What function represents how much more water can be added to the bucket before it overflows? Explain how you solved this problem.

Question

Functions are in the world all around us. You have used functions to model phenomena and to make predictions. Mathematically, you can add, subtract, multiply, and divide functions, and you can also create a composition of functions. A composition of functions occurs when you evaluate one function using another function, or even with the same function. A composition of functions is useful when a change in one function will create a change in another function, which, in turn, affects a third function. For example, suppose that humans build a housing development on land that was previously used as a corn field. This change in land usage causes the number of birds in the area to decrease, which causes the number of mosquitos to increase. You could model the increase in the number of mosquitoes as a function of the number of birds in the area, which is a function of the number of humans that live on that land. You can also use the composition of functions to show that one function is the inverse of another function. Recall that the graphs of inverse functions are reflections of each other over the line . Think about these ideas as you read through the scenario below. 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, t, 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 𝑓𝑓(𝑥𝑥) = 8𝑥^3 + 4x^2 − 6𝑥 + 5. By 10 am, the amount of water in the bucket is 𝑔(𝑥) = 2𝑥^3 + 𝑥^2 − 4𝑥 + 7. Use the information given to explore some of the mathematical concepts you have practiced so far by answering the questions below

Answer 1

I would ask what does the capacity function tell you?

Well, it has an "x" in it so it is variable, meaning it changes. What kind of bucket can have it's capacity change? Isn't the capacity of a bucket fixed? It's a ten-gallon bucket or a 5-gallon bucket or whatever, so it really doesn't change. What the function must be telling us is the remaining capacity of the bucket as it fills up. If you actually graph this function and look only in the first quadrant (the left-hand side of the graph would imply negative time which we should ignore) you'll see it does decrease for a short interval but then increases. I'd say the question is not a good one, and move on to the next one.

It would make more sense if the volume of the bucket were given as a constant since it would be something like 10 gallons or 5 gallons. Then you could subtract the second function from the 10 or 5 to produce a function that would output the remaining volume or amount of water that's left in the bucket.