The base of the Great Pyramid at Giza is a square that is 230 m on each side.
1.Find the height of the pyramid, knowing that when a person is standing at the center of its side he measures the angle of elevation of the apex to be 51 degrees with an error of +/- 0.5 degrees.
2. Build the model of the Great Pyramid at Giza using calculations from #1.
3. Use differentials to estimate the allowable error in the elevation angle that will ensure an error in the height is at most +/- 5m.
1. h = 115 tan51° = 142 m
with error, 139.51 <= h <= 144.57
2. sorry, you'll have to do that yourself!
3.
h = 115 tanθ
dh = 115 sec^2 θ dθ
5 = 115 * 2.525 dθ
dθ = .01722 = 0.986°
just as a check, our error of ±0.5° indicates a height error of
dh = 115*2.525*0.5*π/180 = ±2.53m which agrees quite nicely with the explicit values found above
The answer is 1 in. = 2.5 ft
You're welcome.
1. To find the height of the pyramid, we can visualize a right triangle with the base as one side, the height of the pyramid as the other side, and the line connecting the center of the base to the apex as the hypotenuse.
Using trigonometry, we can use the tangent function to find the height. Let's call the height of the pyramid "h."
tangent(angle) = opposite/adjacent
tan(51°) = h/230
We can rearrange the equation to solve for the height:
h = tan(51°) * 230
Using a calculator, we find that tan(51°) is approximately 1.3077.
So, the height of the pyramid is:
h = 1.3077 * 230
Therefore, the height of the pyramid is approximately 299.71 meters.
2. To build a model of the Great Pyramid at Giza using the calculated height of 299.71 meters, you would construct a scaled-down replica. Let's say you choose a scale of 1:100.
To determine the dimensions of the model, you would divide the real dimensions by the scale factor:
Base side length of the model = 230m / 100 = 2.3m
Height of the model = 299.71m / 100 = 2.9971m
Therefore, the model of the Great Pyramid at Giza would have a base side length of 2.3 meters and a height of 2.9971 meters.
3. To estimate the allowable error in the elevation angle that ensures the height error is at most +/- 5 meters, we can use differentials.
Let's assume that the allowable error in the height of the pyramid is Δh = 5 meters.
We can express the relationship between the height and the elevation angle using the tangent function:
tan(angle) = height/base
To estimate the allowable error in the angle, Δ(angle), we differentiate the tangent function with respect to the angle:
d(tan(angle))/d(angle) = 1/(cos^2(angle))
Since we are given an error of +/- 0.5 degrees in the angle, we can use the differential to estimate the maximum error in the height:
Δh ≈ d(tan(angle))/d(angle) * Δ(angle) * base
Plugging in the values:
Δh ≈ (1/(cos^2(51°))) * (0.5°) * 230
Calculating the cosine squared of 51 degrees:
cos^2(51°) ≈ 0.6090
Therefore, the maximum allowable error in the elevation angle is approximately:
Δ(angle) ≈ Δh / [(1/(cos^2(51°))) * base]
Plugging in the values:
Δ(angle) ≈ 5 / [(1/0.6090) * 230]
Calculating the value, we find:
Δ(angle) ≈ 0.2294 degrees
Therefore, an error in the elevation angle of at most +/- 0.2294 degrees will ensure an error in the height of the pyramid is at most +/- 5 meters.
1. To find the height of the pyramid, we can use trigonometry. Let's denote the height of the pyramid as 'h'. According to the given information, when a person stands at the center of one side of the pyramid, the angle of elevation to the apex is 51 degrees. We need to take into account the error of +/- 0.5 degrees in our calculations.
First, let's consider the right triangle formed by the person, the apex, and the center of the base. The angle of elevation is the angle between the horizontal ground and the line connecting the person to the apex.
Using the tangent function, we can write:
tan(51) = h/d
where 'd' is the distance from the person to the apex. In this case, 'd' is half the length of one side of the square base, which is 230/2 = 115 meters.
So, we have:
tan(51) = h/115
To take the error into account, we will calculate the height for both the maximum and minimum angles of elevation:
For the maximum angle (51 + 0.5 degrees):
tan(51 + 0.5) = h/115
For the minimum angle (51 - 0.5 degrees):
tan(51 - 0.5) = h/115
Now we have three equations that we can solve to find the height of the pyramid.
2. To build a model of the Great Pyramid at Giza using the calculations from step 1, you will need to gather the necessary materials, such as cardboard, ruler, pencil, scissors, and glue. Start by drawing a square base with sides of 230 meters on the cardboard. Use the height value obtained in step 1 to construct the apex of the pyramid. You can then cut out the shape and assemble it by folding along the edges and using the glue to secure the corners. Adjust the proportions accordingly to create a visually accurate model.
3. To estimate the allowable error in the elevation angle that will ensure an error in the height is at most +/- 5 meters, we can use differentials. Recall that differentials measure the approximate change in a function given small changes in the independent variable.
Let's denote the allowable error in the elevation angle as 'x' (in degrees). From step 1, we have:
h = 115 * tan(51 + x)
h = 115 * tan(51 - x)
By taking the differentials of both equations, we get:
dh = 115 * sec^2(51 + x) * dx
dh = 115 * sec^2(51 - x) * dx
To estimate the allowable error in the height, we want to find the maximum change in 'h' that corresponds to a given change in 'x'. In this case, we want the height error to be at most +/- 5 meters. So, |dh| <= 5.
From the differentials equations, we can write:
|115 * sec^2(51 + x) * dx| <= 5
|115 * sec^2(51 - x) * dx| <= 5
By rearranging the equations, we can solve for 'dx':
dx <= (5 / (115 * sec^2(51 + x)))
dx <= (5 / (115 * sec^2(51 - x)))
These inequalities will provide an estimate of the allowable error in the elevation angle that ensures an error in the height is at most +/- 5 meters.