Chapter 8.2, Problem 39PPS

(a)

To determine

To find: the equation for the diameter $d$ of the base of a sequoia tree.

Answer to Problem 39PPS

Theequation for the diameter $d$ of the base of a sequoia tree is $y=1.30{\left(b\right)}^{1.5}$.

Explanation of Solution

Given information:

The diameter of the base of a tree trunk in centimetres varies directly with the $\frac{3}{2}$ power of its height in meters.

Consider the given information.

Substitute $19.1$ for $y$ , $6$ for $b$ , and $1.5$ for $x$ in the general exponential function $y=a{b}^x$.

  $\begin{array}{c}19.1=a{\left(6\right)}^{1.5}\\ a=\frac{19.1}{14.697}\\ =1.30\end{array}$

Substitute $1.30$ for $a$ , $1.5$ for $x$ in the equation $y=a{b}^x$.

  $y=1.30{\left(b\right)}^{1.5}$

Therefore, the equation for the diameter $d$ of the base of a sequoia tree is $y=1.30{\left(b\right)}^{1.5}$.

(b)

To determine

To find: the diameter of the general Sherman tree at its base.

Answer to Problem 39PPS

A diameter of the general Sherman tree at its base is $1000.83$.

Explanation of Solution

Given information:

The diameter of the base of a tree trunk in centimetres varies directly with the $\frac{3}{2}$ power of its height in meters.

Consider the given information.

Substitute $84$ for $b$ in the exponential function $y=1.30{\left(b\right)}^{1.5}$.

  $\begin{array}{c}y=1.30{\left(84\right)}^{1.5}\\ =1.30\left(769.87\right)\\ \approx 1000.83\end{array}$

Therefore, the diameter of the general Sherman tree at its base is $1000.83$.