Chapter 12.6, Problem 28PPS

To determine

Find the diameter of the first tree sphere.

Answer to Problem 28PPS

The diameter of the first tree is $8.63ft$ .

Explanation of Solution

Given: It is given in the question that the spherical tree house, or tree sphere has a diameter of $10.5ft$ and it’s volume is $1.8$ times the volume of the first tree sphere that was built.

Concept Used:

In this, use the concept of volume of a sphere i.e. $V=\frac{4}{3}\pi r^3$ .

Calculation:

Here, $d=10.5ft$ then $r=\frac{10.5}{2}=5.25f$ .

Now,

  $\begin{array}{l}V=\frac{4}{3}\pi r^3\\ $ V=\frac{4}{3}\pi {(5.25)}^3$V=\frac{4}{3}\pi \times 144.7\\ V=\frac{1817.5}{3}\\ V=606f{t}^3\end{array}$

According to question,

New tree house volume = $1.8$ (first tree house volume)

  $\begin{array}{l}606.13=1.8(t)\\ t=\frac{606.13}{1.8}336.7i{n}^2\end{array}$

Since, the volume of first tree house into the formula for volume of a sphere.

  $\begin{array}{l}V=\frac{4}{3}\pi r^3\\ $ 336.7=\frac{4}{3}\pi {(r)}^3$$ \pi {(r)}^3=\frac{336.7\times 3}{4}$\pi r^3=\frac{1010.1}{4}\\ \pi r^3=252.55\\ r^3=80.39\\ r=\sqrt[3]{80.39}\\ r=4.31\\ D=2r=2\cdot 4.31=8.63ft\end{array}$

Conclusion:

The diameter is $8.63ft$ .