To determine how many packages of stickers are needed to cover the surface area of the wooden door stopper shaped like a triangular prism, we need to calculate the total surface area of the prism first.
The surface area \( S \) of a triangular prism can be calculated using the formula:
\[ S = 2A + P \cdot h \]
where:
- \( A \) is the area of the triangular base,
- \( P \) is the perimeter of the triangular base, and
- \( h \) is the height (or length) of the prism.
- Calculate the area \( A \) of the triangular base:
The triangular base is a right triangle with:
- Base \( b = 8.9 \) cm,
- Height \( h_t = 5 \) cm.
The area \( A \) is given by:
\[ A = \frac{1}{2} \times b \times h_t = \frac{1}{2} \times 8.9 \times 5 = 22.25 \text{ cm}^2 \]
- Calculate the perimeter \( P \) of the triangular base:
The last side of the triangle \( c \) can be found using the Pythagorean theorem:
\[ c = \sqrt{(8.9)^2 + (5)^2} \] \[ c = \sqrt{79.21 + 25} = \sqrt{104.21} \approx 10.21 \text{ cm} \]
Now, the perimeter \( P \) is:
\[ P = b + h_t + c = 8.9 + 5 + 10.21 = 24.11 \text{ cm} \]
- Calculate the total surface area \( S \):
The length of the prism (height of the prism) \( h = 15.9 \) cm.
Now plug in the values into the surface area formula:
\[ S = 2A + P \cdot h \] \[ S = 2(22.25) + (24.11)(15.9) \] \[ S = 44.5 + 383.749 = 428.249 \text{ cm}^2 \]
- Determine the number of packages needed:
Each package covers 75 cm². To find the number of packages needed:
\[ \text{Number of packages} = \frac{S}{\text{Area covered by one package}} = \frac{428.249}{75} \approx 5.7 \]
Since packages can only be bought in whole numbers, we round up to the nearest whole number:
So, you need 6 packages of stickers.
Answer: You will need 6 packages of stickers.