QuestionFind the height of a mountain peak given a distance of 27.2193 miles and an angle of elevation of $5.75^{\circ}$ .

Studdy Solution

STEP 1

Assumptions1. The altitude of the observer is14,539 feet. The straight-line distance to the peak of the mountain is27.2193 miles3. The peak's angle of elevation is $5.75^{\circ}$
4. We neglect the curvature of the earth5. We will use the tangent of the angle of elevation to find the height of the mountain6.1 mile =5280 feet

STEP 2

First, we need to convert the distance from miles to feet.
$Distance\, in\, feet = Distance\, in\, miles \\times Conversion\, factor$

STEP 3

Now, plug in the given values for the distance in miles and the conversion factor to calculate the distance in feet.
$Distance\, in\, feet =27.2193\, miles \\times5280\, feet/mile$

STEP 4

Calculate the distance in feet.
$Distance\, in\, feet =27.2193\, miles \\times5280\, feet/mile =143,677.984\, feet$

STEP 5

Now that we have the distance in feet, we can use the tangent of the angle of elevation to find the height of the mountain above the observer's altitude. The formula is $Height\, above\, observer = Distance\, in\, feet \\times \tan(Angle\, of\, elevation)$

STEP 6

Plug in the values for the distance in feet and the angle of elevation to calculate the height above the observer.
$Height\, above\, observer =143,677.984\, feet \\times \tan(5.75^{\circ})$

STEP 7

Calculate the height above the observer.
$Height\, above\, observer =143,677.984\, feet \\times \tan(5.75^{\circ}) =14,363.5\, feet$

STEP 8

Now that we have the height above the observer, we can add this to the observer's altitude to find the total height of the mountain.
$Total\, height = Observer's\, altitude + Height\, above\, observer$

STEP 9

Plug in the values for the observer's altitude and the height above the observer to calculate the total height.
$Total\, height =14,539\, feet +14,363.5\, feet$

STEP 10

Calculate the total height of the mountain.
$Total\, height =14,539\, feet +14,363.5\, feet =28,902.5\, feet$

STEP 11

Round the total height to the nearest foot.
$Total\, height =28,903\, feet$ The height of the mountain is approximately28,903 feet.

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