To solve the problem, we first need to find the surface area of the triangular prism.
A triangular prism consists of two triangular bases and three rectangular faces.
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Area of the triangular base The triangular base is a right triangle.
- Base = 8.9 cm
- Height = 5 cm
The area \( A \) of the triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8.9 \times 5 = 22.25 \text{ cm}^2 \]
Since there are two triangular bases, the total area of the bases is: \[ \text{Total area of bases} = 2 \times 22.25 = 44.5 \text{ cm}^2 \]
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Areas of the rectangular faces We have three rectangular faces:
- Rectangle 1: The side opposite the base (8.9 cm) and the length of the prism (15.9 cm) \[ \text{Area of Rectangle 1} = 8.9 \times 15.9 = 141.81 \text{ cm}^2 \]
- Rectangle 2: The height of the triangle (5 cm) and the length of the prism (15.9 cm) \[ \text{Area of Rectangle 2} = 5 \times 15.9 = 79.5 \text{ cm}^2 \]
- Rectangle 3: The hypotenuse (16.7 cm) and the length of the prism (15.9 cm) \[ \text{Area of Rectangle 3} = 16.7 \times 15.9 = 265.23 \text{ cm}^2 \]
Now, we sum the areas of all three rectangles: \[ \text{Total area of rectangles} = 141.81 + 79.5 + 265.23 = 486.54 \text{ cm}^2 \]
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Total surface area of the prism Adding the total area of the bases and the total area of the rectangles: \[ \text{Total surface area} = 44.5 + 486.54 = 531.04 \text{ cm}^2 \]
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Calculating the number of sticker packages needed Each package of stickers covers 75 cm². The number of packages needed can be calculated by dividing the total surface area by the area covered by one package and rounding up since you can only purchase whole packages: \[ \text{Number of packages} = \lceil \frac{531.04}{75} \rceil = \lceil 7.01 \rceil = 8 \]
Thus, you will need to purchase 8 packages of stickers. Since this was not on the provided response list, it seems the available responses have incorrect calculations or data presented.