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.
v = 1/3 * s^2 * h
multiply dimensions by 5:
V = 1/3 * (5s)^2 * (5h)
= 1/3 * 25s^2 * 5h
= 1/3 *25s^2 * 5h
= 125 * 1/3 * s^2 * h
= 125v
multiplying the dimensions of a figure by k causes
length to multiply by k
area by k^2
volume by k^3
To find the ratio of the volumes of the two pyramids, we first need to determine the volume of each pyramid. The volume of a pyramid can be calculated using the formula:
Volume = (1/3) * base area * height
Since both pyramids have a square base, the base area of the original pyramid and the larger pyramid are the same. Let's denote this base area as B.
The height of the original pyramid is given as 12 inches. Let's denote this height as h.
To calculate the volume of the original pyramid, we substitute the given values into the formula:
Volume_of_original_pyramid = (1/3) * B * h
Now, the carpenter increases the dimensions of the pyramid by a factor of 5. This means that every dimension of the original pyramid is multiplied by 5 to create the larger pyramid.
Therefore, the new height of the larger pyramid is 5 times the height of the original pyramid, which is 5h.
Now, let's calculate the volume of the larger pyramid using the increased dimensions:
Volume_of_larger_pyramid = (1/3) * B * (5h)
Now, to find the ratio of the volumes, we divide the volume of the larger pyramid by the volume of the original pyramid:
Ratio_of_volumes = (Volume_of_larger_pyramid) / (Volume_of_original_pyramid)
Plugging in the values we calculated earlier:
Ratio_of_volumes = [(1/3) * B * (5h)] / [(1/3) * B * h]
The base area B cancels out, as well as the (1/3) factor, simplifying the ratio:
Ratio_of_volumes = [(5h) / h]
Ratio_of_volumes = 5
Therefore, the ratio of the volumes of the two pyramids is 5.