To determine the amount of material needed to make the storage bin, we need to find the total surface area of the rectangular prism. A rectangular prism has six faces, but since the storage bin has a closed top, we need to consider all six faces in our calculation. Here are the detailed steps:
1. Identify the dimensions of the storage bin:
- Length ([tex]\(l\)[/tex]) = 3.5 feet
- Width ([tex]\(w\)[/tex]) = 2 feet
- Height ([tex]\(h\)[/tex]) = 2.5 feet
2. Calculate the area of each type of face:
- The bin has two faces that are [tex]\( length \times width \)[/tex].
- The bin has two faces that are [tex]\( length \times height \)[/tex].
- The bin has two faces that are [tex]\( width \times height \)[/tex].
3. Calculate the area for each pair of faces:
- Area of the top and bottom faces (each is [tex]\( length \times width \)[/tex]):
[tex]\[ 3.5 \times 2 = 7 \, \text{square feet per face} \][/tex]
[tex]\[ 2 \times 7 = 14 \, \text{square feet total for both faces} \][/tex]
- Area of the front and back faces (each is [tex]\( length \times height \)[/tex]):
[tex]\[ 3.5 \times 2.5 = 8.75 \, \text{square feet per face} \][/tex]
[tex]\[ 2 \times 8.75 = 17.5 \, \text{square feet total for both faces} \][/tex]
- Area of the left and right faces (each is [tex]\( width \times height \)[/tex]):
[tex]\[ 2 \times 2.5 = 5 \, \text{square feet per face} \][/tex]
[tex]\[ 2 \times 5 = 10 \, \text{square feet total for both faces} \][/tex]
4. Add together all the areas to find the total surface area of the storage bin:
[tex]\[ 14 + 17.5 + 10 = 41.5 \, \text{square feet} \][/tex]
So, the total amount of material needed to make the storage bin is [tex]\( 41.5 \, \text{square feet} \)[/tex].
Therefore, the correct answer is:
D. [tex]\( 41.5 \, \text{ft}^2 \)[/tex]
