Question

Please help me on these two questions

Answer 1

1. FIRST QUESTION.

M is directly proportional to $r^3$

Suppose you have two variables, say, $x \ and \ y$. We say that $y$ varies directly as $x$ or, in other words, $y$ is directly proportional to $x$ if and only if:

$y=kx$

Where $k$ is a nonzero constant called the constant of proportionality.

Knowing this, we can solve this problem. Here $M$ is directly proportional to $r^3$. So:

$M=kr^{3}$

We know that when $r=4, \ M=160$, hence the constant of proportionality can be found as follows:

$M=kr^{3} \\ \\ 160=k(4^3) \\ \\ 160=k(64) \\ \\ Solving \ for \ k: \\ \\ k=\frac{160}{64} \therefore k=\frac{5}{2}$

So:

$M=\frac{5}{2}r^3$

PART A)

When $r=2$, then:

$M=\frac{5}{2}(2^3) \\ \\ M=\frac{5}{2}(8) \\ \\ \boxed{M=20}$

PART B)

When $M=540$, then:

$540=\frac{5}{2}(r^3) \\ \\ r^3=\frac{2(540)}{5} \\ \\ r^3=216 \\ \\ r=\sqrt[3]{216} \\ \\ \boxed{r=6}$

2. SECOND QUESTION.

M is directly proportional to $r^2$

We know that when $r=2, \ M=14$, hence the constant of proportionality is:

$M=kr^{2} \\ \\ 14=k(2^2) \\ \\ 14=k(4) \\ \\ Solving \ for \ k: \\ \\ k=\frac{14}{4} \therefore k=\frac{7}{2}$

So:

$M=\frac{7}{2}r^2$

PART A)

When $r=12$, then:

$M=\frac{7}{2}(12^2) \\ \\ M=\frac{7}{2}(144) \\ \\ \boxed{M=504}$

PART B)

When $M=224$, then:

$224=\frac{7}{2}(r^2) \\ \\ r^2=\frac{224(2)}{7} \\ \\ r^2=64 \\ \\ r=\sqrt{64} \\ \\ \boxed{r=8}$