Questions: An observer views the space shuttle from a distance of 7 mi from the launch pad as shown in the following figure. (a) Express the height of the space shuttle (in mi ) as a function of the angle of elevation θ. h= mi (b) Express the angle of elevation θ as a function of the height h of the space shuttle. θ=

An observer views the space shuttle from a distance of 7 mi from the launch pad as shown in the following figure.
(a) Express the height of the space shuttle (in mi ) as a function of the angle of elevation θ.
h=
mi
(b) Express the angle of elevation θ as a function of the height h of the space shuttle.
θ=

Transcript text: An observer views the space shuttle from a distance of 7 mi from the launch pad as shown in the following figure. (a) Express the height of the space shuttle (in mi ) as a function of the angle of elevation $\theta$. $h=$ $\square$ mi (b) Express the angle of elevation $\theta$ as a function of the height $h$ of the space shuttle. $\theta=$

Solution

Solution Steps

Step 1: Identify the trigonometric relationship

We have a right triangle formed by the observer's line of sight, the height of the space shuttle (h), and the horizontal distance (7 mi). The angle of elevation θ is the angle between the horizontal distance and the observer's line of sight. In this right triangle, h represents the opposite side to θ, and 7 mi represents the adjacent side to θ. The relevant trigonometric function relating these is tangent: tan(θ) = opposite/adjacent.

Step 2: Express h as a function of θ

Using the tangent function, we have: tan(θ) = h / 7. Solving for h, we get: h = 7 * tan(θ)

Step 3: Express θ as a function of h

Using the same relationship tan(θ) = h / 7, we can solve for θ: θ = arctan(h / 7)