Let f(t) be the number of centimeters of rainfall that has fallen since midnight, where t is the time in hours. Interpret the following in practical terms, giving units.
(a) f(8) = 2.9
A total of 10.9 cm of rain falls on the ground.
8 cm of rain fall at a rate of 2.9 cm per hour.
When t = 8, 2.9 cm of rain has fallen.
When t = 2.9, 8 cm of rain has fallen.
(b) f^-1(2.4) = 8
When t = 2.4, then the rate of rainfall is 8 cm per hour.
When t = 8, then the rate of rainfall is 2.4 cm per hour.
When 2.4 cm of rain have fallen, 8 hours have passed.
When 8 cm of rain have fallen, 2.4 hours have passed.
(c) f'(8) = 0.4
At t = 0.4 hours, 8 cm of rain have accumulated.
At t = 8 hours, the rate of rainfall is 4 times as great as at t = 7.
When t = 8, the rate at which rain is falling is 0.4 cm per hour.
When 0.4 hours have passed, the rate of rainfall is 8 cm per hour.
(d) (f^-1)'(2.4) = 3
When t = 3 hours, the rate of rainfall is increasing by 2.4 cm/hr^2
When t = 2.4 hours, the rate of rainfall is increasing by 3 cm/hr^2
At t = 2.4 hours, the rain is falling at a rate of 3 cm per hour.
When 2.4 cm of rain have fallen, the rain is falling at a rate such that it will take 3 additional hours for another centimeter to fall.