Question

fill in the blank.
a surveyor measures the angle of elevation to a point on a mountain to be 15 °. the point on the mountain is horizontally 6 miles away from the surveyor. the vertical change in elevation from the point where the surveyor is standing to the point on the mountain is ____ miles. (round your answer to the nearest hundredth of a mile.)
answer here

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the horizontal distance (adjacent side) is 6 miles, the vertical change (opposite side) is what we need to find, and the angle of elevation is \(15^\circ\). We use the tangent function, which is defined as \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation

Let \(y\) be the vertical change (opposite side). Then \(\tan(15^\circ)=\frac{y}{6}\).

Step3: Solve for \(y\)

Multiply both sides by 6: \(y = 6\times\tan(15^\circ)\). We know that \(\tan(15^\circ)\approx0.2679\). So \(y = 6\times0.2679\approx1.61\) (rounded to the nearest hundredth).

Answer:

\(1.61\)