Question

A television camera at ground level films the liftoff of a space shuttle at a point 750 meters from the launch pad. Let be the angle of elevation to the shuttle in degrees and let s be the height of the shuttle in meters. (a) Write as a function of s.

Answer 1

Explanation:

To write the angle of elevation θ as a function of the height s of the shuttle, you can use trigonometry. In this case, you have a right triangle formed by the camera, the launch pad, and the shuttle. The height of the shuttle s is the side opposite to the angle θ, and the horizontal distance from the camera to the shuttle is 750 meters.

Using the tangent function, which is defined as:

`\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]`

In this context, the opposite side is s (the height of the shuttle), and the adjacent side is 750 meters. Therefore, you can write:

`\[ \tan(\theta) = (s)/(750) \]`

Now, to express θ as a function of s, you need to isolate θ:

`\[ \theta = \arctan\left((s)/(750)\right) \]`

So, θ is a function of s and can be written as:

`\[ \theta(s) = \arctan\left((s)/(750)\right) \]`

This function gives you the angle of elevation θ in degrees as a function of the height s of the shuttle in meters.