# 007A Sample Midterm 1, Problem 5 Detailed Solution

To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula:

${\displaystyle {\text{BSA}}={\frac {\sqrt {hm}}{60}}}$

where  ${\displaystyle h}$  is the height in centimeters and  ${\displaystyle m}$  is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass of  ${\displaystyle m=85.}$  What is the rate at  ${\displaystyle h=170}$  and  ${\displaystyle h=190?}$  Express your results in the correct units. Does the BSA increase more rapidly with respect to height at lower or higher heights?

Background Information:
Power Rule
${\displaystyle {\frac {d}{dx}}(x^{n})=nx^{n-1}}$

Solution:

Step 1:
First, we have  ${\displaystyle m=85.}$
So, we have
${\displaystyle {\begin{array}{rcl}\displaystyle {\text{BSA}}&=&\displaystyle {\frac {\sqrt {85h}}{60}}\\&&\\&=&\displaystyle {{\frac {{\sqrt {85}}{\sqrt {h}}}{60}}.}\end{array}}}$
Step 2:
Now, using the Power Rule, we get
${\displaystyle {\begin{array}{rcl}\displaystyle {\frac {d({\text{BSA}})}{dh}}&=&\displaystyle {{\frac {\sqrt {85}}{60}}\cdot {\frac {1}{2}}h^{-1/2}}\\&&\\&=&\displaystyle {{\frac {\sqrt {85}}{120{\sqrt {h}}}}.}\end{array}}}$
Step 3:
For  ${\displaystyle h=170,}$  we get
${\displaystyle {\frac {d({\text{BSA}})}{dh}}={\frac {\sqrt {85}}{120{\sqrt {170}}}}{\text{ m}}^{2}{\text{/cm}}.}$
For  ${\displaystyle h=190,}$  we get
${\displaystyle {\frac {d({\text{BSA}})}{dh}}={\frac {\sqrt {85}}{120{\sqrt {190}}}}{\text{ m}}^{2}{\text{/cm}}.}$
Step 4:
Since  ${\displaystyle h}$  is in the denominator of the formula for  ${\displaystyle {\frac {d({\text{BSA}})}{dh}},}$
${\displaystyle {\text{BSA}}}$  increases more rapidly with respect to height at lower heights.

${\displaystyle {\frac {d({\text{BSA}})}{dh}}={\frac {\sqrt {85}}{120{\sqrt {h}}}}}$
${\displaystyle {\frac {\sqrt {85}}{120{\sqrt {170}}}}{\text{ m}}^{2}{\text{/cm}}}$
${\displaystyle {\frac {\sqrt {85}}{120{\sqrt {190}}}}{\text{ m}}^{2}{\text{/cm}}}$
${\displaystyle {\text{BSA}}}$  increases more rapidly with respect to height at lower heights.