A person standing 600 ft from the base of a mountain measures the angle of elevation from the ground to the top of the mountain to be 25°. The person then walks 800 ft straight back and measures the angle of elevation to now be 20°. How tall is the mountain?
let the height of the mountain be h ft
let the distance from the base of the mountain to its vertical line h be x ft (inside the mountain)
You have two right-angled triangles.
From the first: tan25 = h/(x+600)
h = (x+600)tan25
From the 2nd: tan 20 = h/(x+1400)
h = (x+1400)tan20
(x+1400)tan20 = (x+600)tan25
xtan20 + 1400tan20 = xtan25 + 600tan25
xtan25 - xtan20 = 1400tan20 - 600tan25
x(tan25-tan20) = 1400tan20-600tan25
x =(1400tan20-600tan25)/(tan25-tan20)
= ... ***
I will let you do the button-pushing, do not round off
then h = (*** + 600)tan25
= .....
This is the traditional way to do this problem.
Let me know if you want a "slicker" way to do this.
To find the height of the mountain, we can use trigonometry. Let's break down the problem step-by-step:
1. Draw a diagram: Draw a triangle where the base represents the distance from the person to the mountain, and the height represents the height of the mountain. Label the angle of elevation from the first position as 25° and the angle of elevation from the second position as 20°.
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h | \
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|_____\
600 ft 800 ft
2. Identify the trigonometric ratios: In this case, we are given the opposite side (height of the mountain) and the adjacent side (base or distance from the person to the mountain). We also know the values of the angles of elevation. We'll use the tangent ratio since we have the opposite and adjacent sides.
tan(θ) = opposite/adjacent
3. Calculate the height using the first position: Using the angle of elevation of 25°, we can set up the equation as follows:
tan(25°) = h/600 ft
Rearranging the equation to solve for h:
h = 600 ft * tan(25°)
Calculate h:
h ≈ 600 ft * (0.4663) ≈ 279.78 ft (rounded to two decimal places)
Therefore, the height of the mountain from the first position is approximately 279.78 ft.
4. Calculate the remaining distance: The person walks 800 ft straight back, so the remaining distance is the initial distance minus the distance walked back, which is:
Remaining distance = 600 ft - 800 ft = -200 ft (negative value indicates going below the base)
5. Calculate the new height using the second position: The new height can be determined using the angle of elevation of 20° from the second position. Using the same trigonometric ratio (tangent), we have:
tan(20°) = h/(-200 ft)
Rearranging the equation to solve for h:
h = (-200 ft) * tan(20°)
Calculate h:
h ≈ (-200 ft) * (0.3639) ≈ -72.78 ft (rounded to two decimal places)
Therefore, the height of the mountain from the second position is approximately -72.78 ft.
6. Calculate the total height of the mountain: The total height of the mountain can be found by adding the heights from both positions:
Total height = Height from first position + Height from second position
Total height ≈ 279.78 ft + (-72.78 ft) ≈ 207 ft (rounded to two decimal places)
Therefore, the height of the mountain is approximately 207 ft.
So, the mountain is approximately 207 ft tall.
To find the height of the mountain, we can use trigonometry. Let's break down the problem step by step.
Step 1: Draw a diagram
Visualize the situation by drawing a diagram. Draw a right triangle where the base represents the distance from the person to the mountain, the height represents the height of the mountain, and the hypotenuse represents the line of sight.
Step 2: Label the given measurements
Label the distance from the person to the mountain as 600 ft and the angle of elevation as 25°. Also, label the new distance as 800 ft and the new angle of elevation as 20°.
Step 3: Identify the trigonometric ratios
In this case, we are dealing with the tangent ratio. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side (tangent = opposite/adjacent).
Step 4: Solve for the height of the mountain
Using the tangent ratio, we can set up two separate equations:
Equation 1:
tan(25°) = height/600 ft
Equation 2:
tan(20°) = height/800 ft
Step 5: Solve the equations
Let's solve Equation 1 first:
tan(25°) = height/600 ft
Rearrange the equation to solve for the height:
height = tan(25°) × 600 ft
Using a calculator, find the value of tan(25°) and multiply it by 600 ft to get the height.
Next, solve Equation 2:
tan(20°) = height/800 ft
Rearrange the equation to solve for the height:
height = tan(20°) × 800 ft
Using a calculator, find the value of tan(20°) and multiply it by 800 ft to get the height.
Step 6: Compare the results
Check if the height obtained from both equations is the same. If they are, then you have the correct answer for the height of the mountain. If they are different, review your calculations and ensure you input the correct values.
That's it! By following these steps, you can find the height of the mountain based on the given measurements and trigonometric ratios.