To determine the length of each side of a cube that has the same surface area as the given rectangular prism (travel-size cereal box), we must first calculate the surface area of the rectangular prism and then equate it to the surface area of a cube.
### Step 1: Calculate the surface area of the rectangular prism
The given dimensions of the rectangular prism are 3 inches, 1 inch, and 4 inches. The surface area \(SA\) of a rectangular prism is calculated using the formula:
\[ SA = 2(lw + lh + wh) \]
where \( l \), \( w \), and \( h \) are the length, width, and height of the prism, respectively.
Substitute \( l = 3 \) in, \( w = 1 \) in, and \( h = 4 \) in:
\[ SA = 2(3 \cdot 1 + 3 \cdot 4 + 1 \cdot 4) \]
\[ SA = 2(3 + 12 + 4) \]
\[ SA = 2(19) \]
\[ SA = 38 \, \text{square inches} \]
### Step 2: Calculate the side length of the cube
Let \( s \) be the length of each side of the cube. The surface area \( SA \) of a cube is given by:
\[ SA = 6s^2 \]
Since the surface area of the cube should be the same as that of the rectangular prism, we set:
\[ 6s^2 = 38 \]
Solve for \( s \):
\[ s^2 = \frac{38}{6} \]
\[ s^2 = \frac{19}{3} \]
\[ s^2 \approx 6.3333 \]
\[ s \approx \sqrt{6.3333} \]
\[ s \approx 2.52 \, \text{inches} \]
Given the choices:
- 2.3 inches
- 2.5 inches
- 3.5 inches
- 4 inches
The closest approximation is 2.5 inches.
### Therefore, the length of each side of the cube is **2.5 inches**.
7. A travel-size cereal box has dimensions of 3 in. x 1 in. x 4 in.
If it is redesigned to be a cube with the same surface area, what would be the length of each side in the cube?
*
2.3 inches
2.5 inches
3.5 inches
4 inches
1 answer
1 answer