Answer to Question #239088 in Calculus for Fiona

Question

Question 9

In a bottle, the height of the liquid is a function of the volume of the liquid in the bottle, say h = 𝑔(𝑣). Similarly, the volume of the liquid is function of the height of the liquid, say 𝑣 = 𝑓(h). Show that 𝑓and 𝑔 are inverse functions.

Question 10

Scientists have developed a radioisotope-based power supply whose power output in watts is

125 given by the following function: 𝑃 = 75𝑒− 𝑡

where t is the number of days the power supply is used. The experimental power supply has been installed in an unmanned undersea vehicle on an exploration mission. The vehicle requires 7watts to operate at full capacity. Use the given information to find the number of days for which the vehicle can operate at full capacity.

Answer 1

Question 9

To show that 𝑓and 𝑔 are inverse functions we should prove that 𝑔(𝑓(h)) = h and 𝑓(𝑔(𝑣))=𝑣.

We know that f(h) = 𝑣, we also know that by definition 𝑔(𝑣) = h, therefore 𝑔(𝑓(h)) = 𝑔(𝑣) = h. Also 𝑓(𝑔(𝑣)) = 𝑓(h) = 𝑣. Therefore, 𝑓 and 𝑔 are inverse functions.

Question 10

$𝑃 = 75𝑒^{− 𝑡}.$ Let us determine the value of t for which P = 7 W.

$75𝑒^{− 𝑡} = 7, \\ e^{-t} = \dfrac{7}{75}, \\ t = -\log_e \dfrac{7}{75} \approx 2.37.$

Therefore, the vehicle may operate for only two days.

But there is number 125 in the statement of the problem. Let us assume the proper formula is $𝑃 = 75𝑒^{− 𝑡/125}.$ Therefore, the equation becomes

$75𝑒^{− 𝑡/125} = 7, \\ e^{-t/125} = \dfrac{7}{75}, \\ t = -125\log_e \dfrac{7}{75} \approx 296.$

Therefore, the vehicle may operate for 296 days.