Question

A plane flying along a straight path loses altitude at the rate of \(1000 \mathrm{ft}\) for each \(6000 \mathrm{ft}\) covered horizontally. What is the angle of descent of the plane?

Short answer

The angle of descent for the plane is approximately \(9.46°\).

Step-by-step solution

We can model the plane's path as a right triangle, where the 1000 ft altitude loss is the opposite side, the 6000 ft horizontal distance is the adjacent side, and the angle we want to find is between the adjacent side and the hypotenuse (the plane's path). #Step 2: Use the tangent function to find the angle#

To find the angle, we can use the tangent function because we know the length of the opposite and adjacent sides of the triangle, as follows: \[ \tan(\theta) = \frac{opposite}{adjacent} \] #Step 3: Plug in the given values and solve for the angle#

Now, we can plug in the given values for the opposite side (altitude loss) and adjacent side (horizontal distance): \[ \tan(\theta) = \frac{1000}{6000} \] Simplifying the fraction, we get: \[ \tan(\theta) = \frac{1}{6} \] To find the angle, we need to take the inverse tangent (also known as arctangent or atan) of the fraction: \[ \theta = \arctan\left(\frac{1}{6}\right) \] Using a calculator, we find: \[ \theta \approx 9.46° \] #Step 4: State the final answer#

The angle of descent for the plane is approximately 9.46°.