Find the equation of a line, in Slope Intercept Form, that has a slope of 3 and passes through the point (-4, 2).

By substituting the extreme points of the feasible set into the objective function and comparing the values, (x,y)=(1,1) is an optimal solution for the given LP problem.

The given LP problem is to maximize f(x,y)=3x+19y subject to the constraint set of the feasible set. The extreme points of the feasible set are (0,0), (1,0), (0,1), and (1,1).

To prove that (x,y)=(1,1) is an optimal solution, we need to show that f(x,y) is maximized at this point. We can do this by plugging in the extreme points into the objective function and comparing the values.

At (0,0), f(x,y) = 3(0) + 19(0) = 0

At (1,0), f(x,y) = 3(1) + 19(0) = 3

At (0,1), f(x,y) = 3(0) + 19(1) = 19

At (1,1), f(x,y) = 3(1) + 19(1) = 22

From these calculations, we can see that f(x,y) is maximized at (1,1), with a value of 22. Therefore, (x,y)=(1,1) is an optimal solution for this LP problem.

In conclusion, by plugging in the extreme points of the feasible set into the objective function and comparing the values, we have proved that (x,y)=(1,1) is an optimal solution for the given LP problem.

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Answer: 6 cm

solution let the length of AB= x cm the length of BC = (2x-2) cm, and the length of AC = (x+10) cm The perimeter of ABC=32 cm

x+2x-2+x+0=32

4x+8=32

4x=24

x=6

The required polynomial in standard form for the volume of the cylinder is [tex]$48\pi x^3 + 64\pi x^2 + 24\pi x + 4\pi$[/tex].

The formula for the volume of a cylinder is [tex]$V = \pi r^2 h$[/tex], where r is the radius and h is the height.

In this case, the radius is given as 4x + 1, and the height is given as 3x + 4. So we can substitute these values into the formula to get:

[tex]$$V = \pi(4x + 1)^2(3x + 4)$$[/tex]

Simplifying the expression inside the parentheses first, we have:

[tex]$$(4x + 1)^2 = (4x + 1)(4x + 1) = 16x^2 + 8x + 1$$[/tex]

Substituting this expression into the formula for V, we get:

[tex]$$V = \pi(16x^2 + 8x + 1)(3x + 4)$$[/tex]

Expanding the expression using the distributive property, we get:

[tex]$$V = \pi(48x^3 + 64x^2 + 24x + 4)$$[/tex]

Simplifying further, we get:

[tex]$$V = 48\pi x^3 + 64\pi x^2 + 24\pi x + 4\pi$$[/tex]

Therefore, the polynomial in standard form for the volume of the cylinder is [tex]$48\pi x^3 + 64\pi x^2 + 24\pi x + 4\pi$[/tex].

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Answer:

81.55

Step-by-step explanation:

because 4 twentydollar bills = 80 dollars 1 five-dollar bill = 5.00

80 + 5 = 85 dollars 2 quarts = 50 cents and 1 dime equals 10 cent 50 + 10 = 60 together you have 85.60 when you subtract 4.05 minus 85.60 you get 81.50 cent

The problem can be solved using the simplex method, and the solution can be verified using the graphical method. The optimal solution is X1 = 6, X2 = 0, X3 = 6, Z = 24.

The problem can be solved using the simplex method, and verified using the graphical method. Here are the steps:

Convert the problem to standard form by introducing slack variables:
Min Z = -2X1 - 4X2 - 3X3 + 0S1 + 0S2 + 0S3
St. X1 + 3X2 + 2X3 + S1 = 30
X1 + X2 + X3 + S2 = 24
3X1 + 5X2 + 3X3 + S3 = 60
Xi, Si >= 0

Set up the initial simplex tableau:
| 1 3 2 1 0 0 30 |
| 1 1 1 0 1 0 24 |
| 3 5 3 0 0 1 60 |
| 2 4 3 0 0 0 0 |

