Last night it started raining as Hibah was closing up the store where she works. By 3 am, the roof of the store developed a small hole and water began to leak onto the sales floor. The water created a circular puddle on the floor. At any time, 1, in minutes, the radius of the puddle increases by 0.1 cm. The rain continued through the night, but stopped by the time the morning shift arrived at the store at 7 am. When the employees arrived at work, the leak and water puddle were discovered. The employees cleaned up the water puddle and placed a tall circular bucket underneath the leak, which was still slowly dripping. The capacity of the bucket can be expressed as f(x) = 8x³+4x² - 6x + 5. By 10 am, the amount of water in the bucket is g(x) = 2x³ + x² - 4x + 7. Use the information given to explore some of the mathematical concepts you have practiced so far by answering the questions below
1) Write a function that describes the area of the water puddle on the floor as a function of its
radius. Use function notation.
2) Write a function that represents the radius of the puddle at time ??. Use function notation.
3) What is the area of the puddle on the floor when the employees arrive at work at 7 am? Write
a composition of functions to help you, and round your answer to the nearest whole number.
Explain how you found your answer.
4) What function represents how much more water can be added to the bucket before it
overflows? Explain how you solved this problem.
5) At around noon the store roof appears to have stopped leaking, so an employee removes the
bucket that was catching the water and does not replace it. Overnight it begins to rain again, and
water starts leaking from the ceiling onto the floor, again creating a circular puddle. The hole in
the roof is larger this time, so at each time, t, in minutes, the radius of the puddle increases by
0.25 cm. Write a composition of functions to represent the area of the puddle as a function of
time.
6) Look at the function you wrote for question 5. Does this function have an inverse? If so, what
is it? Is the inverse a function? Show your work and explain your reasoning.
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