Question

the altitude of a mountain peak is measured as shown in the figure to the right. at an altitude of 14,576 feet on a different mountain, the straight-line distance to the peak of mountain a is 27.6178 miles and the peak’s angle of elevation is θ = 5.4100°. (a) approximate the height (in feet) of mountain a. (b) in the actual measurement, mountain a was over 100 mi away and the curvature of earth had to be taken into account. would the curvature of earth make the peak appear taller or shorter than it actually is? (a) the height of mountain a is approximately □ feet. (do not round until the final answer. then round to the nearest foot as needed.)

Explanation:

Part (a)

Step 1: Convert miles to feet

We know that 1 mile = 5280 feet. So, we convert the straight - line distance (27.6178 miles) to feet.
Let \(d = 27.6178\) miles. The conversion formula is \(d_{feet}=d_{miles}\times5280\).
\(d_{feet}=27.6178\times5280\) feet.

Step 2: Find the additional height using trigonometry

We can use the sine function to find the additional height (\(h_{additional}\)) above the 14,576 - foot altitude. The formula for the sine of an angle \(\theta\) in a right - triangle is \(\sin\theta=\frac{opposite}{hypotenuse}\). Here, the opposite side to the angle \(\theta = 5.4100^{\circ}\) is the additional height \(h_{additional}\) and the hypotenuse is the straight - line distance \(d_{feet}\). So, \(h_{additional}=d_{feet}\times\sin(\theta)\)
First, calculate \(d_{feet}=27.6178\times5280 = 27.6178\times5280=145822.0\) (approximate value, we will keep more precision for calculation).
Then, \(\theta = 5.4100^{\circ}\), so \(\sin(5.4100^{\circ})\approx\sin(5.41^{\circ})\). Using a calculator, \(\sin(5.41^{\circ})\approx0.0944\) (more precisely, using calculator input: \(\sin(5.4100^{\circ})\approx0.09443\))
\(h_{additional}=27.6178\times5280\times\sin(5.4100^{\circ})\)
\(27.6178\times5280 = 27.6178\times5280 = 145822.0\) (exact value: \(27.6178\times5280=27.6178\times5000 + 27.6178\times280=138089+7733.0=145822\))
\(h_{additional}=145822\times\sin(5.4100^{\circ})\)
\(\sin(5.4100^{\circ})\approx0.09443\)
\(h_{additional}=145822\times0.09443\approx145822\times0.09443 = 145822\times(0.09 + 0.004+0.0004 + 0.00003)=145822\times0.09+145822\times0.004 + 145822\times0.0004+145822\times0.00003=13123.98+583.288+58.3288 + 4.37466 = 13769.97146\)

Step 3: Calculate the total height of Mountain A

The total height \(H\) of Mountain A is the sum of the initial altitude (14,576 feet) and the additional height \(h_{additional}\).
\(H=14576 + h_{additional}\)
\(H = 14576+13769.97146=28345.97146\approx28346\) (rounded to the nearest foot)

When we consider the curvature of the Earth, the line of sight from the observer to the peak of Mountain A is not a straight line in the Euclidean sense (on a flat plane) but a line that follows the curve of the Earth. The Earth's curvature causes the observer's line of sight to "dip" slightly. So, when we calculate the height using the flat - Earth (Euclidean) model, we are over - estimating the additional height (because the actual horizontal distance is slightly longer than the straight - line distance on a flat plane, and the angle of elevation is affected). In other words, the curvature of the Earth makes the peak appear shorter than it actually is because the non - flat nature of the Earth's surface means that the vertical component (the additional height we calculate) is less than what we would get from the flat - triangle model.

Answer:

28346

Part (b)