Question

Total profit from marginal profit. Stevens Bakery has found that its marginal profit, in dollars per wedding cake, is

Answer 1

Answer:

2150 Profit at 50 cakes quantity

Step-by-step explanation:

Marginal Profit [C'(x)]  is Addition to total profit due to increase in quantity 'x'. So it is derivation of 'total profit function' with respect to 'quantity x'= ∂TP/∂x.  Hence, Total Profit is integration of Marginal Profit with respect to x.

Marginal Profit : C'(x) = -0.12x + 40

Total Profit : C (x) = ∫-0.12x + 40  

= -0.12x^2 / 2 + 40x

Total Profit Function = 0.06x^2 + 40x

Total Profit at given quantity = 50 cakes :

0.06 (50)^2 + 40 (50)

0.06 (2500) + 2000

150 + 2000

= 2150 [ Profit at 50 cakes quantity ]

Answer 2

Answer:

The total profit from producing 50 wedding cakes is 2,150 dollars

Step-by-step explanation:

Stevens Bakery has found that its marginal profit, in dollars per wedding cake, is $C'(x) = - 0.12 x + 40$ , where x is the number of wedding cakes produced

From the given information:

C'(x) = - 0.12 x + 40

Integrate both sides we get,

$\int C'(x) dx = \int (-0.12x+40)dx$             $\int x^n dx = \frac{x^{n+1}}{n+1}+C$

$C(x) = -0.12 \times \frac{x^{2}}{2}+40x$

$C(x) = -0.06x^2+40x$                                  ...... (1)

Substitute the value of x=50 wedding cakes in equation (1) as shown below:

$C(50) = -0.06 (50)^2 +40(50) = 150+2000$

          =2,150 dollars

Hence, the total profit from producing 50 wedding cakes is 2,150 dollars.