Chapter 12.7, Problem 11P

To determine

To calculate: The ratio of the volumes of the spheres if the ratio of their radii is 4:3. also given volume of the spheres varies directly as the cube of the radius.

Answer to Problem 11P

The ratio of their volumes is $v_1^{}:v_2^{}=64:27$

Explanation of Solution

Given information:

Volume of the spheres varies directly as the cube of the radius.

Let, Volume of the two cubes is $v_1^{}$ and $v_2^{}$ and there radius is $r_1^{}$ and $r_2^{}$ respectively.

So $v\prec r^3$

Let k be a constant

  $v=k\cdot r^3$

Volume of the first cube is

  $v_1^{}=k\cdot {\left(r_1^{}\right)}^3$...... (1)

Volume of the second cube is

  $v_2^{}=k\cdot {\left(r_2^{}\right)}^3$...... (2)

Divide equation 1 and 2.

  $\frac{v_1^{}}{v_2^{}}={\left(\frac{r_1^{}}{r_2^{}}\right)}^3$

Now, put the value of $r_1^{}$ and $r_2^{}$ .

  $\frac{v_1^{}}{v_2^{}}={\left(\frac{4}{3}\right)}^3$

  $\frac{v_1^{}}{v_2^{}}=\frac{64}{27}$

Solve further

  $v_1^{}:v_2^{}=64:27$