Question

the diagram shows the portion of earth you can see when you are on top of mount rainier at point b on a clear day, where mcd≈4°. what is your height above sea level? round to the nearest hundredth.

Explanation:

Step1: Use the tangent - secant relationship

Since \(BC\) and \(BD\) are tangent to the Earth (a circle with center \(A\)), \(\angle ACB=\angle ADB = 90^{\circ}\). Let \(AB = h + 4000\) (where \(h\) is the height above sea - level, \(AC = AD=4000\) (radius of the Earth), and \(\angle CAD = 4^{\circ}\). The central angle \(\angle CAD\) and the angle at \(B\) are related. We know that \(\angle CAB=\frac{1}{2}\angle CAD = 2^{\circ}\) (because the line from the center of the circle to the external point \(B\) bisects the angle between the two tangent segments).
In right - triangle \(ABC\), \(\cos\angle CAB=\frac{AC}{AB}\).

Step2: Substitute the values

We have \(\cos(2^{\circ})=\frac{4000}{4000 + h}\).
Since \(\cos(2^{\circ})\approx0.9994\), then \(0.9994=\frac{4000}{4000 + h}\).
Cross - multiply: \(0.9994(4000 + h)=4000\).
Expand: \(0.9994\times4000+0.9994h = 4000\).
\(3997.6+0.9994h = 4000\).
Subtract \(3997.6\) from both sides: \(0.9994h=4000 - 3997.6\).
\(0.9994h = 2.4\).
Solve for \(h\): \(h=\frac{2.4}{0.9994}\approx2.40\).

Answer:

\(h\approx2.40\) miles.