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:
Revenue function: [tex]R(x)=100x[/tex]
Profit function: [tex]P(x)=-0.1 x^2 + 92 x - 2750[/tex]
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.
[tex]C(x)=2750 + 8x + 0.1x^2[/tex]
Gymnast Clothing sells the cleats at $100 per pair.
The Revenue (in dollars) for a run of x pairs of cleats.
[tex]R(x)=100x[/tex]
Profit = Revenue - Cost
[tex]P(x)=R(x)-C(x)[/tex]
[tex]P(x)=100x-(2750 + 8x + 0.1x^2)[/tex]
[tex]P(x)=-0.1 x^2 + 92 x - 2750[/tex]
We need to find how many should Gymnast Clothing manufacture to make a profit.
[tex]P(x)\geq 0[/tex]
[tex]-0.1 x^2 + 92 x - 2750\geq 0[/tex]
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.
