
Diect proportion
The price $p of a steel ball is proportional to the cube of its radius x cm It is known that the price is $48 for a teel ball of certain radius R cm. Find the price of a steel ball of radius R/2 cm . Ans is $6.
Please explain to me how to solve why is it $6

Given:
$\displaystyle p = kr^3$ for some constant k.
$\displaystyle 48 = kR^3$
$\displaystyle p = k\left(\frac{R}{2}\right)^3$
Now, let's rearrange the third equation:
$\displaystyle p = k\frac{R^3}{8} \implies 8P = kR^3$
Notice tha $\displaystyle kR^3$ from second equation equals 48, so substitute:
$\displaystyle 8p = 48$
$\displaystyle p = ...$