Question

let $f(t)$ be the number of centimeters of rainfall that has fallen since midnight, where $t$ is the time in hours. match the following statements to their interpretations, given below.
(a) $f(5)=2.5$:
(b) $f^{-1}(4)=5$:
(c) $f(5)=0.4$:
(d) $(f^{-1})(4)=2$:
interpretations:
1: when 4 centimeters of rain have fallen, it is accumulating at a rate of 1/2 cm/hr, so it takes approximately 12 min for the total accumulation to go from 4 to 4.1 cm
2: when 2 centimeters have fallen, rain is accumulating at a rate of 1/4 hr/cm, so it takes approximately 15.0 min for the total accumulation to go from 2 to 3 cm
3: at 2 hours after midnight, rain is falling at a rate of 4 cm/hr, so approximately 0.67 cm of rain falls between 2 am and 2:10 am
4: at 4 hours after midnight, rain is falling at a rate of 2 cm/hr, so approximately 0.33 cm of rain falls in the next ten minutes

Explanation:

• (a) $f(5)=2.5$ means that 5 hours after midnight, 2.5 centimeters of rainfall has fallen.
• (b) $f^{-1}(4) = 5$ means that when 4 centimeters of rainfall has fallen, 5 hours have passed since midnight.
• (c) $f^{\prime}(5)=0.4$ means that 5 hours after midnight, the rate of rainfall is 0.4 cm/hr.
• (d) $(f^{-1})^{\prime}(4)=2$ means that when 4 centimeters of rainfall has fallen, the rate of change of time with respect to rainfall is 2 hours per cm. So when 4 centimeters of rain have fallen, it is accumulating at a rate of 1/2 cm/hr, so it takes approximately 12 min for the total accumulation to go from 4 to 4.1 cm.

Answer:

(a) 5 hours after midnight, 2.5 cm of rainfall has fallen.
(b) When 4 cm of rainfall has fallen, 5 hours have passed since midnight.
(c) 5 hours after midnight, the rate of rainfall is 0.4 cm/hr.
(d) When 4 centimeters of rain have fallen, it is accumulating at a rate of 1/2 cm/hr, so it takes approximately 12 min for the total accumulation to go from 4 to 4.1 cm.