Question

solve each question below and be sure to show all work.
find the lateral and total surface area of the prism.
(image of a rectangular prism with dimensions 2.1 in, 8.4 in, 4 in)

1. lateral surface area:
2. total surface area:

find the lateral and total surface area of the prism.
(image of a triangular prism with dimensions 25 cm, 30 cm, 15 cm, 20 cm)

1. lateral surface area:
2. total surface area:
3. every surface of the block shown will be painted, except for one of the bases. how many square centimeters will be painted?

(image of a cube with side length 3.2 cm)

Explanation:

Step1: Calculate lateral surface area (rect prism)

Lateral surface area of a rectangular prism is $2(lh + wh)$, where $l=8.4$, $w=2.1$, $h=4$.
$\text{Lateral Area} = 2[(8.4 \times 4) + (2.1 \times 4)] = 2[33.6 + 8.4] = 2 \times 42 = 84$ square inches

Step2: Calculate total surface area (rect prism)

Total surface area is lateral area plus $2lw$.
$\text{Total Area} = 84 + 2(8.4 \times 2.1) = 84 + 2(17.64) = 84 + 35.28 = 119.28$ square inches

Step3: Calculate base area (triangular prism)

First find area of the triangular base: $\frac{1}{2} \times 20 \times 15 = 150$ square cm

Step4: Calculate lateral surface area (tri prism)

Lateral surface area is sum of the 3 rectangular faces: $(25 \times 20) + (30 \times 20) + (20 \times 20)$
$\text{Lateral Area} = 500 + 600 + 400 = 1500$ square cm

Step5: Calculate total surface area (tri prism)

Total surface area is lateral area plus $2 \times$ base area.
$\text{Total Area} = 1500 + 2(150) = 1500 + 300 = 1800$ square cm

Step6: Calculate painted area (cube)

Area of one face: $3.2 \times 3.2 = 10.24$ square cm. Painted area is 5 faces.
$\text{Painted Area} = 5 \times 10.24 = 51.2$ square cm

Answer:

1. Lateral surface area: $84$ square inches
2. Total surface area: $119.28$ square inches
3. Lateral surface area: $1500$ square cm
4. Total surface area: $1800$ square cm
5. Painted area: $51.2$ square cm