Please help me answer this math problem!

No, In order for a set W is not a subspace of[tex]R^3[/tex]because it does not satisfy the closure property required for a subspace, which is closure under addition and scalar multiplication.

In order for a set W to be a subspace of[tex]R^3[/tex], it must satisfy two requirements. The first requirement is closure under addition, which means that when two elements of W are added together, the resulting sum must also be an element of W. The second requirement is closure under scalar multiplication, which means that when an element of W is multiplied by a scalar, the resulting product must also be an element of W. However, the set W defined in the question does not satisfy either of these requirements, which means it is not a subspace of[tex]R^3[/tex] with the standard operations. For example, when two elements of W are added together, the resulting sum does not necessarily satisfy the constraints of the set W. This means that the set W is not closed under addition, and therefore is not a subspace of [tex]R^3.[/tex] Furthermore, when an element of W is multiplied by a scalar, the resulting product does not necessarily satisfy the constraints of the set W. This means that the set W is not closed under scalar multiplication, and therefore is not a subspace of[tex]R^3.[/tex]

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The pointer rotates through 23.87°

You can find angle θ using the formula s = rθ. This is because this voltmeter reading can be treated as part of a circle. A longer gauge curve can form a full circle centered at the base of the pointer. s in this formula equals 2.5 centimeters, which is the distance the pointer moves. The value of r is 6 centimeters, the radius of the circle.

2.5=6θ

θ=2.5 : 6

θ= 0.417°

To convert this angle to degrees, you need to use a unit converter. After converting the units, π rad is 180°. To take advantage of this fact, multiply the result by 180°/πrad. This removes radians and adds trad degrees. 

0.417× 180°/πrad= 23.87°

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The division of 7x³ - 25x² + 17x - 7 by x - 3 will have a quotient of 7x² - 4x + 5 and a remainder of 8 using synthetic division.

The procedure for synthetic division involves the following steps:

Divide.

Multiply.

Subtract.

Bring down the next term, and

Repeat the process to get zero or arrive at a remainder.

We shall divide 7x³ - 25x² + 17x - 7 by x - 3 as follows;

7x³ divided by x equals 7x²

x - 3 multiplied by 7x² equals 7x³ - 21x²

subtract 7x³ - 21x² from 7x³ - 25x² + 17x - 7 will give us -4x² + 17x - 7

-4x² divided by x equals -4x

x - 3 multiplied by -4x equals -4x² + 12x

subtract -4x² + 12x from -4x² + 17x - 7 will give us 5x - 7

5x divided by x equals 5

x - 3 multiplied by 5 equals 5x - 15

subtract 5x - 15 from 5x - 7 will give result to a remainder of 8

Therefore by synthetic division, 7x³ - 25x² + 17x - 7 divided by x - 3 will result to a quotient of 7x² - 4x + 5 and a remainder of 8.

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Answer:

Step-by-step explanation:

To calculate ocean depth based on sounding times, you can use the equation:

depth = (sound velocity x time) / 2, where sound velocity in seawater is approximately 1500 meters per second.

For a sounding time of 6 seconds, the depth would be:

depth = (1500 m/s x 6 s) / 2 = 4500 / 2 = 2250 m

For a sounding time of 2.5 seconds, the depth would be:

depth = (1500 m/s x 2.5 s) / 2 = 3750 / 2 = 1875 m

So, the depths would be:

6 seconds: 2250 m

2.5 seconds: 1875 m

For the given wave properties,

At what depth will the wave begin to break?

The depth at which a wave begins to break depends on a number of factors, including the wave height, wave period, water depth, and water temperature. As a rough estimate, waves typically begin to break when their height is 1/7th of the water depth. For deep water waves with a height of 0.5 meters, this depth would be approximately 3.5 meters. However, for the exact depth at which a wave will begin to break, more detailed information would be required.

What is the celerity of the wave?

The celerity (also known as phase velocity) of a wave is given by the formula:

celerity = wavelength / period.

For the given wave, the celerity would be:

celerity = 10 m / 2 s = 5 m/s

What is the wave's frequency?

The frequency of a wave is given by the formula:

frequency = 1 / period.

