To find out how many packages of stickers are needed to cover the surface area of the triangular prism, we first need to calculate the surface area of the prism.
- Surface Area of the Triangular Prism:
The formula for the surface area of a triangular prism is: \[ \text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} \]
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Base Area of the triangular face (Right Triangle)
The base area \( A \) of a right triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Using the values provided: -
Base = 8.9 cm
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Height = 5 cm \[ A = \frac{1}{2} \times 8.9 \times 5 = \frac{44.5}{2} = 22.25 \text{ cm}^2 \]
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Area of the two triangular faces: Since there are two triangular faces: \[ \text{Area of triangular faces} = 2 \times 22.25 = 44.5 \text{ cm}^2 \]
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Lateral Surface Area: The lateral surface area consists of the rectangle sides connecting the triangular faces. The dimensions of these rectangles are:
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One rectangle from the length and one side of the triangle:
- \( 15.9 , \text{(length)} \times 5 , \text{(height)} = 79.5 \text{ cm}^2 \)
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One rectangle from the length and the base of the triangle:
- \( 15.9 , \text{(length)} \times 8.9 , \text{(base)} = 141.51 \text{ cm}^2 \)
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One rectangle from the length and the hypotenuse of the triangle: To find the hypotenuse \( c \) of the right triangle, use the Pythagorean theorem: \[ c = \sqrt{5^2 + 8.9^2} = \sqrt{25 + 79.21} = \sqrt{104.21} \approx 10.21 \text{ cm} \] Thus, the area for this rectangle is: \[ 15.9 \times 10.21 \approx 162.30 \text{ cm}^2 \]
Total lateral surface area: \[ \text{Lateral Surface Area} = 79.5 + 141.51 + 162.30 = 383.31 \text{ cm}^2 \]
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Total Surface Area: Now, we can calculate the total surface area of the prism: \[ \text{Total Surface Area} = 44.5 + 383.31 = 427.81 \text{ cm}^2 \]
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Packages of Stickers Needed: Now we need to know how many packages of stickers are needed. Each package covers 75 cm². Therefore: \[ \text{Number of Packages} = \frac{\text{Total Surface Area}}{\text{Area covered by one package}} = \frac{427.81}{75} \approx 5.7041 \]
Since you can only purchase whole packages, you would round up to the nearest whole number: \[ \text{Number of Packages} = 6 \text{ packages} \]
Thus, the answer is: You will need 6 packages of stickers.