A wooden pyramid, 12 inches tall, has a square base. A carpenter increases the dimensions of the wooden pyramid by a factor of 5 and makes a larger pyramid with the new dimensions. Describe in complete sentences the ratio of the volumes of the two pyramids.
the actual values of the dimensions make no difference.
If the original base sides are s, the height is h, then the volume v is
v = 1/3 s^2 * h
now, multiply each dimension by 5 and your new volume V is
V = 1/3 (5s)^2 * (5h)
= 1/3 * 25s^2 * 5h
= 125 * 1/3 s^2 * h
= 125v
or, 5^3 * v
scaling the dimensions of a figure by a factor of n means
area is scaled by n^2
volume is scaled by n^3
To find the ratio of the volumes of the two pyramids, we can use the fact that the volume of a pyramid is proportional to the cube of its dimensions.
First, let's determine the dimensions of the larger pyramid. Since the carpenter increased the dimensions of the original wooden pyramid by a factor of 5, the dimensions of the larger pyramid would be 5 times the dimensions of the original pyramid.
Since the original pyramid has a height of 12 inches, the larger pyramid would have a height of 5 times 12, which is 60 inches.
The base dimensions of the original pyramid are not given, so we don't have specific values to work with. However, we know that the base of the larger pyramid would be 5 times the base of the original pyramid.
With this information, we can now determine the ratio of the volumes of the two pyramids. Let's denote the ratio of the volumes as R.
R = (Volume of larger pyramid) / (Volume of original pyramid)
Since the volume is proportional to the cube of the dimensions, we can write the ratio of the volumes as:
R = (5^3 * (Base of larger pyramid)^2 * 60) / (1^3 * (Base of original pyramid)^2 * 12)
Simplifying, we have:
R = (5^3 * (5(Base of original pyramid))^2 * 60) / (1 * (Base of original pyramid)^2 * 12)
R = (5^3 * 5^2 * 60) / (12)
Simplifying further, we get:
R = (125 * 25 * 60) / 12
R = 1,500,000 / 12
R = 125,000
Therefore, the ratio of the volumes of the two pyramids is 125,000 to 1.