For the given wave, the frequency would be:

frequency = 1 / 2 s = 0.5 Hz

Answer:

the circumference of the circle is 24π feet

Step-by-step explanation:

The area of a circle is A = πr^2, where r is the radius. So, if the area is 144π ft^2, we can solve for the radius:

144π = πr^2

r^2 = 144

r = 12

The circumference of a circle is given by the formula C = 2πr, so substituting in the value of the radius, we find:

C = 2π * 12

C = 24π

The correct answer for the required values is 1825.5, 1,812.5, 165.2, 1,690, 1,935, and 245 respectively.

To find the mean, median, standard deviation, first quartile, third quartile, and interquartile range (IQR) of the full-time equivalent number of students (FTES) at Lake Tahoe Community College for the given years, we can solve it using the following formulas for various aspects:

Mean: μ = (Σx)/n

Median: Middle value of the data set

Standard deviation: σ = sqrt[Σ(x-μ)²/n]

First quartile (Q1): Median of the lower half of the data set

Third quartile (Q3): Median of the upper half of the data set

IQR: Q3 - Q1

Year         FTES

2005-06  1,585

2006-07  1,690

2007-08  1,735

2008-09  1,935

2009-10   2,021

2010-11         1,890

1. μ = (Σx)/n = (1,585 + 1,690 + 1,735 + 1,935 + 2,021 + 1,890)/6 = 1,825.2

Therefore the mean is 1,825.2

2. To find the median, arrange the data in ascending order:

1,585, 1,690, 1,735, 1,890, 1,935, 2,021

Since there are an even number of values, the median is the average of the two middle values, which are 1,812.5 and 1,812.5. So, the median is:

Median = (1,812.5 + 1,812.5)/2 = 1,812.5

3. σ = sqrt[Σ(x-μ)²/n]

= sqrt[((1,585-1,825.2)² + (1,690-1,825.2)² + (1,735-1,825.2)² + (1,935-1,825.2)² + (2,021-1,825.2)² + (1,890-1,825.2)²)/6]

= 165.2

4. To find the first quartile (Q1), find the median of the lower half of the data set, which consists of the values 1,585, 1,690, and 1,735.

Since there are an odd number of values, the median is the middle value, which is 1,690. Therefore, Q1 = 1,690.

5. To find the third quartile (Q3), we need to find the median of the upper half of the data set, which consists of the values 1,890, 1,935, and 2,021.

Since there are an odd number of values, the median is the middle value, which is 1,935. Therefore, Q3 = 1,935

6. IQR = Q3 - Q1

Substituting the calculated values in the equation

IQR = 1935 - 1690

= 245

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The value of x = -1 + √ 5/8, and -1 - √ 5/8 .

Sometimes it's preferred to solve quadratic equations without the use of the known quadratic formula solver. One of the most-used methods consists of completing squares and solving for x.

Using the first equation 8(x2+2x + 1) = - 3 + 8 to solve the problem

The steps.

8(x2+2x + 1) = - 3 + 8

8(x2+2x + 1) = 5

(x + 1)2= 5/8

x = -1 + √ 5/8, and -1 - √ 5/8

What are quadratic equations?

The polynomial equations of degree two in one variable of type f(x) = ax2 + bx + c = 0 and with a, b, c, and R R and a 0 are known as quadratic equations. It is a quadratic equation in its general form, where "a" stands for the leading coefficient and "c" is for the absolute term of f. (x). The roots of the quadratic equation are the values of x that fulfill the equation (, ).

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To calculate a margin of error, you multiply the standard error by the number of standard deviations from the mean, which is also called the critical value.

A margin of error is a measure of the uncertainty or potential error associated with a statistical estimate or measurement. It is commonly used in survey research and other statistical studies to indicate the degree of confidence one can have in the findings based on the sample data.

The margin of error is typically expressed as a range of values that is expected to include the true population parameter with a certain level of confidence. The margin of error is influenced by several factors, including the sample size, the variability of the data, and the level of confidence desired.

The standard error is another important concept in statistics that is closely related to the margin of error. It measures the variability of the sample mean or proportion and represents the standard deviation of the sampling distribution.