Identify the entering variable (most negative coefficient in the objective row): X2

Identify the leaving variable (smallest ratio of RHS to coefficient of entering variable): S1

Pivot around the intersection of the entering and leaving variables to create a new tableau:
| 0 2 1 1 -1 0 6 |
| 1 0 0 -1 2 0 18 |
| 0 0 0 5 -5 1 30 |
| 2 0 1 -2 4 0 36 |

Repeat steps 3-5 until there are no more negative coefficients in the objective row. The final tableau is:
| 0 0 0 7/5 -3/5 0 18 |
| 1 0 0 -1/5 2/5 0 6 |
| 0 0 1 1/5 -1/5 0 6 |
| 0 0 0 -2 4 0 24 |

The optimal solution is X1 = 6, X2 = 0, X3 = 6, Z = 24.

To verify the solution using the graphical method, plot the constraints on a graph and find the feasible region. The optimal solution will be at one of the corner points of the feasible region. By checking the values of the objective function at each corner point, we can verify that the optimal solution found using the simplex method is correct.

In conclusion, the problem can be solved using the simplex method, and the solution can be verified using the graphical method. The optimal solution is X1 = 6, X2 = 0, X3 = 6, Z = 24.

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The perimeter of the given triangle MNP is 65.

A regular figure with n-sides has n equal sides in it, and they are the only parts of it(that means, nothing more than those equal lengthened n sides).

Suppose that length of each side of that figure be of u units, then we have the perimeter as:

P=u+u+u+u+u+u......=n*u

units.

We are given that;

Side MN=5x-34, QR=25, QS=22, RS=x+4

Now,

5x + x - 34 + 4 = 22

6x - 30 = 22

6x - 30 + 30 = 22 + 30

6x = 52

6x/6 = 52/6

x = 8.67

P= 5*8-34+25+22+8+4

=40-34+47+12

=65

Therefore, the perimeter of the triangle will be 65.

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Answer:

(x, y)→(x,-y)

R(-4,0) S(-1,-3) T(2,-2)

Step-by-step explanation:

The opposite of a number is made by multiplying it by negative 1 (-1)

If it's reflected across the x axis, then the y axis will be the only one to change. (y to -y)

(x, y)→(x,-y)

Our formula for reflection across the x axis is (x,-y)

The rest is simple: Change each set of coordinates to an opposite y value.

The zeros of the polynomial f(x) = x4 - 8x3 + 18x2 + 16x - 40 are 4 + 2i, 4 - 2i, -2, and -8.

Given that 4 + 2i is one of its zeros, we can use the fact that the product of the zeros of a polynomial is equal to the product of the coefficients of the polynomial.

We can use this fact to find all of the zeros of the polynomial:

1. We can calculate the product of the coefficients of the polynomial:
( -40 ) * ( 16 ) * ( 18 ) * ( -8 ) = -442368

2. We can calculate the product of the known zero and its conjugate:
( 4 + 2i ) * ( 4 - 2i ) = 16

3. We can divide the product of the coefficients by the product of the known zero and its conjugate:
-442368 / 16 = -27735

4. This is the product of the other zeros:
-27735 = x^(2) + 8x + 1135

5. We can use the quadratic formula to solve for the remaining zeros:
x = (-8 +/- sqrt(64 - 4*1*1135))/2

x1 = (-8 + sqrt(144 - 4640))/2
x2 = (-8 - sqrt(144 - 4640))/2

Therefore, the remaining zeros of the polynomial f(x) are:
x1 = -5 + i7
x2 = -5 - i7
To find all the zeros of the polynomial f(x) = x4 - 8x3 + 18x2 + 16x - 40, we can use the fact that 4 + 2i is a zero and apply the conjugate root theorem. The conjugate root theorem states that if a polynomial has a complex root a + bi, then it also has a conjugate root a - bi. Therefore, 4 - 2i is also a zero of the polynomial.

