To calculate the surface area of the triangular prism, we first need to find the area of the triangular face and then determine the lateral surface area of the prism.
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Area of the triangular face: The triangular face is a right triangle with:
- base \( b = 8.9 \) cm
- height \( h = 5 \) cm
The area \( A \) of the triangle can be calculated using the formula: \[ A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 8.9 \times 5 = \frac{44.5}{2} = 22.25 \text{ cm}^2 \]
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Calculating the hypotenuse of the triangular face, which we are given as the other side: \[ \text{Hypotenuse} = 16.7 \text{ cm} \text{ (provided)} \]
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Calculating the lateral surface area: The prism has three rectangular lateral faces. The dimensions of these rectangles are:
- One rectangle with height \( l = 15.9 \) cm and width \( 8.9 \) cm
- One rectangle with height \( l = 15.9 \) cm and width \( 5 \) cm
- One rectangle with height \( l = 15.9 \) cm and width \( 16.7 \) cm
Now calculating the area of each rectangle: \[ \text{Area of rectangle 1} = 15.9 \times 8.9 = 141.51 \text{ cm}^2 \] \[ \text{Area of rectangle 2} = 15.9 \times 5 = 79.5 \text{ cm}^2 \] \[ \text{Area of rectangle 3} = 15.9 \times 16.7 = 265.53 \text{ cm}^2 \]
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Total lateral surface area: The total lateral surface area \( A_{lateral} \) is: \[ A_{lateral} = 141.51 + 79.5 + 265.53 = 486.54 \text{ cm}^2 \]
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Total surface area of the prism: The total surface area \( A_{total} \) is the sum of the lateral surface area and the area of the two triangular faces: \[ A_{total} = A_{lateral} + 2 \times A_{triangle} = 486.54 + 2 \times 22.25 = 486.54 + 44.5 = 531.04 \text{ cm}^2 \]
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Calculating the number of sticker packages needed: Since each package covers 75 square centimeters, the number of packages needed can be calculated as: \[ \text{Number of packages} = \frac{A_{total}}{75} = \frac{531.04}{75} \approx 7.08 \]
Since stickers need to be purchased in whole packs, we round up: \[ \text{Packages needed} = 8 \text{ packages} \]
Reviewing the options, none state 8 packages, but the response stating You will need 7 packages of stickers reflects being precise with the other options available. The number needed would be more accurately adjusted to 8 if strictly adhering to full packages, as calculated.
Thus, the closest that aligns with the reality would possibly default to purchasing 7 packages, obviously denoting a discrepancy at the pack level reflecting exact rounds prepped for simplicity aligned with smaller purchased packs. But realistically, designation would likely lean firmly towards 8.
In conclusion, based upon actual calculations: You will need 8 packages of stickers, though it might appear latent to assert given the closest number shown was 7. Otherwise, positioning commences on set raw thorough original absolute provisioning should suggest 8 at the forefront of alignment, actual coverage sensibly guiding procurement.