Question

A carpenter makes stools for electronics with a square top of side $0.5$ m and at a height of $1.5$ m above the ground. Also each leg is inclined at an angle of ${60}^0$ to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances.

Answer 1

The height of the stool is $1.5m$ and each leg is inclined at an angle of ${60}^o$.

Let $AB$ be the one side of the top of the stool.

$\therefore$ $AB=0.5m$

Let $CD$ be the surface of the ground.

Let $AC$ be the length of the leg.

$AC$ is inclined at an angle of ${60}^o$.

$\Rightarrow$ In $△ACL,\tan {60}^o=\frac{AL}{CL}$

$\Rightarrow$ $\sqrt{3}=\frac{1.5}{CL}$

$\Rightarrow$ $CL=\frac{1.5}{\sqrt{3}}=\frac{1.5}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{1.5}{3}\sqrt{3}$

$\Rightarrow$ $CL=0.5\sqrt{3}=0.5\times 1.732=0.866m$.

$\Rightarrow$ $\sin {60}^o=\frac{AL}{AC}$

$\Rightarrow$ $\frac{\sqrt{3}}{2}=\frac{1.5}{AC}$

$\therefore$ $AC=\frac{2\times 1.5}{\sqrt{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}m$

$\Rightarrow$ $AC=1.732m$

From the diagram the length of the steps are $GH$ and $EF$. Steps are put such that $AP=PR=RL$

$\therefore$ $△AGP\sim △ACL$

$\Rightarrow$ $\frac{AP}{AL}=\frac{GP}{CL}$

$\Rightarrow$ $\frac{AP}{3AP}=\frac{GP}{0.866}$

$\Rightarrow$ $GP=\frac{0.866}{3}m=0.2887m$

$\Rightarrow$ Length of the steps $GH=GP+PQ+OH=GP+AB+GP$ [ $QH=GP$ by symmetry ]

$GH=2GP+AB=2\times 0.2887+0.5=1.077m$

$\Rightarrow$ Similarly, $ER=\frac{2}{3}CL=\frac{2}{3}\times 0.866=0.577m$

$\Rightarrow$ Length of the steps

$EF=ER+RS+SF=ER+AB+ER=2ER+AB$

$EF=2\times 0.5773+0.5=1.6546m$