Question

Gymnast Clothing manufactures expensive soccer cleats for sale to college bookstores in runs of up to 500. Its cost (in dollars) for a run of x pairs of cleats is C(x) = 2750 + 8x + 0.1x2 (0 ≤ x ≤ 500). Gymnast Clothing sells the cleats at $100 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?

Answer 1

A. Profit (P):

P = 100 x

 

Revenue (R) is Profit – Cost:

R = P – C

R = 100 x – [2750 + 8x + 0.1x^2]

R = 100x – 2750 – 8x – 0.1x^2

R = -0.1x^2 + 92x – 2750

B. 

To make a profit, R must be zero, R = 0:

0 = -0.1x^2 + 92x – 2750

x^2 – 920x = - 27,500

Completing the square:

x^2 – 920x + 211,600 =  - 27,500 + 211,600

(x – 460)^2 = 184,100

x – 460 = ± 429.07

x = 30.93, - 889.07

 

Therefore sell at least 31.

31<x<500

 

 

Answer 2

Answer:

Revenue function: $R(x)=100x$

Profit function: $P(x)=-0.1 x^2 + 92 x - 2750$

31 to 500 Gymnast Clothing are manufactured to make a profit.

Step-by-step explanation:

The cost (in dollars) for a run of x pairs of cleats.

$C(x)=2750 + 8x + 0.1x^2$

Gymnast Clothing sells the cleats at $100 per pair.

The Revenue (in dollars) for a run of x pairs of cleats.

$R(x)=100x$

Profit = Revenue - Cost

$P(x)=R(x)-C(x)$

$P(x)=100x-(2750 + 8x + 0.1x^2)$

$P(x)=-0.1 x^2 + 92 x - 2750$

We need to find how many should Gymnast Clothing manufacture to make a profit.

$P(x)\geq 0$

$-0.1 x^2 + 92 x - 2750\geq 0$

From the given figure it is clear that P(x) is greater than or equal to 0 for 30.931 ≤ x ≤ 889.069.

The value of x can not be more than 500.

31 ≤ x ≤ 500

31 to 500 Gymnast Clothing are manufactured to make a profit.