Questions: Question 12 The student photo club at the college is planning on selling prints that it makes to raise money. The profit P, in dollars, from selling x prints is given by the function: P(x) = 213x - 2x^2 a) Find the number of prints, to the nearest whole print, that need to be sold to maximize the profit. You must sell prints to maximize the profit. b) The maximum profit is - (No dollar signs or comma's.)

Question 12
The student photo club at the college is planning on selling prints that it makes to raise money.
The profit P, in dollars, from selling x prints is given by the function:
P(x) = 213x - 2x^2
a) Find the number of prints, to the nearest whole print, that need to be sold to maximize the profit.

You must sell prints to maximize the profit.
b) The maximum profit is - (No dollar signs or comma's.)

Transcript text: Question 12 The student photo club at the college is planning on selling prints that it makes to raise money. The profit $P$, in dollars, from selting $x$ prints is given by the function: \[ P(x)=213 x-2 x^{2} \] a) Find the number of prints, to the nearest whole print, that need to be sold to maximize the profit. You must sell $\square$ prints to maximize the profit. b) The maximum profit is $\$$ $\square$ - (No dollar signs or comma's.)

Solution

Solution Steps

To solve this problem, we need to find the number of prints that maximizes the profit function $ P(x) = 213x - 2x^2 $. This is a quadratic function, and the maximum profit occurs at the vertex of the parabola. The x-coordinate of the vertex for a quadratic function $ ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $. We will use this formula to find the number of prints that maximizes the profit. Then, we will calculate the maximum profit by substituting this x-value back into the profit function.

Step 1: Identify the Profit Function

The profit function is given by: \[ P(x) = 213x - 2x^2 \]

Step 2: Determine the Vertex of the Parabola

The profit function is a quadratic equation of the form $ ax^2 + bx + c $, where $ a = -2 $ and $ b = 213 $. The x-coordinate of the vertex, which gives the number of prints to maximize profit, is calculated using: \[ x = -\frac{b}{2a} = -\frac{213}{2 \times (-2)} = \frac{213}{4} = 53.25 \]

Step 3: Round to the Nearest Whole Print

Since the number of prints must be a whole number, we round $ x = 53.25 $ to the nearest whole number: \[ x = 53 \]

Step 4: Calculate the Maximum Profit

Substitute $ x = 53 $ back into the profit function to find the maximum profit: \[ P(53) = 213 \times 53 - 2 \times 53^2 = 11289 - 5618 = 5671 \]