All categories

2 years ago

These are the cost and revenue functions for a line of trumpets sold at a music store:

Mathematics

1 answer:

Maslowich2 years ago

4 0

Answer:

168 trumpets for $1702

Step-by-step explanation:

Profit is the measure to be maximized. We are given revenue and cost relationships as a function of units, x (trumpets). Profit is the difference:

Profit = Revenue[R(x)] - Cost[C(x)]

Profit = (76x – 0.25x^2) - (-7.75x + 5,312.5)

Profit = 76x - 0.25x^2 + 7.75x - 5,312.5

Profit = - 0.25x^2 + 83.75x - 5312.5

At this point we can find the trumpets needed for maximum profit by either of two approaches: algebraic and graphing. I'll do both.

Mathematically

The first derivative will give us the slope of this function for any value of x. The maximum will have a slope of zero (the curve changes direction at that point). Take the first derivative and set that equal to 0 and solve for x.

First derivative:

d(Profit)/dx = - 2(0.25x) + 83.75

d(Profit)/dx = - 0.50x + 83.75

0 = - 0.50x + 83.75

0.50x = 83.75

x = 167.5 trumpets

Graphically

Plot the profit function and look for the maximum. The graph is attached. The maximum is 167.5 trumpets.

Round up or down to get a whole trumpet. I'll go up: 168 trumpets.

Maximum Profit

Solve the profit equation for 168 trumpets:

Profit = - 0.25x^2 + 83.75x - 5312.5

Profit = - 0.25(168)^2 + 83.75(168) - 5312.5

Profit = $1702

You might be interested in

Suppose that θ is an acute angle of a right triangle and that sec(θ)=52. Find cos(θ) and csc(θ).

insens350 [35]

Answer:

$\cos{\theta} = \dfrac{1}{52}$

$\csc{\theta} = \dfrac{52}{\sqrt{2703}}$

Step-by-step explanation:

To solve this question we're going to use trigonometric identities and good ol' Pythagoras theorem.

a) Firstly, sec(θ)=52. we're gonna convert this to cos(θ) using:

$\sec{\theta} = \dfrac{1}{\cos{\theta}}$

we can substitute the value of sec(θ) in this equation:

$52 = \dfrac{1}{\cos{\theta}}$

and solve for for cos(θ)

$\cos{\theta} = \dfrac{1}{52}$

side note: just to confirm we can find the value of θ and verify that is indeed an acute angle by $\theta = \arccos{\left(\dfrac{1}{52}\right)} = 88.8^\circ$

b) since right triangle is mentioned in the question. We can use:

$\cos{\theta} = \dfrac{\text{adj}}{\text{hyp}}$

we know the value of cos(θ)=1\52. and by comparing the two. we can say that:

length of the adjacent side = 1
length of the hypotenuse = 52

we can find the third side using the Pythagoras theorem.

$(\text{hyp})^2=(\text{adj})^2+(\text{opp})^2$

$(52)^2=(1)^2+(\text{opp})^2$

$\text{opp}=\sqrt{(52)^2-1}$

$\text{opp}=\sqrt{2703}$

length of the opposite side = √(2703) ≈ 51.9904

we can find the sin(θ) using this side:

$\sin{\theta} = \dfrac{\text{opp}}{\text{hyp}}$

$\sin{\theta} = \dfrac{\sqrt{2703}}{52}}$

and since $\csc{\theta} = \dfrac{1}{\sin{\theta}}$

$\csc{\theta} = \dfrac{52}{\sqrt{2703}}$

4 0

3 years ago

Find the annual interest rate. I=$16, P=$200, t= 2 years

rjkz [21]

Answer:

The annual interest rate is 4%

7 0

2 years ago

What is the perimeter of a rectangle with length x+4 and a width of 5x-7

AlladinOne [14]

Answer:

12x-6

Step-by-step explanation:

perimeter= 2l+2w

=2(x+4)+2(5x-7)

=2x+8+10x-14

=12x-6

8 0

2 years ago

Which of the following best describes the volume of a cylinder?

Ahat [919]

A. the area of the circular base multiplied by the height of the cylinder B. the circumference of the circular base multiplied by the height of the cylinder C. the sum of the areas of the two circular bases multiplied by the height of the cylinder D.

4 0

3 years ago

Simplify (2/9)^3 A. 8/729 B. 6/9 C. 2/729 D. 8/9 No guessing please

Lunna [17]

Eliminate the exponent
2^3=8
9^3=729

The simplified answer would be 8/729.

Hope this helps!

7 0

2 years ago