Question

lesson 4: ratios in right triangles
cool down: lift off
shortly after takeoff, an airplane is climbing 250 feet for every 1,000 feet it travels. estimate the airplane’s climb angle while this is happening.

Explanation:

Step1: Identify the right triangle

We have a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), opposite side (height) \( AC = 250 \) feet, and adjacent side (horizontal distance) \( AB = 1000 \) feet. The climb angle is \( \angle B \).

Step2: Use the tangent function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. So, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{250}{1000} \).

Step3: Calculate the ratio

Simplify \( \frac{250}{1000} = 0.25 \).

Step4: Find the angle

We need to find \( \theta \) such that \( \tan(\theta) = 0.25 \). Using the arctangent function, \( \theta = \arctan(0.25) \).

Step5: Compute the arctangent

Using a calculator, \( \arctan(0.25) \approx 14.04^\circ \).

Answer:

The airplane's climb angle is approximately \( 14.04^\circ \) (or can be rounded to a reasonable estimate like \( 14^\circ \) or \( 14.0^\circ \)).