Question

2 mark for review the function ( t = f(s) ) models the time, in hours, for a sample of water to evaporate as a function of the size ( s ) of the sample, measured in milliliters. what are the units for ( f^{prime prime}(s) )? a hours per milliliter b milliliters per hour c hours per milliliter per milliliter d milliliters per hour per hour

Explanation:

Step1: Recall the units of the first - derivative

The first - derivative \(f^{\prime}(S)=\frac{dt}{dS}\). Since \(t\) (time) has units of hours and \(S\) (volume) has units of milliliters, the units of \(f^{\prime}(S)\) are hours per milliliter (\(\frac{\text{hours}}{\text{milliliter}}\)).

Step2: Recall the formula for the second - derivative

The second - derivative \(f^{\prime\prime}(S)=\frac{d}{dS}(f^{\prime}(S))\). Let \(y = f^{\prime}(S)\) with \(y\) having units \(\frac{\text{hours}}{\text{milliliter}}\) and the independent variable \(S\) having units of milliliters.
Using the quotient rule for units in differentiation \(\frac{d}{dS}(y)=\frac{\text{units of }y}{\text{units of }S}\).
Substituting the units of \(y\) and \(S\), we get \(\frac{\frac{\text{hours}}{\text{milliliter}}}{\text{milliliter}}=\frac{\text{hours}}{\text{milliliter}\times\text{milliliter}}\) (hours per milliliter per milliliter).

Answer:

C. hours per milliliter per milliliter