Metal Company is a mine for metals. Part of their process involves disposing of waste rock that contains a harmful substance that is water soluable. The waste is deposited into ponds that have clean water flowing into them and water flowing out at the same rate so that the volume of the pond stays the same. A certain type of algae grows in the pond that degrades the harmful substance at a rate proportional to the square of the concentration of the harmful substance in the pond.

Letx(t) be the amount of the harmful substance in the pond at time t andx(0) =x0be the initial amount of the harmful substance in the pond att= 0. Letrinandroutbe the rate of water into and out of the pond. LetVbe the volume of the pond andkbe a proportionality constant for the degradation term.

(a) Assume for a moment that the pond does not have algae. Derive a differential equation to describe how the amount of the harmful substance in the pond changes with time.

Math2250 Lab2 -Page2of5 (b) Solve the differential equation derived in part (a).

(c) Now consider how the differential equation would be different if algae is in the pond. Derive the differential equation.

(d) Solve the differential equation derived in part (c).Hint: Use partial fractions.

(e)

(f) What percentage less of the harmful substance is in the pond with algae than in the pond without algae in the first 100 days.

I have the answers but I want to know how to do them

thank you