Total Profit Calculation in a Toy Store

How do we calculate the total profit from selling puzzles in a toy store?

Given the marginal profit equation and the profit when no puzzles are sold, how can we determine the total profit and the profit from selling a specific quantity of puzzles?

Answers:

a. The total profit equation is P(x) = 36x³ + 22.5x² - 40x.

b. The total profit from selling 400 puzzles is $2504.

To find the total profit equation, we need to integrate the marginal profit equation P'(x) with respect to x. By integrating x(144x² + 45x), we obtain 36x^3 + 22.5x^2 as the antiderivative. However, this only gives us the profit earned from selling x hundred puzzles. Since the profit is -$40 when no puzzles are sold, we need to subtract $40 from the equation.

Therefore, the total profit equation is P(x) = 36x^3 + 22.5x^2 - 40x.

To find the total profit from selling 400 puzzles, we substitute x = 400 into the total profit equation:

P(4) = 36(4)³ + 22.5(4)² - 40(4)

     = 36(64) + 22.5(16) - 160

     = 2,304 + 360 - 160

     = 2,664 - 160

     = $2504.

Thus, the total profit from selling 400 puzzles is $2504.

Integrating a function to find the antiderivative, which represents the total value of the function. Integrating allows us to find the accumulated value over a given range of the independent variable, in this case, the quantity of puzzles sold. By evaluating the antiderivative at specific values, such as substituting x = 4, we can calculate the total profit at that particular quantity.