Question

Ask 1: Electronics Manufacturer

An electronics manufacturer recently created a new version of a popular device. It also created this function to represent the profit, P(x), in tens of thousands of dollars, that the company will earn based on manufacturing x thousand devices: P(x) = -0.16x^2 + 21.6x – 400.
a. The profit function for the first version of the device was very similar to the profit function for the new version. As a matter of fact, the profit function for the first version is a transformation of the profit function for the new version. For the value x = 40, the original profit function is half the size of the new profit function. Write two function transformations in terms of P(x) that could represent the original profit function.
b. Write the two possible functions from part a in simplified form.
c. Which of the two possible functions would represent a better situation for the company at the time the original version was released? Explain.
d. After a natural disaster, the company decided to give 2,000 of the new devices to organizations to use in auctions or raffles to raise money to help those affected. In terms of function transformations and P(x), write a function that would represent the new profit function based on this.
e. Based on this new function, approximately how many devices would the company have to manufacture to avoid losing money?
f. The average rate of change of a function over a given interval can be determined by finding the slope of the line that connects the two points on the function representing the endpoints of the interval. Find the average rate of change of the function from part d over the interval [60, 80].

Answer 1

a. Function Transformations: The possible transformations of the new profit function are given. b. Simplified Form: The simplified forms of the two possible functions are provided. c. Better Situation: The better situation for the company at the time the original version was released is explained.

Step-by-step explanation:

a. Function Transformations:

Since the original profit function is half the size of the new profit function for x = 40, we can write two possible transformations of the new profit function:

1. P(x) = 0.5(-0.16x^2 + 21.6x – 400)
2. P(x) = -0.16(0.5x)^2 + 21.6(0.5x) – 400

b. Simplified Form:

The simplified forms of the two possible functions are:

1. P(x) = -0.08x^2 + 10.8x – 200
2. P(x) = -0.04x^2 + 10.8x – 400

c. Better Situation:

The second possible function represents a better situation for the company at the time the original version was released because it has a smaller negative coefficient for x^2, indicating a slower decline in profit with increasing production.

d. New Profit Function:

If the company gives away 2,000 devices, the new profit function can be represented as:

P(x) = -0.16x^2 + 21.6x – 400 - 2,000

e. Avoid Losing Money:

To avoid losing money, the company would need to manufacture enough devices for the profit function to be greater than or equal to zero. This can be calculated by solving the inequality:

-0.16x^2 + 21.6x – 400 - 2,000 ≥ 0

f. Average Rate of Change:

The average rate of change of the profit function over the interval [60, 80] can be found by calculating the slope of the line connecting the two points P(60) and P(80). This can be calculated using the formula:

Average rate of change = (P(80) - P(60)) / (80 - 60)