Chapter 11, Problem 19CT

To determine

Find the angle between two adjacent sides.

Answer to Problem 19CT

  $\boxed{75.92°}$ and $\boxed{85.35°}$ .

Explanation of Solution

Given information:

A tractor fuel tank has the shape and dimensions shown in the figure. In fabricating the tank, it is necessary to know the angle between two adjacent sides. Find this angle.

  

Calculation:

Consider the given tractor fuel tank.

  

The vertices of tractor fuel tank are,

  $P\left(10,0,0\right),Q\left(10,10,0\right),R\left(8,8,12\right),S\left(0,0,0\right)andT\left(2,2,12\right)$

Now compute the vectors,

  $\begin{array}{l}\overline{QR}=\left(8-10,8-10,12-0\right)\\ =\left(-2,-2,12\right)\\ \overline{QP}=\left(10-10,0-10,0-0\right)\\ =\left(0,-10,0\right)\end{array}$

Now angle between these two vector be calculated by taking dot product of $\overline{QR}$ and $\overline{QP}$ .

  $\begin{array}{l}\overline{QR}\cdot \overline{QP}\left|\overline{QR}\right|\left|\overline{QP}\right|\cos \theta \\ \left(-2\times 0\right)+\left(-2\times -10\right)+\left(12\times 0\right)=10\times 12.33\times \cos \theta \\ {\theta }_{PQR}={\cos }^{-1}0.24\\ {\theta }_{PQR}=75.92°\end{array}$

Now compute the vectors,

  $\begin{array}{l}\overline{ST}=\left(2-0,2-0,12-0\right)\\ =\left(2,2,12\right)\\ \overline{SP}=\left(10-0,0-0,0-0\right)\\ =\left(10,0,0\right)\end{array}$

Now angle between these two vector be calculated by taking dot product of $\overline{ST}$ and $\overline{SP}$ .

  $\begin{array}{l}\overline{ST}\cdot \overline{SP}\left|\overline{ST}\right|\left|\overline{SP}\right|\cos \theta \\ \left(10\times 2\right)+\left(0\times 2\right)+\left(0\times 12\right)=10\times 12.33\times \cos \theta \\ {\theta }_{PST}={\cos }^{-1}0.08\\ {\theta }_{PST}=85.35°\end{array}$

Hence, the angles between walls of the tank are $\boxed{75.92°}$ and $\boxed{85.35°}$ .