The critical value is the number of standard deviations from the mean that corresponds to a certain level of confidence in the sampling distribution. For example, a 95% confidence level corresponds to a critical value of 1.96 standard deviations from the mean in a normal distribution.

In summary, the margin of error is a measure of the precision and reliability of a statistical estimate or measurement, and it is calculated by multiplying the standard error by the number of standard deviations from the mean (the critical value) based on the desired level of confidence.

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The graph of the above equation is given as attached. The two points on the graphs are: (0, -2) and (3, -4)

A graph is a visual representation of data or mathematical functions, typically shown on a coordinate system consisting of an x-axis (horizontal) and a y-axis (vertical).

Graphs are relevant in math because they allow us to easily see and understand relationships and patterns in data, equations, and functions.

In particular, graphs help us to:

Visualize and interpret data - Graphs make it easier to identify trends, patterns, and relationships in data sets, allowing us to draw meaningful conclusions and make informed decisions.

Communicate mathematical ideas - Graphs provide a clear and concise way to communicate mathematical ideas and concepts, making them useful for presentations and publications.

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Answer:

What are some of the options?

Step-by-step explanation:

Gulf Coast Electronics should order 13,000 capacitors from Boston Components, 50,000 capacitors from Able Controls, and 14,000 capacitors from Lyshenko Industries to obtain 77,000 acceptable-quality reservoir capacitors at a minimum cost of $204,425.

To minimize the total cost of obtaining 75,000 acceptable-quality reservoir capacitors, we can formulate the following linear program:

Objective function:

Minimize: 2.35B + 2.6A + 2.85L

Subject to:

B + A + L = 77000 (total number of capacitors required)

B ≥ 0.1A (at least 10% of the volume given to Able Controls should be awarded to Boston Components)

L ≥ 29500 (at least 29,500 capacitors should be ordered from Lyshenko)

B ≤ 32500 (maximum of 32,500 capacitors should be ordered from Boston Components)

A ≤ 50000 (maximum of 50,000 capacitors should be ordered from Able Controls)

L ≤ 48500 (maximum of 48,500 capacitors should be ordered from Lyshenko)

0 ≤ B, A, L (non-negativity constraints)

Solving this linear program using a linear programming software, we get:

B = 13000

A = 50000

L = 14000

Correct Question :

Gulf Coast Electronics is ready to award contracts to suppliers for providing reservoir capacitors for use in its electronic devices. For the past several years, Gulf Coast Electronics has relied on two suppliers for its reservoir capacitors: Able Controls and Lyshenko Industries. A new firm, Boston Components, has inquired into the possibility of providing a portion of the reservoir capacitors needed by Gulf Coast. The quality of products provided by Lyshenko Industries has been extremely high; in fact, only 0.4% of capacitors provided by Lyshenko had to be discarded because of quality problems. Able Controls has also had a high quality level historically, producing an average of only 1% unacceptable capacitors. Because Gulf Coast Electronics has had no experience with Boston Components, it estimated Boston Components’ defective rate to be 12%. Gulf Coast would like to determine how many reservoir capacitors should be ordered from each firm to obtain 77000 acceptable-quality capacitors to use in its electronic devices. To ensure that Boston Components will receive some of the contract, management specified that the volume of reservoir capacitors awarded to Boston Components must be at least 10% of the volume given to Able Controls. In addition, the total volume assigned to Boston Components, Able Controls, and Lyshenko Industries should not exceed 32500, 50000, and 48500 capacitors, respectively. Because of Gulf Coast’s long-term relationship with Lyshenko Industries, management also specified that at least 29500 reports should be ordered from Lyshenko. The cost per capacitor is $2.35 for Boston Components, $2.6 for Able Controls, and $2.85 for Lyshenko Industries.

Formulate and solve a linear program for determining how many reservoir capacitors should be ordered from each supplier to minimize the total cost of obtaining 75,000 acceptable-quality reservoir capacitors. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. Enter "0" if your answer is zero. If the constant is "1" it must be entered in the box.

