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!