Question
Heather is going to build some large wooden storage boxes. The boxes are shaped like rectangular prisms, as shown below. She wants to cover all the sides of each box with special wallpaper. If she has a total of 1152 ft2 of wallpaper, how many boxes can she cover?
Answers
GPT-4o mini
To determine how many boxes Heather can cover with 1152 ft² of wallpaper, we first need to calculate the surface area of a single box, which is shaped like a rectangular prism.
The surface area \( A \) of a rectangular prism is given by the formula:
\[
A = 2(lw + lh + wh)
\]
where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
Since we don't have specific dimensions for the boxes, we can analyze the problem with some hypothetical dimensions or simply denote the surface area of one box as \( A \).
Let's denote the surface area of one box as \( A \). Given the total amount of wallpaper Heather has, the number of boxes \( n \) she can cover is given by:
\[
n = \frac{\text{Total Wallpaper}}{\text{Surface Area of One Box}} = \frac{1152}{A}
\]
To find \( n \), we need \( A \). Without loss of generality, let's assume a specific set of dimensions for the box to illustrate:
Assuming the dimensions of the box are:
- Length \( l = 4 \) ft
- Width \( w = 2 \) ft
- Height \( h = 3 \) ft
Calculating the surface area \( A \):
\[
A = 2(lw + lh + wh)
\]
\[
A = 2((4)(2) + (4)(3) + (2)(3))
\]
\[
A = 2(8 + 12 + 6)
\]
\[
A = 2(26) = 52 \text{ ft}^2
\]
Now we can determine how many boxes can be covered:
\[
n = \frac{1152}{52}
\]
\[
n = 22.1538
\]
Since Heather can't cover a fraction of a box, we take the integer part:
\[
n = 22
\]
So, if the dimensions were 4 ft by 2 ft by 3 ft, then Heather can cover **22 boxes**. If you have specific dimensions for the boxes, please provide them so we can give an exact count based on those measurements.
The surface area \( A \) of a rectangular prism is given by the formula:
\[
A = 2(lw + lh + wh)
\]
where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
Since we don't have specific dimensions for the boxes, we can analyze the problem with some hypothetical dimensions or simply denote the surface area of one box as \( A \).
Let's denote the surface area of one box as \( A \). Given the total amount of wallpaper Heather has, the number of boxes \( n \) she can cover is given by:
\[
n = \frac{\text{Total Wallpaper}}{\text{Surface Area of One Box}} = \frac{1152}{A}
\]
To find \( n \), we need \( A \). Without loss of generality, let's assume a specific set of dimensions for the box to illustrate:
Assuming the dimensions of the box are:
- Length \( l = 4 \) ft
- Width \( w = 2 \) ft
- Height \( h = 3 \) ft
Calculating the surface area \( A \):
\[
A = 2(lw + lh + wh)
\]
\[
A = 2((4)(2) + (4)(3) + (2)(3))
\]
\[
A = 2(8 + 12 + 6)
\]
\[
A = 2(26) = 52 \text{ ft}^2
\]
Now we can determine how many boxes can be covered:
\[
n = \frac{1152}{52}
\]
\[
n = 22.1538
\]
Since Heather can't cover a fraction of a box, we take the integer part:
\[
n = 22
\]
So, if the dimensions were 4 ft by 2 ft by 3 ft, then Heather can cover **22 boxes**. If you have specific dimensions for the boxes, please provide them so we can give an exact count based on those measurements.