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Chapter 10: Problem 37

A plane rises from take-off and flies at an angle of \(10^{\circ}\) with the horizontal runway. When it has gained 500 feet in altitude, find the distance, to the nearest foot, the plane has flown.

Short Answer

Expert verified
The distance to the nearest foot, the plane has flown can be calculated using the ratio \(\frac{500\ ft}{sin(10^{\circ})}\)

Step by step solution

01

Understand the problem

First, let's understand what's given: we have a right triangle where one angle is \(10^{\circ}\) and its opposite side (the height of the plane) is 500 feet. We need to find the hypotenuse (the total distance flown by the plane). This is a standard trigonometric problem, where we can use the definition of the sine function, which is \(sin(\theta) = \frac{opposite\ side}{hypotenuse}\).
02

Substitute the given values

Using the definition of sine, we get the equation \(sin(10^{\circ}) = \frac{500\ ft}{hypotenuse}\). We know that \(sin(10^{\circ})\) and the height of the plane, we want to solve for the hypotenuse.
03

Solve for the hypotenuse

To get the hypotenuse on its own, we can rearrange the equation to get \(hypotenuse = \frac{500\ ft}{sin(10^{\circ})}\). This gives us the total distance the plane has flown.
04

Calculate the final answer

By inserting the values into the calculable equation \(hypotenuse = \frac{500\ ft}{sin(10^{\circ})}\), we can find the final answer.

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