The problem contains three sub-problems: a) Find the value of $y$ given the equation $11y = (18)^2 - (15)^2$. b) Find the perimeter of a circle with radius $35$ cm, given $\pi = \frac{22}{7}$. c) i) Make $r$ the subject of the formula $m = \frac{r-s}{2nr}$. ii) Find the value of $r$ when $s=117$, $m=2$, and $n=-3$.

1. Problem Description

The problem contains three sub-problems:

a) Find the value of $y$ given the equation $11y = (18)^2 - (15)^2$.

b) Find the perimeter of a circle with radius $35$ cm, given $\pi = \frac{22}{7}$.

c) i) Make $r$ the subject of the formula $m = \frac{r-s}{2nr}$.

ii) Find the value of $r$ when $s=117$, $m=2$, and $n=-3$.

2. Solution Steps

a)

We are given $11y = (18)^2 - (15)^2$.

$11y = 324 - 225$

$11y = 99$

$y = \frac{99}{11}$

$y = 9$

b)

The perimeter (circumference) of a circle is given by the formula

$C = 2\pi r$, where $r$ is the radius.

Given $r = 35$ cm and $\pi = \frac{22}{7}$,

$C = 2 \times \frac{22}{7} \times 35$

$C = 2 \times 22 \times 5$

$C = 44 \times 5$

$C = 220$ cm

c) i)

We are given $m = \frac{r-s}{2nr}$. We want to isolate $r$.

Multiply both sides by $2nr$:

$2nmr = r - s$

Rearrange to get all terms with $r$ on one side:

$2nmr - r = -s$

Factor out $r$:

$r(2nm - 1) = -s$

$r = \frac{-s}{2nm - 1}$

$r = \frac{s}{1 - 2nm}$

c) ii)

We are given $s=117$, $m=2$, $n=-3$. Using the rearranged formula:

$r = \frac{s}{1 - 2nm}$

$r = \frac{117}{1 - 2(2)(-3)}$

$r = \frac{117}{1 - (-12)}$

$r = \frac{117}{1 + 12}$

$r = \frac{117}{13}$

$r = 9$

3. Final Answer

a) $y = 9$

b) $220$ cm

c) i) $r = \frac{s}{1-2nm}$

ii) $r = 9$