Chapter 8.6, Problem 17WE

To determine

The difference in heights of the ladder when it placed at ${70}^{\circ }$ and ${60}^{\circ }$ angles

Answer to Problem 17WE

To the nearest tenth of a meter, the ladder is $0.4m$ higher when it placed at ${70}^{\circ }$ angle with the ground than when it placed at ${60}^{\circ }$ angle instead.

Explanation of Solution

Given information:

The height of the ladder is $6m$

From the two right angle triangles, we observe that the both triangles have same hypotenuse length $=6m$

From the sine rule, we have $\begin{array}{l}\sin \left({60}^{\circ }\right)=\frac{\text{altitude}\left(h_1\right)}{\text{hypotenuse}}\text{and}\sin \left({70}^{\circ }\right)=\frac{\text{altitude}\left(h_2\right)}{\text{hypotenuse}}\\ \Rightarrow \text{altitude}\left(h_1\right)=\sin \left({60}^{\circ }\right)\times \text{hypotenuse}and\text{altitude}\left(h_2\right)=\sin \left({70}^{\circ }\right)\times \text{hypotenuse}\\ \Rightarrow \text{altitude}\left(h_1\right)=0.8660\times \text{6}\text{and}\text{altitude}\left(h_2\right)=0.93969\times 6\\ \therefore \text{altitude}\left(h_1\right)=5.20mand\text{altitude}\left(h_2\right)=5.64m\end{array}$

Thus, the difference in heights is $5.64-5.20=0.44m$

Hence, to the nearest tenth of a meter, the ladder is $0.4m$ higher when it placed at ${70}^{\circ }$ angle with the ground than when it placed at ${60}^{\circ }$ angle instead.