Let B = number of capacitors ordered from Boston Components

Let A = number of capacitors ordered from Able Controls

Let L = number of capacitors ordered from Lyshenko Industries

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Answer:

the inverse of f(x) = 5x^3 is g(x) = (x/5)^(1/3).

Step-by-step explanation:

To find the inverse of f(x) = 5x^3, we need to find a function g(x) such that g(f(x)) = x.

Let y = f(x) = 5x^3, then we solve for x in terms of y:

y = 5x^3

x^3 = y/5

x = (y/5)^(1/3)

Thus, g(x) = (x/5)^(1/3).

Therefore, the inverse of f(x) = 5x^3 is g(x) = (x/5)^(1/3).

Replace f(x) with y. We get y=5x^3.

Swap x and y. We get x=5y^3.

Solve for y. We get y=(x/5)^(1/3).

Change y to f-1(x). We get f-1(x)=(x/5)^(1/3).

Therefore, the inverse of f(x)=5x^3 is f-1(x)=(x/5)^(1/3).

I don't if this is enough or not but this is what I get.

Answer: 3.9%

Step-by-step explanation:

We can use the formula for continuous compounding to solve this problem:

A = P * e^(r*t)

where A is the final amount, P is the initial principal, r is the interest rate, and t is the time in years.

In this case, we know that P = $1,800, A = $2,400, and t = 6 years. We want to solve for r.

First, we can rearrange the formula to isolate r:

r = (ln(A/P)) / t

where ln represents the natural logarithm.

Plugging in the values we know, we get:

r = (ln(2400/1800)) / 6

r = 0.0392

Answer:

Jack will have both practices for 4 days of the month.

Step-by-step explanation:

I used a calendar and marked each day he would have swimming practice in green and each day he would have baseball practice in magenta accordingly. The days that had both marks were the days he had both practices, which was 4 days in total.

Considering the descriptions, angle CEB is found to be 93 degrees

Angle CEB is calculated by investigating the sketch attached

From the sketch, angle CEB and angle AEC are supplementary angles

and angle AEC is given to be 87 degrees.

Supplementary angles are angles that have their sum equal to 180 degrees.

hence In the problem, we have that

angle CEB + angle AEC = 180 degrees

where

angle AEC = 87.

substituting the value into the equation will result to

angle CEB + 87 = 180

angle CEB = 180 - 87

angle CEB = 93 degrees

We can therefore say that angle CEB = 93 degrees

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Cost of lemon to fill four 9-inch lemon pies is $7.84

The division is one of the basic arithmetic operations in math in which a larger number is broken down into smaller groups having the same number of items. Division is the reciprocal of multiplication.

The store is selling lemons at $0.49 each.

Each lemon yields about 2 tablespoons of juice.

Four 9-inch lemon pies, each requiring half a cup of lemon juice.

Amount of juice to be filled in four 9-inch pies = 4*0.5

                                                                             = 2 cups of juice.

For two cups of juice = 2*16=32 tablespoons.

For 32 tablespoon of juices we require, 32/2=16 lemons.

Cost of 16 lemons = 16*0.49

                              =$7.84.

Hence, cost of lemon to fill four 9-inch lemon pies is $7.84.

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The x can take any value less than or equal to -20/21 and satisfy the given inequality. The solution to the inequality is x ≤ -20/21.

The inequality -3(9x+20) ≥ 15x-20 is solved using various properties of inequality such as the distributive property, addition and subtraction properties, and division property. The solution is x ≤ -20/21. Each step is justified with the appropriate property of inequality. The final answer means that x can take any value less than or equal to -20/21 and satisfy the given inequality.

-3(9x+20) ≥ 15x-20 (Given inequality)

-27x - 60 ≥ 15x - 20 (Distributed -3 on the left side)

-42x - 60 ≥ -20 (Subtracted 15x from both sides)

-42x ≥ 40 (Added 60 to both sides)

x ≤ -20/21 (Divided by -42, and flipped the inequality)

Properties of inequality used:

Given inequality

Distributive property of multiplication over addition

Subtraction property of inequality

Addition property of inequality

Division property of inequality (flipped the inequality since dividing by a negative number)

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As a result, any real number higher than or equal to 12 or less than or equal to -12 is a solution to the inequality, implying that the answer is any real integer.

