The angle of elevation of the top of the spire from the foot of a building is 60°.The angle of depression of the top of the building from the top of the spire is 47.7°.Given that the height is 15 metres, calculate to one decimal place:
(a)the distance between the spire and the building(4marks)
(b)The difference in height between the spire and the building(2marks)
(c)The height of the spire(2marks)
(d)The angle of depression of the spire from the top of the building(2marks)
when you say "the height is 15 m" I assume you mean the height of the building, since you later ask for the height of the spire. SO, if we let
h = height of spire
x = distance between
Then things are pretty straightforward.
h/x = tan60°
(h-15)/x = tan47.7°
eliminate x, and you have
h/tan60° = (h-15)/tan47.7°
That gives you
h = 41.03m
x = 23.69m
Now the questions can easily be answered. Post your work if you get stuck.
Sure, let me use my clownish math skills to help you with these calculations. Get ready for some laughter and answers!
(a) To find the distance between the spire and the building, we need to use trigonometry. Since the angle of elevation is 60°, we can assume a right-angled triangle. Let's label the distance as "x." Using the tangent function, we can write: tan(60°) = height of the building / x. Given that the height of the building is 15 meters, we can solve for x using the equation: x = height of the building / tan(60°). Plugging in the values and doing some funny math, we get x = 15 / tan(60°) ≈ 8.2 meters.
(b) To find the difference in height between the spire and the building, we simply subtract the height of the building from the height of the spire. Since the height of the building is given as 15 meters, we can say the difference is the height of the spire minus 15 meters. Since we don't know the height of the spire yet, please hold your laughter until we get to the next question!
(c) To find the height of the spire, we can use the angle of depression of 47.7°. Again, let's assume a right-angled triangle, with the height of the spire being "h." We can use the tangent function: tan(47.7°) = height of the building / x. Plugging in the angle and the value we found for x in part (a), we get tan(47.7°) ≈ (height of the spire - 15) / 8.2. By rearranging the equation, we can find the height of the spire: height of the spire = tan(47.7°) * 8.2 + 15. Doing some funny math, the height of the spire is approximately 21.1 meters.
(d) The angle of depression of the spire from the top of the building can be found by using the same right-angled triangle we used in part (c). The angle we need to find is opposite to the height of the spire, so we use the tangent function again: tan(θ) = height of the spire / x. We know the height of the spire is 21.1 meters, and from part (a), we found that x is approximately 8.2 meters. Plugging in the values, we get tan(θ) ≈ 21.1 / 8.2. By taking the inverse tangent, we can find θ. Get ready for some hilarious math: θ ≈ arctan(21.1 / 8.2) ≈ 68.7°.
I hope you had a good laugh while solving these problems!
To solve this problem, we can use trigonometric ratios such as tangent and sine. Let's start with part (a).
(a) To find the distance between the spire and the building, we can use the tangent function. Let's call this distance "d".
Using the given information:
The angle of elevation from the foot of the building to the top of the spire is 60°.
Therefore, the tangent of this angle is equal to the opposite side (the height of the building) divided by the adjacent side (the distance between the spire and the building).
tan(60°) = height of the building / d
√3 = 15 / d
To solve for "d", we can cross-multiply:
d * √3 = 15
d = 15 / √3
d ≈ 8.7 meters
Therefore, the distance between the spire and the building is approximately 8.7 meters.
(b) To find the difference in height between the spire and the building, we can subtract the height of the building from the height of the spire.
Given that the height of the building is 15 meters, we need to find the height of the spire. Let's call the height of the spire "h".
Using the tangent function again:
The angle of depression from the top of the building to the top of the spire is 47.7°.
Therefore, the tangent of this angle is equal to the opposite side (the difference in heights) divided by the adjacent side (the distance between the spire and the building).
tan(47.7°) = difference in heights / d
0.9721 = (h - 15) / 8.7
To solve for "h", we can cross-multiply and solve the equation:
0.9721 * 8.7 = h - 15
8.4574 = h - 15
h = 8.4574 + 15
h ≈ 23.5 meters
So, the height of the spire is approximately 23.5 meters.
Now we can calculate the difference in height between the spire and the building:
difference in heights = height of the spire - height of the building
difference in heights = 23.5 - 15
difference in heights ≈ 8.5 meters
Therefore, the difference in height between the spire and the building is approximately 8.5 meters.
(c) We have already calculated the height of the spire in part (b), which is approximately 23.5 meters.
(d) To find the angle of depression of the spire from the top of the building, we can use the tangent function once again.
Given that the height of the building is 15 meters and the distance between the spire and the building is 8.7 meters, we want to find the angle θ.
tan(θ) = opposite side (height of the building) / adjacent side (distance between spire and building)
tan(θ) = 15 / 8.7
To find the angle θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(15 / 8.7)
θ ≈ 59.7°
Therefore, the angle of depression of the spire from the top of the building is approximately 59.7°.
To solve this problem, we can use the principles of trigonometry. Let's assign some variables:
- Let x represent the distance between the spire and the building.
- Let h represent the difference in height between the spire and the building.
(a) To find the distance between the spire and the building:
Using the definition of the tangent function, we have:
tan(60°) = h / x
√3 = h / x (since tan(60°) = √3)
To solve for x, we need to isolate it:
x = h / √3
(b) To find the difference in height between the spire and the building:
We are given that the height is 15 meters, so we have:
h = 15 meters
(c) To find the height of the spire:
The height of the spire is given as 15 meters, which is the value of h.
(d) To find the angle of depression of the spire from the top of the building:
Using the definition of the tangent function, and the fact that the angle of depression is the complement of the angle of elevation, we have:
tan(90° - 47.7°) = h / x
tan(42.3°) = 15 / x
To solve for x, we need to isolate it:
x = 15 / tan(42.3°)
Now, let's calculate the values:
(a) The distance between the spire and the building:
x = 15 meters / √3
x ≈ 8.7 meters
(b) The difference in height between the spire and the building:
h = 15 meters
(c) The height of the spire:
h = 15 meters
(d) The angle of depression of the spire from the top of the building:
alpha = arctan(15 / x) = arctan(15 / 8.7)
alpha ≈ 59.4°
Therefore, to one decimal place:
(a) The distance between the spire and the building is approximately 8.7 meters.
(b) The difference in height between the spire and the building is 15 meters.
(c) The height of the spire is 15 meters.
(d) The angle of depression of the spire from the top of the building is approximately 59.4°.