To estimate the original volume of the pyramid, we need to first find the volume of the frustrum.
The height of the frustrum can be calculated by subtracting the height of the top from the height of the original pyramid:
h = h_original - h_top = h_original - 12m
Using the formula for the volume of a frustrum:
V_frustrum = (1/3) * h * (A_top + A_base + sqrt(A_top * A_base))
where A_top and A_base are the areas of the top and base of the frustrum, respectively.
A_top = 4m x 4m = 16m^2
A_base = 20m x 20m = 400m^2
Substituting in the values and simplifying:
V_frustrum = (1/3) * (h_original - 12m) * (16m^2 + 400m^2 + sqrt(16m^2 * 400m^2))
V_frustrum = (1/3) * (h_original - 12m) * (16m^2 + 400m^2 + 6400m^2)
V_frustrum = (1/3) * (h_original - 12m) * 6816m^2
V_frustrum = 2272m^2 * (h_original - 12m)
To estimate the original volume of the pyramid, we assume that the frustrum was created by cutting off the top of the original pyramid with a plane parallel to the base. This means that the ratio of the heights of the two sections is the same as the ratio of their volumes:
V_original / V_frustrum = (h_original / (h_original - 12m))^3
Rearranging and substituting in the value for V_frustrum:
V_original = V_frustrum * (h_original / (h_original - 12m))^3
V_original = 2272m^2 * (h_original - 12m) * (h_original / (h_original - 12m))^3
Since we only need an estimate, we can round off the height of the original pyramid to the nearest meter and use 8m (20m - 12m) as the height of the frustrum. This gives:
V_original = 2272m^2 * 8m * (h_original / 8m)^3
V_original = 18176m^3 * (h_original / 8m)^3
Using the base area of the original pyramid (20m x 20m) and the estimated height of the pyramid, we can check if our estimate is reasonable:
V_check = (1/3) * 20m * 20m * h_original
V_check = 13333m^3 * h_original
If we assume that the original pyramid was a regular pyramid with a square base, then the height would be given by:
h_original = sqrt(20m^2 - (4m^2 / 2))
h_original = sqrt(396)m
h_original ≈ 19.9m
Substituting this into our estimate for V_original:
V_original ≈ 18176m^3 * (19.9m / 8m)^3
V_original ≈ 287500m^3
This is reasonably close to our estimate using the base area and height, which suggests that our assumption of a regular pyramid was reasonable.
Estimate the original volume of the pyramid in fig.15.28, given that its frustrum has a 4 m by 4m square top which is 12 m vertically above the square base which is 20 m by 20m
1 answer