The vertical height of a large square based solid pyramid is 30m. Its mass is 16000 tonnes. If the mass of the smaller pyramid is 2000 tones, calculate its vertical height.
Is the mass taken as volume or area the formula im trying to use is AREA/area =k^2
volume is dependent on the mass
the volume of similar solids varies directly as the cube of their heights, so
16000/2000 = 30^3/h^3
8 = 27000/h^3
take cube root of both sides
2 = 30/h
h = 15 ------> the height of the smaller is 15 m
check:
let the base of larger be x by x
(1/3)x(x)(30) = 16000
x^2 = 1600
x = appr 40
ratio of heights = 30 : 15 = 2 : 1
so ratio of base sides = 40 : s
2/1 = 40/s
2s = 40
s = 20
volume of larger = (1/3)(40^2)(30) = 1600
volume of smaller = (1/3)(20^2)(15) = 2000
To calculate the vertical height of the smaller pyramid, we can use the concept of similar shapes. Similar shapes have proportional sides.
Given:
- The vertical height of the large pyramid is 30m.
- The mass of the large pyramid is 16000 tonnes.
- The mass of the smaller pyramid is 2000 tonnes.
We need to find the vertical height of the smaller pyramid.
Step 1: Find the ratio of masses.
The ratio of masses is equal to the ratio of volumes since density (mass/volume) remains constant. Therefore, the ratio of masses is equal to the ratio of the cubes of the corresponding sides.
mass ratio = (mass of large pyramid) / (mass of small pyramid)
mass ratio = 16000 tonnes / 2000 tonnes
mass ratio = 8
Step 2: Find the ratio of heights.
Since the height of the large pyramid is 30m, we can assume the sides of the large pyramid are also in a ratio of 8:1 (height ratio).
height ratio = 8:1
Step 3: Calculate the vertical height of the smaller pyramid.
Let the vertical height of the smaller pyramid be 'h'.
Using the height ratio, we can write the proportion: 30m / h = 8 / 1
Therefore, 30 / h = 8 / 1
Cross-multiplying, we get 30 = 8h
Dividing both sides by 8, we get h = 30 / 8
Simplifying, h = 3.75m
So, the vertical height of the smaller pyramid is 3.75m.