Chapter 12.7, Problem 1P

To determine

To calculate: The diameter of the ball if 1027 $c{m}^2$ of material is used to cover it, if 12 cm diameter ball needs 452 $c{m}^2$ of the material to cover it . also given amount of the material needs to cover a ball is directly propositional to the square of its diameter.

Answer to Problem 1P

The diameter of the ball is $d=17.9cm$ .

Explanation of Solution

Given information:

Amount of the material needs to cover a ball is directly propositional to the square of its diameter.

Let, Material = m

Diameter= d

And $m\prec d^2$

Let k be a constant

  $m=k\cdot d^2$...... $\left(1\right)$

Also given, 12 cm diameter ball needs 452 $c{m}^2$ of the material to cover it.

If 12 cm diameter ball needs 452 $c{m}^2$ of the material to cover it.

Put the value of diameter and material to get value of k .

  $\begin{array}{l}m=k\cdot d^2\\ 452c{m}^2=k\cdot {\left(12cm\right)}^2\end{array}$

Solve further to get k .

  $k=\frac{452c{m}^2}{\begin{array}{l}144c{m}^2\\ k=3.1388\end{array}}$

Now, put the value of m and k in equation 1 to get the value of d .

  $\begin{array}{l}m=k\cdot d^2\\ 1017c{m}^2=\mathrm{3.1388.}{d}^2\\ d^2=\frac{117c{m}^2}{3.1388}\\ d=\sqrt{\frac{1017}{3.1388}}c{m}^2\end{array}$

Solve further to get d .

  $\begin{array}{l}d=\sqrt{318.9287}cm\\ d=17.85cm\end{array}$

Value of d to the nearest tenth

  $d=17.9cm$