Now, we can use synthetic division to divide the polynomial by (x - 4 - 2i) and (x - 4 + 2i) to find the other zeros. The result of the synthetic division will be a quadratic polynomial, which we can then solve using the quadratic formula.

Synthetic division with (x - 4 - 2i):

4 + 2i | 1 -8 18 16 -40
      | 0 4+2i -4+14i -44-8i 56+40i
      ----------------------------
      1 -4+2i 14+14i -28-8i 16+40i

Synthetic division with (x - 4 + 2i):

4 - 2i | 1 -4+2i 14+14i -28-8i 16+40i
      | 0 4-2i -4-14i 44+8i -56-40i
      ----------------------------
      1 0 10 16 0

The result of the synthetic division is the quadratic polynomial x2 + 10x + 16. We can solve this using the quadratic formula:

x = (-10 ± √(102 - 4(1)(16)))/(2(1))

x = (-10 ± √(100 - 64))/2

x = (-10 ± √36)/2

x = (-10 ± 6)/2

The two solutions are x = -2 and x = -8.

Therefore, the zeros of the polynomial f(x) = x4 - 8x3 + 18x2 + 16x - 40 are 4 + 2i, 4 - 2i, -2, and -8.

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The value of angle e is 55⁰.

The value of angle f is 55⁰.

The value of angle e is calculated by applying the following principles of angles on a straight line.

The vertical angle between e⁰ and 100⁰ = 25⁰ ( vertically opposite angles are equal)

The value of angle e is calculated as;

e⁰ + 25⁰ + 100⁰ = 180⁰ ( sum of angles on a straight line )

e⁰ = 180⁰ - 125⁰

e⁰ = 55⁰

The missing base angle of the triangle on the same line as e is calculated as;

? + 110⁰ = 180⁰ (sum of angles on a straight line )

? = 180 - 110

? = 70⁰

The value of angle f is calculated as;

f⁰ + e⁰ + ? = 180⁰ ( sum of angles in a triangle )

f⁰ + 55⁰ + 70⁰ = 180

f⁰ = 180 - 125⁰

f⁰ = 55⁰

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The given equation y = (1/2)x -2 represents a linear function because it has an independent and a dependent variable, each with an exponent of 1.

A linear function is defined as an equation in which the highest exponent of the variable is always one.

To determine the equation "y equals one-half times x minus 2" represents a linear or nonlinear function.

The algebraic form of this phrase "y equals one-half times x minus 2" is :

y = (1/2)x -2

Since it has an independent "x" and a dependent variable "y" with an exponent of one, the above equation y = (1/2)x -2 defines a linear function.

Therefore, the correct answer would be an option (C).

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Answer: 18

Step-by-step explanation

​

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 27x, the least common multiple of 27,x.

x × 6=27 × 4

Multiply 27 and 4 to get 108.

x × 6=108

Divide both sides by 6.

x= 108/6

​

Divide 108 by 6 to get 18.

x=18

The quotient is [tex]x^{3}[/tex]+2[tex]x^{2}[/tex]+7x+20 and the remainder is 62.

To find the quotient and remainder using synthetic division, we can follow these steps:

1. Write down the coefficients of the dividend, which are 1, -1, 1, -1, and 2.

2. Write down the value of x from the divisor, which is 3.

3. Bring down the first coefficient, 1, to the bottom row.

4. Multiply the value of x, 3, by the first coefficient in the bottom row, 1, and write the result, 3, in the second column of the top row.

5. Add the second coefficient in the dividend, -1, to the value in the second column of the top row, 3, and write the result, 2, in the second column of the bottom row.

6. Repeat steps 4 and 5 for the remaining columns.

7. The bottom row will contain the coefficients of the quotient, and the last value in the bottom row will be the remainder.
The synthetic division will look like this:
3|1-11-12|362160|1272062

Therefore, the quotient is  [tex]x^{3}[/tex]+2[tex]x^{2}[/tex]+7x+20 and the remainder is 62. The final answer is ([tex]x^{3}[/tex]+2[tex]x^{2}[/tex]+7x+20)+62 / (x-3).

