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

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

Answer 1

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 2

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 3

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