Question

A company has determined that its profit for a product can be described by a linear function. The profit from the production and sale of 210 units is $490. The profit from 310 units is $1040.

a) Find the rate of change of profit for this product when between 210 and 310 units are sold.
b) Write the equation of the profit function for this product.
c) Use the profit function equation to predict the profit from the sale of 350 units.

Answer 1

The rate of change of profit between 210 and 310 units is $5.50 per unit. The equation for the profit function is P(x) = $5.50x - $665. Using this function, the predicted profit for 350 units sold is $1260.

Step-by-step explanation:

To find the rate of change of profit, we can use the profit values at 210 units and 310 units and apply the formula for the slope of a line since the profit function is linear. The formula is (change in profit)/(change in units sold) = (profit2 - profit1)/(units2 - units1).

a) Using the given values, the rate of change is ($1040 - $490)/(310 - 210) = $550/100 = $5.50 per unit.

b) The profit function can be represented in the form of P(x) = mx + b, where m is the rate of change we just calculated and b is the y-intercept. To find b, we can use one of the given points, for example, (210, $490), and solve: $490 = $5.50(210) + b. This gives us b = $490 - $1155 = -$665.

So, the profit function equation is P(x) = $5.50x - $665.

c) To predict the profit from the sale of 350 units, we substitute x with 350 into the profit function: P(350) = $5.50(350) - $665 = $1925 - $665 = $1260.

Therefore, the predicted profit from the sale of 350 units is $1260.