Question
A rectangular prism has a square base. The height of the prism is equal to twice the edge length of its base. The base can be completely covered, with no overlap, by 9 squares that each have an edge length of 1 inch.
What is the total surface area of the rectangular prism?
.
.
.
Question content area top right
Part 1
Question content area bottom
Part 1
The surface area of the figure is
enter your response here in squared.
What is the total surface area of the rectangular prism?
.
.
.
Question content area top right
Part 1
Question content area bottom
Part 1
The surface area of the figure is
enter your response here in squared.
Answers
GPT-4o mini
To determine the total surface area of the rectangular prism, we first need to find the dimensions of the prism based on the given information.
1. **Determine the edge length of the base**:
- The base is square and can be covered completely with 9 squares that each have an edge length of 1 inch.
- Therefore, the area of the base:
\[
\text{Area of base} = 9 \times (1 \text{ inch})^2 = 9 \text{ square inches}
\]
- Since the base is square, let \(s\) represent the edge length of the base. Then:
\[
s^2 = 9 \text{ square inches}
\]
- Solving for \(s\):
\[
s = \sqrt{9} = 3 \text{ inches}
\]
2. **Determine the height of the prism**:
- The height \(h\) of the prism is equal to twice the edge length of its base:
\[
h = 2s = 2 \times 3 \text{ inches} = 6 \text{ inches}
\]
3. **Calculate the total surface area**:
- The surface area \(A\) of a rectangular prism can be calculated using the formula:
\[
A = 2lw + 2lh + 2wh
\]
- For our prism, we have:
- Length \(l = s = 3\) inches
- Width \(w = s = 3\) inches
- Height \(h = 6\) inches
- Plugging these values into the formula:
\[
A = 2(3)(3) + 2(3)(6) + 2(3)(6)
\]
- Calculating each term:
\[
A = 2(9) + 2(18) + 2(18)
\]
\[
= 18 + 36 + 36
\]
\[
= 90 \text{ square inches}
\]
Thus, the total surface area of the rectangular prism is \( \boxed{90} \) square inches.
1. **Determine the edge length of the base**:
- The base is square and can be covered completely with 9 squares that each have an edge length of 1 inch.
- Therefore, the area of the base:
\[
\text{Area of base} = 9 \times (1 \text{ inch})^2 = 9 \text{ square inches}
\]
- Since the base is square, let \(s\) represent the edge length of the base. Then:
\[
s^2 = 9 \text{ square inches}
\]
- Solving for \(s\):
\[
s = \sqrt{9} = 3 \text{ inches}
\]
2. **Determine the height of the prism**:
- The height \(h\) of the prism is equal to twice the edge length of its base:
\[
h = 2s = 2 \times 3 \text{ inches} = 6 \text{ inches}
\]
3. **Calculate the total surface area**:
- The surface area \(A\) of a rectangular prism can be calculated using the formula:
\[
A = 2lw + 2lh + 2wh
\]
- For our prism, we have:
- Length \(l = s = 3\) inches
- Width \(w = s = 3\) inches
- Height \(h = 6\) inches
- Plugging these values into the formula:
\[
A = 2(3)(3) + 2(3)(6) + 2(3)(6)
\]
- Calculating each term:
\[
A = 2(9) + 2(18) + 2(18)
\]
\[
= 18 + 36 + 36
\]
\[
= 90 \text{ square inches}
\]
Thus, the total surface area of the rectangular prism is \( \boxed{90} \) square inches.