Chapter 13.2, Problem 45E

To determine

To find: The distance above the wheel must at which a lamp is placed so it will not be showered with sparks.

Answer to Problem 45E

The lamp must be placed at as distance of $3.313\text{cm}$ above wheel.

Explanation of Solution

Given:

A knife is held at $45°$ .

The diameter of sharpening wheel is 16 inch.

Calculation:

Draw the following figure with the given information.

  

Assume the lamp must be place at the point $P$ to avoid the sparks.

From the figure,

  $CP=8+x$

The triangle $\Delta CAP$ is right angled isosceles.

  $\begin{array}{c}CA=8\\ =CQ\end{array}$ 8+x

By using Pythagoras Theorem,

  $\begin{array}{c}C{P}^2=C{A}^2+A{P}^2\\ {\left(8+x\right)}^2=8^2+8^2\\ {\left(8+x\right)}^2=128\\ 8+x=8\sqrt{2}\end{array}$

Simplify further.

  $\begin{array}{c}x=8\sqrt{2}-8\\ =3.313\text{cm}\end{array}$

Therefore, the lamp must be placed at as distance of $3.313\text{cm}$ above wheel.