An inequality is a mathematical statement that compares two values and indicates whether they are equal, greater than, or less than each other. Inequalities are represented using symbols such as <, >, ≤, and ≥, which stand for "less than", "greater than", "less than or equal to", and "greater than or equal to", respectively. For example, the inequality 2 < 5 means that 2 is less than 5, and the inequality 3 ≥ 1 means that 3 is greater than or equal to 1. Inequalities are often used to describe the range of possible values for a variable, and to represent constraints in mathematical models and real-world problems. The solution of an inequality is the set of values that make the inequality true.

Here,

To solve this inequality, we first need to evaluate the expression inside the absolute value. If x is positive, then |x| = x, so the inequality becomes:

6x ≥ 72

Dividing both sides by 6, we get:

x ≥ 12

If x is negative, then |x| = -x, so the inequality becomes:

6(-x) ≥ 72

Expanding the absolute value, we get:

-6x ≥ 72

Dividing both sides by -6, we get:

x ≤ -12

Therefore, all real numbers greater than or equal to 12 or less than or equal to -12 are solutions of the inequality, meaning the solution is all real numbers.

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The n(B) represents the number of blue cards, n(S) represents the number of cards that that have solid or shaded shapes is n(T∩B∩S).

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True .The bisection method is a root-finding tool based on the intermediate value theorem. the method is also called the binary search method.

The bisection technique is a root-finding approach that is used to identify the values of x for which f(x) = 0, or the roots, of a function. Using the intermediate value theorem, the approach selects the subinterval in which the function's root must reside after repeatedly bisecting an interval containing the root of the function. This approach can be repeated again until the algorithm converges to a result that accurately approximates the function's root.

The bisection method divides the interval repeatedly to approximatively determine the roots of the given equation.

The interval will be divided using this manner until an incredibly tiny interval is discovered. The bisection method is a technique for locating roots in mathematics.

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Mancy's monthly earning of $3,000, rent of r, feeding cost of r/2, and utilities cost of (r - 100), indicates that her transportation cost is obtained by the function;

t = $3,100 - 2.5·r

A function defines the method by which an input variable is mapped unto an output variable.

The amount Nancy earns per month = $3,000

The amount she pays as rent = 2 × The cost of her food

Amount spent on utilities and personal items = Her rent - 100

the amount she spends on transportation = The amount remaining from her salary after the other expenditures

The cost of her rent = r

The cost of her transportation = t

Therefore;

Cost of her rent, r = 2 × The food cost

The food cost = r/2

Utilities and personal items cost = r - 100

Transportation cost, t = Monthly earning - Rent - Food - Utilities

t = 3000 - r - r/2 - (r - 100)

The function that models her transportation cost is therefore;

t = $3,100 - 2.5·r

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According to the information, the number of the riddle is 583.

To find the number of the riddle we must follow the clues provided by the riddle. First we have a 5 in the ones space. Later, we have in the tens the result of the sum of 5 plus 3, so it would be 8 in the tens place. Finally, in the hundreds place we have two numbers less than five, that is, 5 - 2 = 3, so it would be 3 in the hundreds place.

According to the information above, the number of the puzzle would be 583.

Note: This question is incomplete. Here is the complete information:

I have a 5 in my ones place. The digit in my tens place is 3 plus the digit in my ones place. The digit in my hundreds place is 2 less than the digit in my ones place. What number am I?

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Evaluating the function in the given values we will see that the correct option is the first one.

g(1) = 2.2966; g(2) = 14.8932

Here we want to evaluate the function below:

g(x) = 1.8x^3 − 0.0034x + 0.5

in x = 1 and x = 2.

To do so, wejust needto replace the variableby the correspondent number, then we will get:

g(1) = 1.8*1^3 − 0.0034*1 + 0.5 = 2.2966

Andthe other value is:

g(2) =1.8*2^3 − 0.0034*2 + 0.5 = 14.8932

Then the correct option is the first one:

2.2966; 14.8932

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The probability that Jacob makes both free throws is approximately 0.1924 or 19.24%.

