Bob is driving along a straight and level road towards a mountain. At some point, he
measures the angle of elevation to the top of the mountain and finds it to be 22.5 degrees. He
then drives 1 mile (5280 ft) and measures the angle of elevation to be 34.4 degrees. Find the
height of the mountain to the nearest tenth of a foot.
Thank you!
draw a side view. You can see that if the mountain has height h,
h cot22.5° - h cot33.4° = 5280
To find the height of the mountain, we can use the tangent function and create a right triangle with the angle of elevation as one of the angles.
Let's define the following variables:
- h: height of the mountain
- x: horizontal distance from Bob's initial position to the base of the mountain
Now, we can set up the following equations:
Equation 1: tan(22.5 degrees) = h / x
Equation 2: tan(34.4 degrees) = h / (x + 5280 ft)
To solve these equations, we can isolate h in Equation 1 and substitute it into Equation 2:
h = x * tan(22.5 degrees)
Substituting this expression for h into Equation 2:
tan(34.4 degrees) = (x * tan(22.5 degrees)) / (x + 5280 ft)
Now we can solve for x:
tan(34.4 degrees) * (x + 5280 ft) = x * tan(22.5 degrees)
Simplifying:
(1) tan(34.4 degrees) * x + tan(34.4 degrees) * 5280 ft = x * tan(22.5 degrees)
Rearranging:
(2) x * (tan(34.4 degrees) - tan(22.5 degrees)) = tan(34.4 degrees) * 5280 ft
Now we can solve for x:
x = (tan(34.4 degrees) * 5280 ft) / (tan(34.4 degrees) - tan(22.5 degrees))
Let's calculate the value of x and use it to find the height of the mountain.
To find the height of the mountain, we can use trigonometry. Let's break down the problem and use the information provided step by step.
1. We know that Bob measures the angle of elevation to the top of the mountain from two different points along the road. Let's call the distance between the first measurement point and the second measurement point "x". We are given that x = 1 mile or 5280 feet.
2. In order to use trigonometry, we need to create a right triangle. To do this, we can consider the horizontal distance covered by Bob's car as the base of the triangle, and the height of the mountain as the height of the triangle.
3. Based on the first measurement, we know that the angle of elevation is 22.5 degrees. This angle is the angle between the line of sight from Bob to the top of the mountain and the horizontal line.
4. From trigonometry, we know that the tangent of an angle is equal to the ratio of the height of the triangle to the base of the triangle.
So, in this case, tan(22.5) = height of the mountain / 0 (the first measurement point is considered as the base of the triangle since the height at that point is zero).
As tan(22.5) is a known value, we can use a calculator to find its numerical value.
5. Now, let's move to the second measurement. After driving the distance of x (1 mile or 5280 feet), Bob measures the angle of elevation to be 34.4 degrees. This angle is also between the line of sight from Bob to the top of the mountain and the new horizontal line represented by the base of the triangle.
6. Since the base of the triangle has changed to x (5280 feet), we can use the tangent of the new angle to find the height of the mountain.
So, tan(34.4) = height of the mountain / x.
This equation can be simplified to height of the mountain = x * tan(34.4).
7. Now, we substitute the value of x (5280 feet) into the equation and calculate the numerical value of tan(34.4).
8. Finally, we multiply x (5280 feet) by the numerical value of tan(34.4) to find the height of the mountain. Rounding the result to the nearest tenth of a foot will give us the final answer.
By following these steps and using trigonometry, we can find the height of the mountain to the nearest tenth of a foot.