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The solution for \(\tan \theta\) is \(\frac{v\sqrt{3}}{u\sqrt{1-\frac{1}{3} v^{2}}}\).

To solve for \(\tan \theta\), we need to simplify the equation by combining the fractions and simplifying the square roots.

First, let's combine the fractions on the right side of the equation:
\[ \tan \theta=\frac{v}{\sqrt{3}} \cdot \frac{3}{\sqrt{9-3 v^{2}}} \cdot \frac{\sqrt{3}}{u} \]

Next, we can simplify the square roots:
\[ \tan \theta=\frac{v}{\sqrt{3}} \cdot \frac{3}{\sqrt{9}\sqrt{1-\frac{1}{3} v^{2}}} \cdot \frac{\sqrt{3}}{u} \]

\[ \tan \theta=\frac{v}{\sqrt{3}} \cdot \frac{3}{3\sqrt{1-\frac{1}{3} v^{2}}} \cdot \frac{\sqrt{3}}{u} \]

Now we can simplify the fractions:
\[ \tan \theta=\frac{v}{\sqrt{3}} \cdot \frac{1}{\sqrt{1-\frac{1}{3} v^{2}}} \cdot \frac{\sqrt{3}}{u} \]

Finally, we can combine the terms to get the final expression for \(\tan \theta\):
\[ \tan \theta=\frac{v\sqrt{3}}{u\sqrt{1-\frac{1}{3} v^{2}}} \]

Therefore, the solution for \(\tan \theta\) is \(\frac{v\sqrt{3}}{u\sqrt{1-\frac{1}{3} v^{2}}}\).

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Ticket sales fell $960 below the goal.

What is system of equations?

A system of linear equations can be solved graphically, by substitution, by elimination, and by the use of matrices.

Since we know that the goal is to raise $3,300 each night and that the price of an adult ticket is $7.50 and the price of a student ticket is $4.50, we can write:

7.5d + 4.5s = 3300

We also know that the auditorium holds 600 people, so the total number of tickets sold must be:

d + s = 600

total number of tickets sold was 480. We can use this information to set up another system of equations:

s = 3d (since there were three times as many student tickets sold as adult tickets)

d + s = 480 (since the total number of tickets sold was 480)

Now we can solve the first system of equations to find the values of d and s that satisfy the constraints:

7.5d + 4.5s = 3300

d + s = 600

Multiplying the second equation by 4.5 and subtracting it from the first equation, we get:

3d = 1650

So, d = 550. Substituting this value back into the equation d + s = 600, we get:

550 + s = 600

s = 50

Therefore, 550 adult tickets and 50 student tickets must have been sold to raise exactly $3,300.

To answer the second part of the question, we can use the second system of equations to find the values of d and s for that performance:

s = 3d

d + s = 480

Substituting the first equation into the second equation, we get:

d + 3d = 480

So, 4d = 480 and d = 120. Substituting this value back into the first equation, we get:

s = 3d = 360

Therefore, 120 adult tickets and 360 student tickets were sold at that performance.

To calculate how much below the goal of $3,300 ticket sales fell, we can plug in the values for d and s from this performance into the equation:

7.5d + 4.5s = revenue

7.5(120) + 4.5(360) = $2,340

So, ticket sales fell $960 ($3,300 - $2,340) below the goal.

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The number of students who are expected to be wearing the spirit wear  out of 500 total students are 200 students.

Since it is given that there are 500 students studying in a school and 4 out of 10 students are expected to wear spirit wear. This means that the ratio of  students wearing spirit wear to students wearing normal clothes is 4:10. Now, if we consider the ratio for total number of students that is 500 and the expected number of students wearing spirit dress are equal to x, then following relation is obtained.

