Question

(a) find the following side lengths for the net. a = 3 cm b = 10 cm c = 4 cm d = 5 cm (b) use the net to find the lateral surface area of the prism. neither the top nor bottom is included. (c) use the net to find the total surface area of the prism.

Explanation:

Step1: Identify the prism type

This is a triangular prism. The lateral surface area (LSA) of a triangular prism is calculated as the perimeter of the triangular base multiplied by the length of the prism.
First, find the perimeter of the triangular base. The triangle has sides \( A = 3\space cm \), \( C = 4\space cm \), and we can find the third side (hypotenuse) using Pythagoras, but wait, actually from the net, the triangular base has sides 3, 4, and 5 (since \( 3^2 + 4^2 = 5^2 \), so it's a right triangle). Wait, no, the length of the prism is \( B = 10\space cm \)? Wait, no, the lateral faces: the triangular prism's lateral faces are rectangles. The triangular base has sides: let's check the given values. \( A = 3\space cm \), \( C = 4\space cm \), \( D = 5\space cm \) (wait, maybe \( D \) is the hypotenuse? Wait, the problem says "lateral surface area" and "neither the top nor bottom is included". So lateral surface area of a triangular prism is the sum of the areas of the three rectangular faces.
The triangular base has sides: let's see, the triangle has base \( A = 3\space cm \), height \( C = 4\space cm \), and hypotenuse \( D = 5\space cm \). The length of the prism (the distance between the two triangular bases) is \( B = 10\space cm \)? Wait, no, the lateral faces: each lateral face is a rectangle with one side as the side of the triangle and the other as the length of the prism. Wait, maybe the prism's length is 10 cm, and the triangular base has sides 3, 4, 5? Wait, no, the given values: \( A = 3\space cm \), \( B = 10\space cm \), \( C = 4\space cm \), \( D = 5\space cm \). So the triangular base has sides 3, 4, 5 (since 3-4-5 triangle). Then the lateral surface area is the sum of the areas of the three rectangles:
- Rectangle 1: side \( A = 3\space cm \) and length \( B = 10\space cm \)? Wait, no, maybe the length of the prism is 10 cm, and the three sides of the triangle are 3, 4, 5. Wait, no, the lateral surface area of a triangular prism is \( LSA = (\text{Perimeter of triangular base}) \times \text{length of the prism} \). Wait, the perimeter of the triangular base: \( 3 + 4 + 5 = 12\space cm \). Then the length of the prism (the distance between the two triangles) is 10 cm? Wait, no, maybe I got it wrong. Wait, the net: the triangular parts are the bases, and the rectangles are the lateral faces. The lateral faces: one rectangle with dimensions \( 3 \times 10 \), one with \( 4 \times 10 \), and one with \( 5 \times 10 \)? Wait, no, the problem says "neither the top nor bottom is included", so lateral surface area is the area of the three rectangular faces. Wait, but the given \( B = 10\space cm \), \( A = 3\space cm \), \( C = 4\space cm \), \( D = 5\space cm \). Wait, maybe the length of the prism is 10 cm, and the three sides of the triangle are 3, 4, 5. So the lateral surface area is \( (3 + 4 + 5) \times 10 \)? Wait, no, that would be if the length is 10, but wait, maybe the length is 10, and the sides of the triangle are 3, 4, 5. Wait, but let's check the problem again: "Use the net to find the lateral surface area of the prism. Neither the top nor bottom is included." So lateral surface area of triangular prism: sum of areas of the three rectangles. The three rectangles have dimensions:
- One rectangle: \( 3 \times 10 \) (side A and length B)
- One rectangle: \( 4 \times 10 \) (side C and length B)
- One rectangle: \( 5 \times 10 \) (side D and length B)
Wait, but that would be \( 3\times10 + 4\times10 + 5\times10 = (3 + 4 + 5)\times10 = 12\times10 = 120 \). But wait, maybe I made a mistake. Wait, the triangular base: the two legs are 3 and 4, hypotenuse 5. The length of the prism (the distance between the two triangles) is 10. So the lateral faces are three rectangles: each with height 10 and width equal to the sides of the triangle. So area of each rectangle: 3*10, 4*10, 5*10. Sum: 30 + 40 + 50 = 120. Wait, but let's confirm. Alternatively, the formula for LSA of triangular prism is \( LSA = \text{Perimeter of base} \times \text{height of prism} \). The base perimeter is 3 + 4 + 5 = 12, height of prism (the length) is 10. So \( 12 \times 10 = 120 \space cm^2 \). Wait, but maybe the length is not 10? Wait, the given \( B = 10\space cm \), which is the length of the prism. So yes, that makes sense.

Step2: Calculate the lateral surface area

Perimeter of triangular base \( P = 3 + 4 + 5 = 12\space cm \)
Length of the prism \( l = 10\space cm \)
Lateral Surface Area \( LSA = P \times l = 12 \times 10 = 120\space cm^2 \)

Wait, but wait, the problem says "neither the top nor bottom is included", so lateral surface area is correct. But let's check the given values again. The user provided \( A = 3\space cm \), \( B = 10\space cm \), \( C = 4\space cm \), \( D = 5\space cm \). So the three sides of the triangle are 3, 4, 5 (right triangle), and the length of the prism (the distance between the two triangles) is 10 cm. So the lateral surface area is the sum of the areas of the three rectangles:
- Area of rectangle with side 3 and length 10: \( 3 \times 10 = 30 \)
- Area of rectangle with side 4 and length 10: \( 4 \times 10 = 40 \)
- Area of rectangle with side 5 and length 10: \( 5 \times 10 = 50 \)
Sum: \( 30 + 40 + 50 = 120 \space cm^2 \)

Answer:

120