To find the volume [tex]\( V \)[/tex] of a closed rectangular box with given dimensions, we use the formula for the volume of a rectangular prism:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Given the dimensions:
- Length [tex]\( l = 6 \)[/tex] feet
- Width [tex]\( w = 4 \)[/tex] feet
- Height [tex]\( h = 5 \)[/tex] feet
Plug these values into the volume formula:
[tex]\[ V = 6 \, \text{ft} \times 4 \, \text{ft} \times 5 \, \text{ft} \][/tex]
So, the volume [tex]\( V \)[/tex] of the rectangular box is:
[tex]\[ V = 120 \, \text{cubic feet} \][/tex]
To find the surface area [tex]\( S \)[/tex] of a closed rectangular box, we use the formula for the surface area of a rectangular prism:
[tex]\[ S = 2(\text{length} \times \text{width} + \text{width} \times \text{height} + \text{height} \times \text{length}) \][/tex]
Again, using the given dimensions:
[tex]\[ S = 2(6 \, \text{ft} \times 4 \, \text{ft} + 4 \, \text{ft} \times 5 \, \text{ft} + 5 \, \text{ft} \times 6 \, \text{ft}) \][/tex]
Calculating each term inside the parentheses:
[tex]\[ 6 \, \text{ft} \times 4 \, \text{ft} = 24 \, \text{square feet} \][/tex]
[tex]\[ 4 \, \text{ft} \times 5 \, \text{ft} = 20 \, \text{square feet} \][/tex]
[tex]\[ 5 \, \text{ft} \times 6 \, \text{ft} = 30 \, \text{square feet} \][/tex]
Adding these together:
[tex]\[ 24 \, \text{sq ft} + 20 \, \text{sq ft} + 30 \, \text{sq ft} = 74 \, \text{square feet} \][/tex]
Finally, multiply by 2:
[tex]\[ S = 2 \times 74 \, \text{sq ft} = 148 \, \text{square feet} \][/tex]
Therefore, the volume [tex]\( V \)[/tex] of the rectangular box is [tex]\( 120 \, \text{cubic feet} \)[/tex] and the surface area [tex]\( S \)[/tex] is [tex]\( 148 \, \text{square feet} \)[/tex].
