Suppose the marginal profit from the sale of x hundred items is:
P'(x)=4-6x=3x^2
and the profit of 0 items is -$40
A. Find the profit function.
B. Find the profit from selling 800 items
I assume the P' is 4-6x+3x^2
Then P= 4x - 3x^2 + x^3 + C
where C is the constant of integration. You have P when x=0, so C= -40
Put in x=800 and you have it.
the profit is $510,083.16
To find the profit function, we need to integrate the marginal profit function, P'(x), with respect to x.
A. To find the profit function, integrate P'(x):
∫(4 - 6x + 3x^2) dx = 4x - 3/2x^2 + x^3/3 + C
Since the profit of 0 items is -$40, we can substitute x=0 and P= -40 into the profit function equation to solve for the constant C:
-40 = 0 + 0 + 0 + C
C = -40
Now we can substitute the value of C back into the profit function equation to get the final profit function:
P(x) = 4x - 3/2x^2 + x^3/3 - 40
B. To find the profit from selling 800 items, substitute x=800 into the profit function:
P(800) = 4(800) - 3/2(800)^2 + (800)^3/3 - 40
Calculate the expression to get the answer.