Chapter 9.4, Problem 22E

Multivariable Calculus

Total Differentials and Approximations

LIFE SCIENCE APPLICATIONS

Horn Volume The volume of the horns from bighorn sheep was estimated by researchers using the equation

$V=\frac{h\pi }{3}\left(r^2{}_1+r_1{r}_2+r^2{}_2\right)$

where $h$ is the length of a horn segment (in centimeters) and $r_1$ and $r_2$ are the radii of the two ends of the horn segment (in centimeters). Source: Conservation Biology.

a. Determine the volume of the segment of horn that is $40\text{ cm}$ long with radii of $5\text{ cm}$ and $3\text{ cm,}$ respectively.

b. Use the total differential to estimate the volume of the segment of horn if the horn segment from part a was actually $42\text{ cm}$ long with radii of $5.1\text{ cm}$ and $2.9\text{ cm,}$ respectively. Compare this with the actual volume.