a man made lake is losing volume at an approximate rate of 8% per year

Question

a.) At this rate how many years will it take for the lake to lose more then one quarter of its original volume?

b.) If the lake has a volume of 0.4 cubic miles how much will it have in 6 years in the rate stays the same?

c.) At initial volume .4 cubic miles, would the lake lose more volume over 4 years if it lost 8% per year or if it lost .03 per year cubic miles per year?

Answer 1

The lake will lose volume each year based on keeping 92% of its volume at the end of the year!

Let V_o = the original volume.

End of year 1: V = .92V_o

year 2: V = .92(.92V_o) = .92²V_o so you can see where the trend is for a general formula

Then, Part a: .75V_o = .92ⁿV_o or .75 = .92ⁿ or log(.75) = nlog(.92) or n = 3.45 years

Part b: at the end of 6 years, V = .92⁶(.4) = .2425 cubic miles

Part c: After 4 years, the volume is decreased by .V = .92⁴(.4) = .2866 cubic miles (8% formula)

If you lost .03 cubic miles per year, you would lose .12 cubic miles in 4 years. .4 -.12 = .28 would be the new volume. So you lose more if you lost .03 cubic miles per year.