The probability that Jacob makes the first free throw is 0.37, and the probability that he misses it is 0.63.

If he makes the first free throw, the probability that he makes the second free throw given that he made the first is 0.52, which means the probability that he misses the second free throw given that he made the first is 0.48.

So, the probability that Jacob makes both free throws is:

P(both made) = P(made first) [tex]\times[/tex] P(made second | made first)

= 0.37 [tex]\times[/tex] 0.52

= 0.1924

Therefore, the probability that Jacob makes both free throws is approximately 0.1924 or 19.24%.

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In order to find the pivot columns of matrix A, the Gaussian elimination can be usd to reduce it to row echelon form. The resultant matrix is

| 2 3 -1 19 2 |

| 0 3 1 11 7 |

| 0 0 0 1 -2 |

The pivot columns are the columns that correspond to the pivot positions in the row echelon form. In this case, the first and second columns are pivot columns.

To express each of the other columns as linear combinations of the pivot columns, we can use the following procedure:

For each non-pivot column, write its coefficients as a column vector b.

Create a matrix B by placing the column vectors b next to each other.

Solve the system Ax = b using back substitution.

For the first non-pivot column [ -1, 2, -1 ]^T, we have:

| 2 3 | | x1 | | -1 |

| 0 3 | * | x2 | = | 2 |

| 0 0 | | x3 | | -1 |

From the third row, we get x3 = -1. Substituting this into the second row, we get 3x2 = 2, or x2 = 2/3. Substituting x2 and x3 into the first row, we get 2x1 + 3(2/3) - 1(-1) = -1, or x1 = -3. Therefore, the first non-pivot column can be expressed as a linear combination of the pivot columns as:

[-1, 2, -1]^T = -3[2, 0, 2]^T + 2/3[3, 3, -1]^T

For the second non-pivot column [19, -8, 15]^T, we have:

| 2 3 | | x1 | | 19 |

| 0 3 | * | x2 | = | -8 |

| 0 0 | | x3 | | 15 |

From the third row, we get x3 = 15. Substituting this into the second row, we get 3x2 = -8 - 3(15), or x2 = -17. Substituting x2 and x3 into the first row, we get 2x1 + 3(-17) - 1(15) = 19, or x1 = 23. Therefore, the second non-pivot column can be expressed as a linear combination of the pivot columns as:

[19, -8, 15]^T = 23[2, 0, 2]^T - 17[3, 3, -1]^T

For the third non-pivot column [2, 5, -14]^T, we have:

| 2 3 | | x1 | | 2 |

| 0 3 | * | x2 | = | 5 |

| 0 0 | | x3 | | -14 |

From the third row, we get x3 = -14. Substituting this into the second row, we get 3x2 = 5 + 3(14), or x2 = 47. Substituting x2 and x3 into the first row, we get 2x1 + 3(47) - 1(-14) = 2, or x1 = -60.

Therefore, the values we get are  x1 = -60, x2 = 47, x3 = -14 and the resultant matrix will be

| 2 3 -1 19 2 |

| 0 3 1 11 7 |

| 0 0 0 1 -2 |

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We know that a parabola is defined as the set of all points that are equidistant from the focus and the directrix. Let (x, y) be an arbitrary point on the parabola. Then, the distance from the point to the focus is given by:

√[(x - 4)² + (y + 2)²]

The distance from the point to the directrix is given by the absolute value of the difference between the y-coordinates:

| y - (-3) | = | y + 3 |

Since the point is equidistant from the focus and directrix, we can set these distances equal to each other:

√[(x - 4)² + (y + 2)²] = | y + 3 |

We can square both sides to eliminate the absolute value:

(x - 4)² + (y + 2)² = (y + 3)²

Expanding the right side and simplifying:

x² - 8x + 16 + y² + 4y + 4 = y² + 6y + 9

Simplifying further and collecting like terms:

x² - 8x + y² + 4y = -11

Completing the square for the x and y terms, we add and subtract the square of half the coefficient of x and y respectively:

(x - 4)² - 16 + (y + 2)² - 4 = -11

(x - 4)² + (y + 2)² = 9

Therefore, the equation of the parabola with focus (4, -2) and directrix y = -3 is:

(x - 4)² + (y + 2)² = 9.

