Luke and Mason are standing 23 feet apart on the football field. Luke throws a high, slow pass to Mason. While the ball is in the air between them, each person notes the angle of elevation from himself to the ball. The angle of elevation from Luke is to the ball is 9°; from Mason to the ball is 13°. How far is the ball from Mason?
Draw triangle LBM where the ball is at B; angle B is 158°.
Using the law of sines, if the ball's distance from M is x, then
x/sin9° = 23/sin158°
Would it be 9.6 feet?
looks like our calculator agree!
To find the distance of the ball from Mason, we can use trigonometry and the tangent function.
Let's denote the distance from Luke to the ball as "x" (in feet). We already know that the distance between Luke and Mason is 23 feet.
Now, we can set up a right triangle with Luke, the ball, and a point on the ground beneath the ball. The angle of elevation from Luke to the ball is 9°.
Since the tangent of an angle is equal to the ratio of the opposite side to the adjacent side, we can write the following equation:
tan(9°) = x / 23
Now, we can solve this equation to find the value of "x" (the distance from Luke to the ball):
x = 23 * tan(9°)
Using a scientific calculator, the value of x is approximately 3.431 feet.
Now that we know the distance from Luke to the ball, we can find the distance from Mason to the ball. Let's denote this distance as "y" (in feet).
We can set up a similar right triangle with Mason, the ball, and a point on the ground beneath the ball. The angle of elevation from Mason to the ball is 13°.
Using the tangent function again, we can write the following equation:
tan(13°) = y / (23 + x)
Substituting the value of x that we found earlier, the equation becomes:
tan(13°) = y / (23 + 3.431)
Now, we can solve this equation to find the value of "y" (the distance from Mason to the ball):
y = (23 + 3.431) * tan(13°)
Using a scientific calculator, the value of y is approximately 5.688 feet.
Therefore, according to our calculations, the ball is approximately 5.688 feet away from Mason.