Question

A company that manufactures racing boats estimates that the profit for selling its brand is given by the following equation.

Answer 1

Answer:

$b = 10$

Step-by-step explanation:

$P(b) = -25b^3 + 750b^2 + 2500b$

This equation shows the income obtained based on the number of boats sold. We want to know the minimum number of boats that must be sold to obtain $ 75 000.

Then we must equalize the equation to 75 000 and clear b.

3 solutions will be obtained (because it is a polynomial of degree 3), and the lowest value will be taken.

$P(b) = 75000$

$-25b^3 + 750b^2 + 2500b = 75000$

Now we need to solve the equation, for that we seek to factor the polynomial

$-25b^3 + 750b^2 + 2500b - 75000 = 0$

$b^3 -30b^2 - 100b + 3000 = 0$ we divide the equation by -25

$b^2(b-30) - 100(b-30) = 0$ We take common factor b

$(b^2 -100)(b-30) = 0$ We take out common factor (b-30)

Finally the solutions are:

$b = 30\\\\b^2 -100 = 0\\\\b^2 = 100$

b = 10 and b = -10

We take the least positive solution (because you can not produce -10 boats)

Finally b = 10

The smallest number of boats that must be produced to make a profit of $ 75,000 is 10 boats.