Question

The function P(x) = –0.015x^2 + 1.2x – 11.5 gives the profit, in thousands of dollars, when a company sells a new product at x dollars per unit. Which of the following statements best interprets the average rate of change of this function from x = 50 to x = 55?
The profit increases by $375 for every $1 increase in the selling price.
The profit decreases by $375 for every $1 increase in the selling price.
The profit increases by $1875 for every $1 increase in the selling price.
The profit decreases by $1875 for every $1 increase in the selling price.

Answer 1

Answer:

The profit decreases by $ 375 for every $ 1 increase in the selling price.

Step-by-step explanation:

From the definition of the secant line we get that the average rate of change of $P(x) = -0.015\cdot x^{2}+1.2\cdot x -11.5$, where $x$ is the selling price of the product, measured in dollars per unit, is:

$r = \frac{P(55)-P(50)}{55-50}$ (1)

Now we evaluate the function at each bound:

x = 50

$P(50) = -0.015\cdot (50)^{2}+1.2\cdot (50)-11.5$

$P(50) = 11$

x = 55

$P(55) = -0.015\cdot (55)^{2}+1.2\cdot (55)-11.5$

$P(55) = 9.125$

Then, the average rate of change is:

$r = \frac{9.125-11}{55-50}$

$r = -0.375$

Hence, the profit decreases by $ 375 for every $ 1 increase in the selling price.

Answer 2

The statements that best interprets the average rate of change from x = 50 to x = 55 is (b) The profit decreases by $375 for every $1 increase in the selling price.

The profit function is given as:

$P(x) = -0.015x^2 + 1.2x - 11.5$

Calculate P(x), when x = 50.

So, we have:

$P(50) = -0.015(50)^2 + 1.2(50) - 11.5$

$P(50) = 11$

Calculate P(x), when x = 55.

So, we have:

$P(55) = -0.015(55)^2 + 1.2(55) - 11.5$

$P(55) = 9.125$

The average rate of change from x = 50, to 55 is then calculated using:

$m = \frac{P(55) - P(50)}{55-50}$

So, we have:

$m = \frac{9.125 - 11}{55-50}$

$m = \frac{-1.875}{5}$

Divide

$m = -0.375$

The function is in 1000 units.

So, we have:

$m = -0.375\times 1000$

$m = -375$

-375 implies a decrease of $375 for every $1 increase in sales

Hence, the correct statement is (b)

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