Question
Julian buys a sandcastle building kit to gift to his niece. In the kit there are two similar cylindrical shapes of two different sizes. If the first cylinder has a surface area of 14 square inches and the larger cylinder has a surface area of 126 square inches, what is the scale factor between the two figures?
9
1/9
6
3
9
1/9
6
3
Answers
GPT-4o mini
To find the scale factor between the two cylinders based on their surface areas, we use the relationship between surface area and the scale factor for similar three-dimensional shapes.
The surface area of similar shapes scales by the square of the scale factor. Let \( k \) be the scale factor. Then the relationship between the surface areas is given by:
\[
\frac{\text{Surface Area of Larger Cylinder}}{\text{Surface Area of Smaller Cylinder}} = k^2
\]
Substituting the given surface areas:
\[
\frac{126}{14} = k^2
\]
Calculating the left side:
\[
\frac{126}{14} = 9
\]
So we have:
\[
9 = k^2
\]
To find \( k \), we take the square root of both sides:
\[
k = \sqrt{9} = 3
\]
Thus, the scale factor between the two figures is \( \boxed{3} \).
The surface area of similar shapes scales by the square of the scale factor. Let \( k \) be the scale factor. Then the relationship between the surface areas is given by:
\[
\frac{\text{Surface Area of Larger Cylinder}}{\text{Surface Area of Smaller Cylinder}} = k^2
\]
Substituting the given surface areas:
\[
\frac{126}{14} = k^2
\]
Calculating the left side:
\[
\frac{126}{14} = 9
\]
So we have:
\[
9 = k^2
\]
To find \( k \), we take the square root of both sides:
\[
k = \sqrt{9} = 3
\]
Thus, the scale factor between the two figures is \( \boxed{3} \).