Number of students wearing spirit dress out of 10 = 4

Number of students wearing spirit dress out of 1 = 4/10

Number of students wearing spirit dress out of 500 = (4/10)*500

∴ Number of students wearing spirit dress out of 500 = 200 students

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The 14th term of the given geometric sequence is -40,960.

A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a constant factor called the common ratio. It is a type of exponential growth or decay.

The given sequence is a geometric sequence with the first term (a₁) as 5 and the common ratio (r) as -2. To find the 14th term, we can use the formula for the nth term of a geometric sequence, which is:

[tex]a_n = a_1 \times r^{(n-1)}[/tex]

Substituting the values of a₁ and r, we get:

[tex]a_n = 5\times -2^{(n-1)}[/tex]

To find the 14th term, we can substitute n = 14 and simplify:

[tex]a_{14} = 5 \times (-2)^{(14-1)}[/tex]

[tex]a_{14} = 5 \times (-2)^{(13)}[/tex]

[tex]a_{14} = 5 \times -8192[/tex]

a₁₄ = -40,960

Therefore, the 14th term of the given geometric sequence is -40,960.

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A poset on [54] = {1, 2, ... ,54} with comparisons a < b if and only if a divides b is a partially ordered set where elements are related if one divides the other. The height of this poset is 6 (1, 2, 4, 8, 16, 32, 54), and the width is 10 (all the elements in the set).

A poset on [54] is a partially ordered set that is defined with the comparison a < b if and only if a divides b. In this poset, the elements are ordered based on the divisibility relation, meaning that an element a is considered to be less than another element b if and only if a divides b.

The height of this poset is the maximum number of elements in a chain, which is 6. This can be seen by considering the chain {1, 2, 4, 8, 16, 32}.

The width of this poset is the maximum number of elements in an antichain, which is 10. This can be seen by considering the antichain {3, 5, 6, 7, 10, 14, 15, 21, 22, 35}.

Therefore, the height and width of this poset are:
Height = 6
Width = 10

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Answer:

19,690 since it is the nearest ten thousandth it would be 20,000

Step-by-step explanation:

subtract 78,920-59,230 to get 19,690 since we have to round up it would be 20,000

Answer: The difference ( rounded to 10,000) is 20,000

Step-by-step explanation: 78,920 and 59,230 both rounded to the nearest 10,000 is 80,000 and 60,000. The difference between the two is 20,000  

The vectors v1 = {1,2,0}, v2 = {1,0,1}, and v3 = {1,a,4} are linearly independent if a = 8/3, t and  linearly dependent for a = 2, and linearly independent for a = 8/3.

For the vectors to be linearly independent, we need to check if the following system of equations has a unique solution:

c1v1 + c2v2 + c3v3 = 0

where c1, c2, c3 are constants, and 0 is the zero vector.

Substituting the given vectors, we get the following system of equations:

c1 + c2 + c3 = 0 (1)

2c1 + ac3 = 0 (2)

c2 + 4c3 = 0 (3)

If we can find values of a for which this system of equations has a non-trivial solution, then the vectors are linearly dependent. Otherwise, they are linearly independent.

To find such values of a, we need to solve the system of equations and find the conditions under which it has non-trivial solutions.

From equations (2) and (3), we get:

c2 = -4c3 (4)

2c1 + ac3 = 0 (5)

Substituting equations (1) and (4) into equation (5), we get:

2(-c2 - c3) + ac3 = 0

Simplifying, we get:

(-2 + a)c3 - 2c2 = 0

Substituting equation (4), we get:

(-2 + a)c3 + 8c3 = 0

Solving for c3, we get:

c3 = 0 if a = 2

c3 = 0 if a = 8/3

For a = 2, the system reduces to:

c1 + c2 = 0

2c1 = 0

c2 + 4c3 = 0

This system has a non-trivial solution: c1 = 0, c2 = 1, c3 = -1/4.

