MAT 117 Arizona State University Revenue Function Algebra Problem Set 3

MAT 117 – SPRING 2021PROBLEM 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 = 𝟐𝟐𝟒 − 𝟏𝟒𝒑 where 𝑞 represents the number of mosquitos sold
each week based on a price, 𝑝, in dollars. The cost to produce each mosquito is $3. (So, 𝑪 = 𝟑𝒒)
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 → 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 (𝑥 ) = 5𝑥 2 − 4𝑥 + 3
Find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
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 “Skeleton Crew” there is a story called
“The 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 = 𝝅𝒓𝟐 . 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.