In traveling across flat land, you notice a mountain in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain.
Two rt triangles are formed. The common ver. side represents the ht. of
the mountain. The 3.5 deg angle of elevation lies bet. the hyp and hor side of the larger triangle. The 9 deg
angle is located bet. the hyp and hor side of the smaller triangle.
tan3.5 = h / (x + 13),
h = (x + 13)tan3.5,
h = (x + 13)0.0612,
Eq1: h = 0.0612x + 0.7951,
tan9 = h/x,
Cross multiply:
h = xtan9,
Eq2: h = 0.1584x.
Substitute 0.1584x for h in Eq1:
0.1584x = 0.0612x + 0.7951,
0.1584x - 0.0612x = 0.7951,
0.0962x = 0.7951,
x = 0.7951 / 0.0962 = 8.27.
h = 0.1584 * 8.27 = 1.31 mi = ht of mt.
Emily chooses two numbers.
She adds the two numbers together and divides the rusult by 2.
Her answer is 44
One Emily's numbers is 12
what is Emily's other numbers?
To approximate the height of the mountain, we can use trigonometry and the concept of similar triangles.
Let's denote the height of the mountain as "h" and the original distance between you and the mountain as "x".
From your vantage point, you form a right triangle with the mountain, where the height of the mountain is the opposite side and the distance between you and the mountain is the adjacent side.
Using trigonometry, we can write the following equation for the first observation (angle of elevation = 3.5 degrees):
tan(3.5°) = h / x
Similarly, after driving 13 miles closer to the mountain, the new distance between you and the mountain is (x - 13). Now, the angle of elevation is 9 degrees:
tan(9°) = h / (x - 13)
We can set up a system of equations to solve for both h and x.
First, rearrange the equations:
h = x * tan(3.5°)
h = (x - 13) * tan(9°)
Since both equations equal to h, we can set them equal to each other:
x * tan(3.5°) = (x - 13) * tan(9°)
Now, we can solve for x.
x * tan(3.5°) = x * tan(9°) - 13 * tan(9°)
x * (tan(3.5°) - tan(9°)) = -13 * tan(9°)
x = (-13 * tan(9°)) / (tan(3.5°) - tan(9°))
Now that we have found the value of x, we can substitute it back into one of the original equations to find the height of the mountain (h):
h = x * tan(3.5°)
Calculating x gives us:
x ≈ 89.437 miles (rounded to three decimal places)
Substituting this value for x into the equation for h:
h ≈ 89.437 miles * tan(3.5°)
h ≈ 6.536 miles (rounded to three decimal places)
Therefore, the approximate height of the mountain is 6.536 miles.