Therefore, the vectors are linearly dependent for a = 2.

For a = 8/3, the system reduces to:

c1 + c2 + c3 = 0

(8/3)c3 = 0

c2 + 4c3 = 0

This system has only the trivial solution: c1 = c2 = c3 = 0.

Therefore, the vectors are linearly independent for a = 8/3.

In summary, the given vectors are linearly dependent for a = 2, and linearly independent for a = 8/3.

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First, we need to convert the maximum weight that the mail carrier is allowed to carry from pounds to ounces:

30 pounds = 30 x 16 ounces = 480v ounces

Then, we can divide the maximum weight by the weight of one letter:

480 ounces / 12 ounces per letter = 40 letters

Therefore, the mail carrier is able to carry 40 letters.

So, the correct answer is option C: 40 letters.

Answer:

C 40

Step-by-step explanation:

Answer:

There are a total of 15 customers, so the probability of selecting any one customer at random is 1/15.

If the first customer selected is a Pike resident, there are 6 Pike residents remaining out of a total of 14 customers remaining. So the probability of the second customer being a Pike resident, given that the first customer was a Pike resident, is 6/14.

Therefore, the probability of both customers selected being Pike residents is:

(6/15) * (6/14) = 0.1714 or approximately 0.17 (rounded to two decimal places).

So the probability that both customers selected are Pike residents is approximately 0.17.

The numbers that should be considered possible outliers from the above figure would be = 31, 87, and 90. That is option B.

An outlier is defined as the term given to an observation which lies in an abnormal distance from other values in a random sample from a population.

On the field of statistics, an outlier is also called an extreme value.

For example in the scores 25,29,2,32,86,33,27,28 both 2 and 86 are "outliers".

From the illustration given above, the values that are at the extreme that didn't enter the box plot are the outliers and they include 31,87, and 90.

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Answer:

Hey there! [tex]x + y = 500 \ 215x + 615y = 187500[/tex]Answer is the first option. Hope this helps.

Step-by-step explanation:

just did it

Answer:

Below

Step-by-step explanation:

Slope, m, is equal to 'rise'  (change in y) divided by 'run' ( change in 'x')

when going L to R

change in x is from -2 to 2  is a change of   +4

change in y is 24 to 8     a change of  - 16

slope = -16/4 = - 4

Answer:

$6236.56

Step-by-step explanation:

Compound Formula

[tex]A = P (1+\frac{r}{n} )^n^t\\[/tex]

I plugged in the numbers of P, r, and t.

n means the number of times the interest is applied per period. It is compounded monthly, and the time is given 3 years.

This means 12 months x 3 years = 36 months.

[tex]A = $5700(1+.03/36)^3^6^(^3^)[/tex]

[tex]A=6236.559672[/tex]

[tex]A=6236.56[/tex]

The sum of the circumferences of the two circles is equal to [tex]$2\pi \sqrt 2.$[/tex]

Circumference is the distance around a two-dimensional shape, such as a circle or ellipse. It can be calculated by multiplying the circumference of the shape by its diameter. The formula for calculating the circumference of a circle is 2πr, where π is the constant 3.14 and r is the radius of the circle. The circumference of an ellipse is more complicated and requires knowledge of the length of its major and minor axes.

The two circles are externally tangent to each other, which means that the distance between them is equal to the sum of their radii. Since the circles are tangent to the sides of the square, the length of one side of the square is equal to the sum of their radii. Since the perimeter of the square is given to be [tex]$2+\sqrt 2,[/tex] we can calculate the length of each side of the square to be [tex]$\sqrt 2.$[/tex] Hence, the sum of the radii of the two circles is equal to [tex]$\sqrt 2.$[/tex]
Therefore, the sum of the circumferences of the two circles is equal to [tex]$2\pi \sqrt 2.$[/tex]

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Answer:

Step-by-step explanation:

You want the objective function, its maximum value, and the variable values that give that maximum based on the model shown in the graph.

