Questions: The profit for a company that produces discount sneakers can be modeled by the quadratic function P(x)=4x^2-24x+220, where x is the number of pairs of sneakers sold. (a) For how many pair(s) of sneakers did the profit reach its lowest point? The low point for the profit is when [] pair(s) of sneakers were sold.

The profit for a company that produces discount sneakers can be modeled by the quadratic function P(x)=4x^2-24x+220, where x is the number of pairs of sneakers sold.

(a) For how many pair(s) of sneakers did the profit reach its lowest point?

The low point for the profit is when [] pair(s) of sneakers were sold.

Transcript text: The profit for a company that produces discount sneakers can be modeled by the quadratic function $P(x)=4x^2-24x+220$, where x is the number of pairs of sneakers sold. (a) For how many pair(s) of sneakers did the profit reach its lowest point? The low point for the profit is when [] pair(s) of sneakers were sold.

Solution

Solution Steps

To find the number of pairs of sneakers at which the profit reaches its lowest point, we need to determine the vertex of the quadratic function. The vertex of a parabola given by $ P(x) = ax^2 + bx + c $ is at $ x = -\frac{b}{2a} $. This will give us the number of pairs of sneakers that result in the minimum profit.

Step 1: Identify the Quadratic Function

The profit function is given by $ P(x) = 4x^2 - 24x + 220 $, where $ x $ represents the number of pairs of sneakers sold.

Step 2: Determine the Vertex

To find the number of pairs of sneakers at which the profit reaches its lowest point, we calculate the vertex of the quadratic function using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Substituting the values $ a = 4 $ and $ b = -24 $: \[ x = -\frac{-24}{2 \cdot 4} = \frac{24}{8} = 3 \]