Chapter 8.7, Problem 13WE

a.

To determine

To find larger angle out of angles A and B.

Answer to Problem 13WE

Angle A is larger

Explanation of Solution

Given information:

  

Given diagram shows a goalpost.

Distance of B from in front of right pole is 40 foot.

Distance of A from the center of goal is 40 feet.

Width of goal is 24 feet.

To find the angles of A and B, trigonometric ratios can be applied in the small triangle formed by A, B and the goalpost.

  

Calculation:

  $\begin{array}{l}\tan \theta =\frac{24}{40}\\ \tan \theta =0.6\end{array}$

Hence, angle B will be less than $\frac{\pi }{4}radian$ .

For angle A another triangle can be formed as

  

  $\tan 2\phi =\frac{2\tan \phi }{1-{\tan }^2\phi }$

Here, $\tan \phi =\frac{12}{40}$

  $\tan 2\phi =\frac{2*\frac{12}{40}}{1-{(\frac{12}{40})}^2}$

  $\tan 2\phi =\frac{\frac{12}{20}}{1-\frac{144}{1600}}$

  $\tan 2\phi =\frac{\frac{12}{20}}{\frac{1456}{1600}}$

  $\begin{array}{l}\tan 2\phi =\frac{12*1600}{20*1456}\\ \tan 2\phi =0.6593\end{array}$

By comparing tangent of angles formed by A and B it can be said that angle A is larger than B.

Conclusion:

Therefore, Angle A turns out to be larger than B by the use of trigonometric functions.

b.

To determine

To find the point from where there will be more chances of kicking the ball into goal.

Answer to Problem 13WE

From point A

Explanation of Solution

Given information:

  

While kicking a ball into goal, a player must have a wider view to increase the chances of scoring.

As found out in the previous result that angle A is larger than B, the player at point A will have wider view of the goal as compared to B.

So, there will be more chances of kicking the ball into the goal from point A.

Conclusion:

Therefore, player at point A will have a higher chance of kicking the ball into goal because of a wider view of goal as compared to point B.