To solve the problem, we need to analyze the rectangular prism and its net. Given the dimensions specified, we can label the sides of the prism.
Given:
- Length (l) = 10 feet
- Width (w) = 2 feet
- Height (h) = 6 feet
(a) Finding Side Lengths for the Net
The net of a rectangular prism consists of six rectangles (the faces of the prism), and we need to label the lengths as follows:
-
A: This will be the length of the base (10 feet).
-
B: This is the height of the prism (6 feet).
-
C: This is the width of the base (2 feet).
-
D: This is also the height of the prism (6 feet), since we have two lateral faces that are rectangles with the height and width.
Thus, we have:
- A = 10 ft
- B = 6 ft
- C = 2 ft
- D = 6 ft
(b) Lateral Surface Area of the Prism
The lateral surface area (LSA) of a rectangular prism is given by the formula:
\[ \text{LSA} = 2h(l + w) \]
Substituting the values we have:
\[ \text{LSA} = 2 \cdot 6 \cdot (10 + 2) = 2 \cdot 6 \cdot 12 = 144 \text{ ft}^2 \]
(c) Total Surface Area of the Prism
The total surface area (TSA) of a rectangular prism can be calculated using the formula:
\[ \text{TSA} = 2lw + 2lh + 2wh \]
Using the given dimensions, we can substitute the values:
\[ \text{TSA} = 2 \cdot 10 \cdot 2 + 2 \cdot 10 \cdot 6 + 2 \cdot 2 \cdot 6 \]
Calculating each term:
- \(2 \cdot 10 \cdot 2 = 40\)
- \(2 \cdot 10 \cdot 6 = 120\)
- \(2 \cdot 2 \cdot 6 = 24\)
Thus,
\[ \text{TSA} = 40 + 120 + 24 = 184 \text{ ft}^2 \]
Final Answers:
- (a) A = 10 ft, B = 6 ft, C = 2 ft, D = 6 ft
- (b) Lateral Surface Area = 144 ft²
- (c) Total Surface Area = 184 ft²