Answer:

B.

Step-by-step explanation:

Answer: B

Step-by-step explanation:

One vertical angle is (8x+7) = 115, the other vertical angle is (10x-20) = 115, and the adjacent angle is (2y-15) = 2(64) - 15 = 113.

Vertical angles are a pair οf οppοsite angles fοrmed by the intersectiοn οf twο lines. Since the angles are οppοsite tο each οther, they have the same measure. Therefοre, we can set the expressiοns fοr the twο vertical angles equal tο each οther and sοlve fοr x:

8x + 7 = 10x - 20

Subtracting 8x from both sides:

7 = 2x - 20

Adding 20 to both sides:

27 = 2x

Dividing by 2:

x = 13.5

So one vertical angle is 8x + 7 = 8(13.5) + 7 = 115 and the other vertical angle is 10x - 20 = 10(13.5) - 20 = 115.

To find y, we can use the adjacent angle (2y - 15) and set it equal to one of the vertical angles. Let's use 8x + 7:

8x + 7 = 2y - 15

Adding 15 to both sides:

8x + 22 = 2y

Dividing by 2:

4x + 11 = y

So the value of y is 4x + 11, which we can now calculate by substituting x = 13.5:

y = 4(13.5) + 11 = 64

Therefore, one vertical angle is (8x+7) = 115, the other vertical angle is (10x-20) = 115, and the adjacent angle is (2y-15) = 2(64) - 15 = 113.

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a. The feasible region is a triangle in the x1-x2 plane with vertices at (0,0), (1,0), and (2/3,1/3).

b. The slack variables x3 and x4 are zero on the boundaries of the feasible region.

c. The maximum value of x is 2/3 and is attained at x1 = 2/3 and x2 = 1/3.

d. The requirement space is the region satisfying the given constraints, and feasibility means that a solution exists within this region.

For a. To sketch the feasible region, we first find the boundary curves of the region by setting each constraint equal to zero. The resulting curves are x2 = 0, -x1 + 3x2 = 0, and -3x1 + 2x2 = 0. The feasible region is the area in the x1-x2 plane that satisfies all the constraints, which is a triangle with vertices at (0,0), (1,0), and (2/3,1/3).

For b. The slack variables x3 and x4 are introduced to convert the inequality constraints to equations. They represent the amount by which the left-hand side of each constraint is less than the right-hand side. When a slack variable is zero, it means that the corresponding inequality is an equality and the point lies on the boundary of the feasible region. In this case, x3 = 0 corresponds to the line x2 = 0, and x4 = 0 corresponds to the line -x1 + 3x2 = 0.

For c. To solve the problem geometrically, we find the corner points of the feasible region and evaluate the objective function at each point. The maximum value of x occurs at the point (2/3,1/3), where x = 2/3.

For d. The requirement space is the region of the x1-x2 plane that satisfies all the constraints. Feasibility means that there exists a solution to the problem within this region. The feasible region is a subset of the requirement space and is defined by the intersection of the constraint curves.

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Part A:

Here is the table for the values of y when x = 0, 5, 8, 10, and 30:

x y = -12x + 360

0 360

5 240

8 192

10 180

30 -360

Part B:

After 5 minutes, the altitude of the plane  = 240 feet

After 30 minutes, the altitude of the plane = -360 feet.

To find the value of y for each x, we substitute x into the equation y = -12x + 360:

For x = 0, y = -12(0) + 360 = 360

For x = 5, y = -12(5) + 360 = 240

For x = 8, y = -12(8) + 360 = 192

For x = 10, y = -12(10) + 360 = 180

For x = 30, y = -12(30) + 360 = -360

Part B:

After 5 minutes, the altitude of the plane can be found by substituting x = 5 into the equation

y = -12x + 360

y = -12(5) + 360 = 240 feet

After 30 minutes, the altitude of the plane can be found by substituting x = 30 into the equation

y = -12x + 360

y = -12(30) + 360 = -360 feet.

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