Profit function

MAT 117 – SPRING 2021 PROBLEM SET #3 NAME: ________________________ (1 point) Directions: Complete the problems on this sheet. All problems must be completed by the end of class to earn full credit. Show all work and include appropriate units where applicable. Box all final answers. 1. Miss Kito is a huge fan of the Jurassic Park movies. When the first movie was released, she decided to try to capitalize off the success of the movie by selling novelty mosquitos trapped in amber. She called her entrepreneurial endeavor Miss Kito’s Mosquitos. A market study showed that the demand for her item was based on the equation: 𝒒 = πŸπŸπŸ’ βˆ’ πŸπŸ’π’‘ where π‘ž represents the number of mosquitos sold each week based on a price, 𝑝, in dollars. The cost to produce each mosquito is $3. (So, π‘ͺ = πŸ‘π’’) Miss Kito wants to determine at what price point she should sell the mosquitos in order to maximize her weekly Profit. (Note that Profit = Revenue – Cost. Revenue is the money made strictly from sales of the item. So, Revenue = price*quantity sold β†’ 𝑹 = 𝒑𝒒) a) [4 pts] Create the Profit function strictly in terms of price, p. b) [5 pts] Determine both the price at which Profit is maximized and the maximum Profit. You must demonstrate an algebraic method for finding this Maximum point (but you can use your calculator to check your answer.) Provide your final answer for both the price and the Profit using a complete sentence with appropriate units. 2. [6 pts] Let 𝑓 (π‘₯ ) = 5π‘₯ 2 βˆ’ 4π‘₯ + 3 Find 𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯) β„Ž and completely simplify your answer. 3. Mr. Fishman thoroughly enjoys reading, with an inclination towards mystery and horror. One of his favorite authors is Stephen King. In the short story collection β€œSkeleton Crew” there is a story called β€œThe Raft” about four college friends who swim out to a raft in an isolated lake. Upon reaching the raft they discover a circular, black substance (much like an oil slick) floating in the water. They soon learn that the substance eats anything swimming in the lake and will not let them leave the raft. When they first discover the circular substance, it has a radius of 2 feet. Every minute that passes, the radius grows by 0.75 inches. a) [4 pts] Create a linear function which gives the radius of the circle in inches after t minutes. Call this function 𝒓(𝒕). b) [2 pts] The Area of a circle is 𝑨 = π…π’“πŸ . With this knowledge, use composition of functions to represent the Area of the circular substance in square inches after t minutes. Call this function 𝑨(𝒕). c) [3 pts] Use your answer to part b to find the area of the circular substance after a half hour has passed. Round your final answer to the nearest whole number and include appropriate units.
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