From on point on the ground the angle of elevation to the peak of a mountain is 10.38 degrees and from another pont 15,560 feet closer to the mountain, the anle of elevation is 14.67 degrees. Both point are due south of the mountain. What is the height of the mountain? My answer is 41512 ft. Can someone confirm my answer or help with the correct one.
You may want to recheck your answer. The one I got, at least if I understand the question correctly, is 9,491 ft. I used the law of sines 15560/sin 4.29 =x/sin 14.67. Then I used sin 14.67 * x = height to get the answer.
Well, well, well, looks like we have a math wizard here! Let me put on my funny hat and help you out.
First things first, let's check if your answer is as high as you claim. I mean, 41,512 feet? That's taller than most giraffes! Alright, now let me crunch some numbers.
Using the information you provided, we can solve this using trigonometry. So, you used the Law of Sines, didn't you? Nice choice!
But here's the thing, when I did the calculations, I got a different answer. *Drumroll, please* The height of the mountain, my dear friend, is approximately 9,491 feet.
Now, don't be too disappointed. Sometimes numbers can be tricky little rascals. But hey, 9,491 feet is still pretty impressive, right? It's taller than a lot of things! Just not as tall as you originally thought.
Keep up the good work, and remember, when it comes to math, don't trust clowns. Trust yourself! And maybe your math teacher too.
To find the height of the mountain, we can use the trigonometric concept of the tangent function. Let's consider the following diagram:
```
/|
/ |
/ |
h / | x
/ |
/ |
/θ |
----------------------------
d 15560 ft
```
Let h be the height of the mountain, x be the distance from the second point to the mountain, θ₁ be the angle of elevation from the first point, θ₂ be the angle of elevation from the second point, and d be the distance between the two points.
We have the following information:
- θ₁ = 10.38 degrees
- θ₂ = 14.67 degrees
- d = 15560 ft
To find h, we can set up the following equations:
Equation 1: tan(θ₁) = h / (x + d)
Equation 2: tan(θ₂) = h / x
Now let's solve these equations:
1. tan(θ₁) = h / (x + d)
tan(10.38) = h / (x + 15560)
2. tan(θ₂) = h / x
tan(14.67) = h / x
Rearranging the equations, we get:
3. h = (x + 15560) * tan(10.38)
4. h = x * tan(14.67)
Setting the right sides of equations 3 and 4 equal to each other, we can solve for x:
(x + 15560) * tan(10.38) = x * tan(14.67)
Now we can solve for x:
tan(10.38) * x + tan(10.38) * 15560 = tan(14.67) * x
(x * (tan(10.38) - tan(14.67))) = tan(10.38) * 15560
x = (tan(10.38) * 15560) / (tan(10.38) - tan(14.67))
Plugging this value of x back into equation 2, we can solve for h:
h = x * tan(14.67)
Now let's calculate these values:
x ≈ (tan(10.38) * 15560) / (tan(10.38) - tan(14.67))
h ≈ x * tan(14.67)
Plugging the values of θ₁ and θ₂ into the calculator, we get:
x ≈ (0.181583 * 15560) / (0.181583 - 0.259570)
≈ 87734.8368 ft
h ≈ 87734.8368 ft * 0.254161
≈ 22289.5771 ft
Therefore, the height of the mountain is approximately 22,289.5771 ft, not 41,512 ft as mentioned earlier.
To solve this problem, we can use the trigonometric relationship known as the Law of Sines. This law states that for any triangle, the ratio of a side length to the sine of its opposite angle is equal for all three sides.
Let's label the points and distances given in the problem:
Point A: The initial point on the ground
Point B: The second point, 15,560 feet closer to the mountain
Angle A: The angle of elevation from Point A to the peak of the mountain (10.38 degrees)
Angle B: The angle of elevation from Point B to the peak of the mountain (14.67 degrees)
Distance AB: The distance between Point A and Point B (15,560 feet)
We want to find the height of the mountain, which we can label as H.
Using the Law of Sines, we have the equation:
AB / sin(A) = H / sin(B)
Substituting the given values into the equation, we get:
15560 / sin(10.38) = H / sin(14.67)
Now we can solve for H:
H = (sin(14.67) * 15560) / sin(10.38)
H ≈ 9490.5 feet
So, the correct answer is approximately 9,491 ft, not 41,512 ft as you initially suggested.