An animal sanctuary is constructing a large tank for its exotic fish collection. The tank will be a rectangular prism that is set into a large wall, and it will feature two, equally sized hemispheres, one on each side of the tank. Visitors will be able to sit inside these hemispheres so they can feel like they are inside the tank with the fish. The polynomial (2 − 30) represents the length of the new tank, the polynomial (2 − 20) represents the width of the new tank, and the polynomial (2 − 60) represents the height of the new tank.

Use the information given to explore some of the mathematical concepts you have practiced so far by answering the questions below.

1. Find a polynomial that represents the volume of the fish tank. Explain how you used the properties of exponents to determine your expression.
HINT: The formula for the volume of a rectangular prism is = ℎ.

2. The volume of each hemisphere is represented by the polynomial ^3 − 70^2 + 360 − 1800. Explain how to rewrite your answer for question 1 to reflect the volume of the fish tank after the hemispheres are installed. Then carry out your plan. Show your work.

3. Show that the binomial that represents the length of the fish tank is a factor of the polynomial you wrote in question 1.

4. Is the binomial that represents the length of the fish tank a factor of the polynomial that represents the volume of the fish tank after the hemispheres are installed? Support your answer mathematically.

5. The sanctuary currently has 125 exotic fish. The average amount of the tank allotted for each fish is represented by the binomial (2^2 − 1). Are the dimensions of the new habitat adequate for these 125 fish? Explain.