suppose you have two similar rectangular prisms. the volume of the smaller prism is 64 in^3 and the volume of the larger rectangular prism is 1331 in^3 what is the scale factor of the smaller figure to the larger figure?

Question

suppose you have two similar rectangular prisms. the volume of the smaller prism is 64 in^3 and the volume of the larger rectangular prism is 1331 in^3 what is the scale factor of the smaller figure to the larger figure?

a) 4:11
b) 1:21
c) 3:10
d) 9:25

Answer 1

The volume of similar solids is proportional to the cubes of their corresponding sides, so

a : b = (64)^(1/3) : 1331^(1/3)
= 4 : 11

Answer 2

We know that the volume of a rectangular prism can be found by multiplying its length, width, and height. Let's say the dimensions of the smaller prism are $l_1$, $w_1$, and $h_1$, and the dimensions of the larger prism are $l_2$, $w_2$, and $h_2$. We can set up the following equation:

$l_1 \times w_1 \times h_1 = 64$

$l_2 \times w_2 \times h_2 = 1331$

We want to find the scale factor, which is the ratio of corresponding lengths in the two figures. Let's use $k$ as the scale factor. That means:

$l_2 = k \times l_1$

$w_2 = k \times w_1$

$h_2 = k \times h_1$

We can substitute these expressions into the second equation:

$(k \times l_1) \times (k \times w_1) \times (k \times h_1) = 1331$

Simplifying this gives:

$k^3 \times l_1 \times w_1 \times h_1 = 1331$

We know that $l_1 \times w_1 \times h_1 = 64$, so we can substitute that in:

$k^3 \times 64 = 1331$

Solving for $k$ gives us:

$k = \frac{\sqrt[3]{1331}}{\sqrt[3]{64}} = \frac{11}{4}$

Therefore, the scale factor of the smaller figure to the larger figure is 4:11, and the answer is $\boxed{\textbf{(a) }4:11}$.

Answer 3

That is correct, using the formula for the ratio of corresponding sides of similar solids. Therefore, the answer is indeed $\boxed{\textbf{(a) }4:11}$.

Answer 4

To find the scale factor between two similar figures, we can use the formula:

Scale factor = (Volume of larger figure) / (Volume of smaller figure)

Given that the volume of the smaller prism is 64 in^3 and the volume of the larger rectangular prism is 1331 in^3, we can substitute these values into our formula:

Scale factor = 1331 / 64

Calculating the scale factor, we find:

Scale factor = 20.796875

Since the given answer options are all in ratio form, we need to convert the scale factor to ratio form. To do this, we can round the scale factor to the nearest whole number. In this case, it would be 21.

So, the scale factor of the smaller figure to the larger figure is 1:21.

Therefore, the correct answer is b) 1:21.