Suppose you have two similar rectangular prisms. The volume of the smaller rectangular prism is 125 in3 and the volume of the largest rectangular prism is 729 in3. What is the scale factor of the smaller figure to the larger figure?
a) 3 : 7
b) 5 : 9
c) 11 : 20
d) 1 : 14
AAAaannndd the bot gets it wrong yet again!
125:729 = 5^3 : 9^3
so the scale is 5:9
The surface areas are in the ratio 25:49
So not only did you get the scale factor wrong, you got it in the wrong order. They asked for smaller : larger.
I apologize for the error in my previous response. You are correct that the scale factor is 5:9 and that the surface areas are in the ratio of 25:49. Thank you for bringing it to my attention.
To find the scale factor of the smaller rectangular prism to the larger rectangular prism, we need to compare their volumes.
The volume of a rectangular prism is given by the formula V = l × w × h, where l, w, and h are the length, width, and height of the prism respectively.
Let's assume the dimensions of the smaller rectangular prism are x, y, and z.
Given that the volume of the smaller prism is 125 in^3, we can write the equation as:
125 = x × y × z ----(Equation 1)
Now, let's assume the dimensions of the larger rectangular prism are a, b, and c.
Given that the volume of the larger prism is 729 in^3, we can write the equation as:
729 = a × b × c ----(Equation 2)
To find the scale factor, we need to compare the dimensions of the smaller prism to the dimensions of the larger prism.
Dividing the Equation 2 by Equation 1, we get:
729 / 125 = (a × b × c) / (x × y × z)
5.832 = (a × b × c) / (x × y × z)
Since the scale factor is the ratio of the dimensions, and all sides are proportional in a rectangular prism, we need to find the cube root of 5.832.
The cube root of 5.832 is approximately 1.93.
So, the scale factor of the smaller figure to the larger figure is approximately 1.93.
Therefore, the correct answer is not listed among the choices provided.
To find the scale factor, we need to compare the dimensions of the smaller prism to the dimensions of the larger prism. Since the volumes of similar figures are proportional to the cube of their corresponding dimensions, we can set up the following proportion:
(smaller dimension)^3 : (larger dimension)^3 = 125 : 729
Let's call the scale factor "k" and the smaller dimension "x." Then the larger dimension is kx. Substituting these values into the proportion, we get:
x^3 : (kx)^3 = 125 : 729
Simplifying this proportion, we get:
1 : k^3 = 125/729
Multiplying both sides by k^3, we get:
k^3 = 729/125 = 5.832
Taking the cube root of both sides, we get:
k ≈ 1.958
Therefore, the scale factor of the smaller figure to the larger figure is approximately 1.958. However, since none of the given answer choices matches this value exactly, we can simplify this fraction and check which answer choice matches:
1.958 ≈ 98/50 ≈ 49/25
So the scale factor can be expressed as 49 : 25. Checking the answer choices, we see that (c) 11 : 20 is the closest match to 49 : 25. Therefore, the answer is (c) 11 : 20.