Question

(a) find the following side lengths for the net.
a = 6 cm
b = 5 cm
c = 4 cm
d = 3 cm
(b) use the net to find the lateral surface area of the prism. neither the top nor bottom is included.

\boxed{\space} cm^2
(c) use the net to find the total surface area of the prism.

\boxed{\space} cm^2

Explanation:

Step1: Identify Lateral Surfaces

The lateral surface area of a triangular prism (excluding top and bottom) consists of the three rectangular faces. The dimensions of the rectangles: two with sides \( B = 5 \, \text{cm} \) and \( A = 6 \, \text{cm} \), and one with sides \( C = 4 \, \text{cm} \) and \( A = 6 \, \text{cm} \), and wait, no—wait, the net: the lateral faces (excluding the triangular bases) are the rectangles. Wait, the prism has a triangular base with legs \( 3 \, \text{cm} \) (D) and \( 4 \, \text{cm} \) (C), hypotenuse \( 5 \, \text{cm} \) (B), and length \( A = 6 \, \text{cm} \). Lateral surface area (LSA) of a prism is perimeter of the base (excluding top/bottom, but here the base is triangle, so lateral faces are rectangles with height \( A \) and widths equal to the sides of the triangle. Wait, no: the lateral surface area (excluding top and bottom triangles) would be the area of the three rectangles? Wait, no—the problem says "Neither the top nor bottom is included". Wait, the net: the two triangular faces (top and bottom) are excluded. So the lateral faces are the three rectangles? Wait, no, looking at the net: the middle rectangle is length \( A = 6 \), height \( B = 5 \)? Wait, no, the given values: \( A = 6 \, \text{cm} \), \( B = 5 \, \text{cm} \), \( C = 4 \, \text{cm} \), \( D = 3 \, \text{cm} \). Wait, the triangular base has legs \( D = 3 \) and \( C = 4 \), hypotenuse \( B = 5 \). The length of the prism is \( A = 6 \). So the lateral surface area (excluding top and bottom triangles) would be the area of the three rectangles: one with dimensions \( A \times B \), one \( A \times C \), one \( A \times D \)? Wait, no, looking at the net: the net has three rectangles? Wait, the left net: top rectangle (A x D?), middle rectangle (A x B?), bottom rectangle (A x D?) No, wait the given values: \( A = 6 \), \( B = 5 \), \( C = 4 \), \( D = 3 \). Wait, the lateral surface area (excluding top and bottom triangles) is the sum of the areas of the three rectangular faces. Wait, the formula for lateral surface area of a prism is perimeter of the base (the triangle) times the length of the prism? Wait, no, perimeter of the triangular base (3 + 4 + 5) = 12, times length \( A = 6 \)? But that would be if we include all three lateral faces. Wait, but the problem says "Neither the top nor bottom is included". So the lateral faces are the three rectangles: each with height \( A = 6 \) and width equal to the sides of the triangle (3, 4, 5). Wait, but let's check the net: the net has three rectangles? Wait, the left net: top rectangle (height D = 3, length A = 6), middle rectangle (height B = 5, length A = 6), bottom rectangle (height D = 3, length A = 6)? No, that doesn't make sense. Wait, maybe the triangular base is a right triangle with legs 3 and 4, hypotenuse 5. The prism has length 6. So the lateral faces are: two rectangles with dimensions 6 x 3, one rectangle 6 x 4, and one rectangle 6 x 5? No, no, a triangular prism has two triangular bases and three rectangular lateral faces. So the three lateral faces have areas: (3 x 6) + (4 x 6) + (5 x 6). Wait, that's (3 + 4 + 5) x 6 = 12 x 6 = 72? But wait, the problem says "Neither the top nor bottom is included", so that's exactly the lateral surface area (excluding the two triangular bases). Wait, but let's check the given values: \( A = 6 \), \( B = 5 \), \( C = 4 \), \( D = 3 \). So the three rectangular faces: one with length A and width B (5 x 6), one with A and C (4 x 6), one with A and D (3 x 6). Wait, but in the net, maybe the two rectangles with D and one with B? No, the triangular base has sides 3, 4, 5. So the three lateral faces are 3x6, 4x6, 5x6. So sum them up: (3*6) + (4*6) + (5*6) = 18 + 24 + 30 = 72. Wait, but let's confirm the net. The net shows three rectangles? Wait, the left net: top rectangle (A x D), middle rectangle (A x B), bottom rectangle (A x D)? No, that would be two rectangles with D and one with B, but the triangle has sides 3,4,5. Wait, maybe the triangular base is right-angled with legs 3 and 4, hypotenuse 5. So the three lateral faces are: 3x6 (along the 3 side), 4x6 (along the 4 side), and 5x6 (along the 5 side). So adding those: 3*6=18, 4*6=24, 5*6=30. Total: 18+24=42, 42+30=72. So the lateral surface area is 72 cm².

Step2: Calculate Each Rectangle's Area

- Area of rectangle with sides \( A = 6 \) and \( B = 5 \): \( 6 \times 5 = 30 \, \text{cm}^2 \)
- Area of rectangle with sides \( A = 6 \) and \( C = 4 \): \( 6 \times 4 = 24 \, \text{cm}^2 \)
- Area of rectangle with sides \( A = 6 \) and \( D = 3 \): \( 6 \times 3 = 18 \, \text{cm}^2 \)

Step3: Sum the Areas

Sum the three areas: \( 30 + 24 + 18 = 72 \, \text{cm}^2 \)

Answer:

72