1. ## equation problem

From the set of equations shown below, find the positive ratio of t to r rounded to two decimal places.
p = 3q
p r = q s
½ p r ^2 + ½ q s^ 2 = ½ q t^ 2

2. Hello, yzc717!

From the set of equations below,
find the positive ratio $\frac{t}{r}$ to two decimal places.

. . $\begin{array}{ccc}p \;=\; 3q & {\color{blue}[1]} \\ \\[-4mm]
pr \;=\; qs & {\color{blue}[2]}\\ \\[-4mm]
\frac{1}{2}pr^2 + \frac{1}{2}qs^2 \;=\; \frac{1}{2}qt^ 2 & {\color{blue}[3]}\end{array}$

[1] becomes: . $\frac{p}{q} \:=\:3\quad{\color{blue}[4]}$

[2] becomes: . $\frac{p}{q} \:=\:\frac{s}{r}$

Substitute [4]: . $3 \:=\:\frac{s}{r} \quad\Rightarrow\quad s \:=\:3r\quad{\color{blue}[5]}$

Multiply [3] by $\frac{2}{q}\!:\quad \frac{p}{q}\!\cdot\!r^2 + s^2 \:=\:t^2$

Substitute [4] and [5]: . $3\!\cdot\!r^2 + (3r)^2 \;=\;t^2 \quad\Rightarrow\quad 3r^2 + 9r^2 \;=\;t^2$

Hence: . $12r^2 \:=\:t^2 \quad\Rightarrow\quad \frac{t^2}{r^2} \:=\:12 \quad\Rightarrow\quad \frac{t}{r} \:=\:\sqrt{12} \:=\:2\sqrt{3}$

Therefore: . $\frac{t}{r} \;\approx\;3.46$