A man 1.9m tall observes the angle of elevation of the top and bottom of a mast as 65° and 41° respectively.If the mast is moed on a house 12m tall,find correct to 3 significant figures I.distance of the man from the house and the length of the pillors
To solve this problem, we can utilize the properties of similar triangles and trigonometric ratios.
Let's begin by understanding the diagram. We have a man (M) who is 1.9m tall observing a mast (T) on a house (H) which is 12m tall. The angle of elevation from M to the top of the mast is 65°, and the angle of elevation from M to the bottom of the mast is 41°.
Now, let's calculate the distance of the man from the house (MH) and the length of the mast (TH).
Step 1: Calculate the distance of the man from the house (MH):
To find MH, we can use the tangent ratio. The tangent of an angle is equal to the opposite side divided by the adjacent side.
First, let's find the opposite side. Since we have the height of the house (OH) and the height of the man (OM), the opposite side is given by OH - OM:
Opposite side (OH - OM) = 12m - 1.9m = 10.1m
Now, let's find the adjacent side, which is MH.
Using the tangent ratio:
tan(41°) = opposite side / adjacent side
tan(41°) = 10.1m / MH
Now, rearrange the equation to solve for MH:
MH = 10.1m / tan(41°)
Calculating MH using the above formula:
MH ≈ 10.1m / tan(41°) ≈ 10.1m / 0.869 ≈ 11.618m
Therefore, the distance of the man from the house, correct to 3 significant figures, is approximately 11.618m.
Step 2: Calculate the length of the mast (TH):
To find TH, we can use the tangent ratio similar to step 1. The opposite side is now the height of the mast (TH), and the adjacent side is still MH.
Using the tangent ratio:
tan(65°) = opposite side / adjacent side
tan(65°) = TH / 11.618m
Now, rearrange the equation to solve for TH:
TH = 11.618m * tan(65°)
Calculating TH using the above formula:
TH ≈ 11.618m * tan(65°) ≈ 11.618m * 2.144 ≈ 24.7m
Therefore, the length of the mast, correct to 3 significant figures, is approximately 24.7m.
In summary, the distance of the man from the house is approximately 11.618m, and the length of the mast is approximately 24.7m.
If the distance is x, then
10.1/x = tan41°
x = 10.1/tan41°
If the mast's height is h, then
(10.1+h)/x = tan65°
so h = x tan65° - 10.1
h = 10.1/tan41° * tan65° - 10.1 = 10.1 (tan65°/tan41° - 1)
no idea what the "pillors" are