Be sure to help with all the problems Task 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) = -0.16x2 + 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.

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b. Write the two possible functions from part a in simplified form.

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c. Which of the two possible functions would represent a better situation for the company at the time the original version was released? Explain.

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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.

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e. Based on this new function, approximately how many devices would the company have to manufacture to avoid losing money?

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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].

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