280 mm^2

■■■■■■■■■■■■■■■■■■■■■■■■■■

Let A be the area of the trapezoid.

A= A'+A"

A= 42+280

A= 322 mm^2

Answer:

[8 -6 2 -4].

Step-by-step explanation:

[4 -4 0 0]+[4 -2 2 -4]

= [4+4 -4+-2 0+2 0+-4]

= [8 -6 2 -4]

Answer:

62.17 days

Step-by-step explanation:

The computation of the number of days  is shown below:

Given that

A(t)=10 × 0.5^(t ÷ 18.72)

1 = 10 × 0.5 ^ (t ÷ 18.72)

0.1 = 0.5 ^ (t ÷ 18.72)

Now consider the log in both sides

ln(0.1) = ln(0.5)^ (t ÷ 18.72)

ln(0.1)  ÷  ln(0.5) = (t ÷ 18.72)

3.321928095  = (t ÷ 18.72)

So, t = 62.17 days

With the help of log, we can easily compute it

Answer: x = 12

Step-by-step explanation:

\(4x-20=-2(-3x+22)\)

Begin by distributing -2

\(4x-20=6x-44\)

Subtract 6x

\(4x-6x-20=-44\)

Add 20

\(4x-6x=-44+20\)

Combine like terms;

\(-2x=-24\)

Divide by -2

\(x=\frac{-24}{-2}\\ x=12\)

Answer:

C Yes, because the cost is always three times the number of pounds.

Step-by-step explanation:

Because 2*3=6, 4*3=12, and 8*3=24

Answer:  C) Yes, because the cost is always three times the number of pounds.

Step-by-step explanation:

Took the assignment on Time4Learning

Answer:

T=(b+4)+(c+10)

Sorry but I really don’t know but I tried

Answer:

x = 2/3

y = 5

Step-by-step explanation:

6x + y = 9

3x -4y = -18

Times the second equation by -2

6x + y = 9

-6x + 8y = 36

9y = 45

y = 5

Now put 5 in for y and solve for x

6x + 5 = 9

6x = 4

x = 2/3

Let's check

6(2/3) + 5 = 9

4 + 5 = 9

9 = 9

So, x = 2/3 and y = 5 is the correct answer.

Answer:

All real numbers

Step-by-step explanation:

Since x=2y+5, and 2x-4y=10, substitute 2y+5 for x into 2x-4y=10

2(2y+5)-4y=10

Use distributive prop.

4y+10-4y=10

10=10

Since both sides of the equation are always equivalent,

All real numbers work, since the two equations are the same line.

5/12 is the answer but 1/3 x 5/4 is the reduced version of your question and the alternate form of the answer is 0.416

Answer:

a sin(bθ−c)+d

→ Using this, find the amplitude based off of the given formula. Use the expression as a reference to help you identify a.

a = 3

b = 4

c = 0(no defined value)

d = 0(impossible to identify)

→ Find a.

→ Since a is already given, we know that a = 3, therefore, the amplitude is 3

Here, M₂2 → R be the linear transformation defined by T(A) = tr(A). Solution is (a) The matrix [1 2; -1 5] is in ker(T). (b) The scalar 0 is in range(T).

(a) To determine if a matrix A is in the kernel of T, we need to check if T(A) = tr(A) equals zero. For the matrix [1 2; -1 5], the trace is 1 + 5 = 6, which is not zero. Therefore, it is not in the kernel of T.

(b) To determine if a scalar c is in the range of T, we need to find a matrix A such that T(A) = tr(A) = c. For the scalar 0, we can choose the zero matrix [0 0; 0 0], which has a trace of 0. Hence, 0 is in the range of T.

Ker(T) refers to the kernel or null space of the linear transformation T. In this case, ker(T) consists of all matrices A such that tr(A) = 0. These matrices have a trace of zero, meaning the sum of their diagonal elements is zero. It forms a subspace of M₂2.

Range(T) refers to the range or image of the linear transformation T. In this case, range(T) consists of all scalars c for which there exists a matrix A such that tr(A) = c. The range of T is the set of all possible values that the trace function can take, which is the set of all real numbers.

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