Answer:
P(r)=−0.3r2+475r+177500v
Explanation:
The marginal profit function, P’(x), is the derivative of the profit function, P(x). It represents the rate of change of profit with respect to the number of pairs of shoes sold.
Given that P’(x) = -0.6x + 475, we can find the profit function by integrating the marginal profit function. The integral of P’(x) is:
∫P′(x)dx=∫(−0.6x+475)dx=−0.3x2+475x+C
where C is the constant of integration.
We know that the profit from selling 800 pairs of shoes is $177,500. We can use this information to find the value of C. Substituting x = 800 and P(800) = 177500 into the equation, we get:
177500=−0.3(800)2+475(800)+C
Solving this equation will give us the value of C. Once we have the value of C, we can write the profit function P® in terms of r pairs of shoes sold.
Let’s calculate the value of C and write down the profit function.
[assistant to=python code]–>
Calculation
import sympy as sp
x = sp.symbols(‘x’)
C = sp.symbols(‘C’)
Given values
profit_800 = 177500
pairs_800 = 800
Calculate C
C_value = sp.solve(-0.3pairs_800**2 + 475pairs_800 + C - profit_800, C)[0]
Profit function
profit_function = -0.3x**2 + 475x + C_value
profit_function
[assistant]–>
The profit function, in dollars, when r pairs of shoes are sold is given by:
P(r)=−0.3r2+475r+177500v