The problem statement tells you the profit function is ...

  1.00x +1.50y . . . . . . objective function

Since the objective is to maximize profit, this is the objective function.

The integer values nearest the vertex of the feasible region farthest from the origin are (x, y) = (69, 70). These are the numbers of 'economy' and 'best' brushes that maximize the profit.

The maximum profit for these numbers of brushes will be ...

  p = x +1.5y = 69 +1.5(70) = 69 +105

  p = 174 . . . . maximum profit

The maximum profit of the situation is $174

From the question, we have the following parameters that can be used in our computation:

Profit function = $1 for x and $1.50 for y

This means that the objective function is

P(x, y) = x +1.5y

Also, the graph is given where we have:

Optimal point, (x, y) = (69, 70)

Substitute these points in the profit function

So, we have

P(x, y) = 69 +1.5 * 70

Evaluate

P(x, y) = 174

Hence, the maximum profit is $174

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Answer:

A

Step-by-step explanation:

c^2 = a^2 + b^2 - 2ab cos(C)

c^2 = 4^2 + 6^2 - 2(4)(6)(-1/4)

c^2 = 16 + 36 + 12

c^2 = 64

c = √64

c = 8

The binomial theorem holds for any integer \( n \geq 1 \) and any real numbers \( x \) and \( y \).

The binomial theorem states that for any positive integer \( n \) and any real numbers \( x \) and \( y \), \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k\]where \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.

To prove the identity \[ \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k = x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + y^n \] we can use differentiation, integration, and multiplication by \( x \) or \( y \).

First, let's differentiate both sides of the equation with respect to \( x \): \[\frac{d}{dx} \left( \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k \right) = \frac{d}{dx} \left( x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + y^n \right)\]Using the power rule for differentiation, we get \[\sum_{k=0}^{n} \binom{n}{k}(n-k)x^{n-k-1}y^k = nx^{n-1} + n(n-1)x^{n-2}y + \frac{n(n-1)(n-2)}{2}x^{n-3}y^2 + \cdots\]Next, we can multiply both sides of the equation by \( x \): \[\sum_{k=0}^{n} \binom{n}{k}(n-k)x^{n-k}y^k = nx^{n} + n(n-1)x^{n-1}y + \frac{n(n-1)(n-2)}{2}x^{n-2}y^2 + \cdots\]Finally, we can integrate both sides of the equation with respect to \( x \): \[\int \left( \sum_{k=0}^{n} \binom{n}{k}(n-k)x^{n-k}y^k \right) dx = \int \left( nx^{n} + n(n-1)x^{n-1}y + \frac{n(n-1)(n-2)}{2}x^{n-2}y^2 + \cdots \right) dx\]Using the power rule for integration, we get \[\sum_{k=0}^{n} \binom{n}{k}\frac{(n-k)}{n-k+1}x^{n-k+1}y^k = \frac{n}{n+1}x^{n+1} + \frac{n(n-1)}{n+1}x^{n}y + \frac{n(n-1)(n-2)}{2(n+1)}x^{n-1}y^2 + \cdots\]Simplifying the coefficients and combining like terms, we get \[\sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k = x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + y^n\]which is the identity we were trying to prove. Therefore, the binomial theorem holds for any integer \( n \geq 1 \) and any real numbers \( x \) and \( y \).

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The dimension of a vector space is equal to the number of linearly independent vectors in its basis.

In the given basis B = {(0,2,1),(0,2,1),(1,0,0)}, we can see that the first two vectors (0,2,1) and (0,2,1) are identical and therefore not linearly independent. Therefore, the dimension of the vector space is equal to the number of linearly independent vectors in the basis, which is 2.

So, the dimension of the vector space whose basis are B = {(0,2,1),(0,2,1),(1,0,0)} is equal to 2.

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