Question

Suppose a manufacturer's profit (in thousands of Ksh) from the sale of pool tables is given by the function P(x)=400(15−x)(x−2), where x is the price at which the pool tables are sold. Find the price that maximizes profit.

a) $6,000
b) $7,500
c) $10,000
d) $12,000

Answer 1

The price that maximizes profit, found by determining the vertex of the profit function vertex, is $8,500, which is not listed in the given options.

Step-by-step explanation:

To find the price that maximizes profit for the manufacturer's sale of pool tables, we need to analyze the given profit function P(x) = 400(15 - x)(x - 2). This is a quadratic equation where the profit is represented as a function of the price x. To maximize profit, we need to determine the vertex of the parabola that this equation represents since it's a concave down parabola (the coefficient of x^2 is negative).

The vertex of a parabola in the form ax^2 + bx + c can be found using the formula -b/(2a). In this case, expanding our profit function gives us -400x^2 + 6800x - 8000. The coefficient a is -400, and the coefficient b is 6800. Applying the vertex formula, we get x = -6800/(2(-400)), which simplifies to x = 8.5.

However, the prices are given in thousands, so the price that maximizes profit is $8,500. This price is not listed in the options provided, which could mean there is a mistake in the given alternatives or an error in interpreting the function. Therefore, none of the options provided is the correct answer based on the profit function given.