A closed box has a square base with side length l feet and height h feet. Given that the volume of the box is 29 cubic feet, express the surface area of the box in terms of l only.

Answer:

The surface area of the box in terms of l is [tex]2l^2 +116/l[/tex]

Step-by-step explanation:

The length of the box = [tex]l[/tex]

The height of the box = [tex]h[/tex]

The volume of the box can be obtained by multiplying the surface area of the base by the height of the box.

The surface area of the base of the box is a square, so it will be obtained by multiplying [tex]l \times l = l^2[/tex]

hence volume = [tex]l^2 \times h =29[/tex]

Notice that we are told to give our answer in terms of l, so we will make h the Subject of the formula in the above equation.

[tex]h =29/l^2[/tex]

In the next phase, we are to find the surface area of the box.

The box has a square base, hence it will be made up of

2 squares with 4 rectangles. This is assuming the top is closed.

This means the surface area will be

[tex]2(l^2) +4 (l\times h)[/tex]

recall [tex]h = 29/l^2[/tex]

Hence, surface area =

[tex]2(l^2) +4(l \times 29/l^2)\\2l^2 +116/l[/tex]

The surface area of the box in terms of l is [tex]2l^2 +116/l[/tex]

The surface area of the closed box in terms of l only, which has the square base is,

[tex]2l^2+\dfrac{116}{l}\;\rm ft^2\\[/tex]

Volume of cuboid or box is the amount of quantity, which is obtained by it in the 3 dimensional space.

The volume of cuboid or box can be given as,

[tex]V=l\times b\times h[/tex]

Here, (l) is the length, (b) is the width of the box and (h) is the height of the box.

A closed box has a square base with side length l feet and height h feet.  Therefore, the length and width will be the same.

The volume of the box is 29 cubic feet. Thus,

[tex]29=l\times (l)\times h\\h=\dfrac{29}{l^2}[/tex]

The surface area of the square base box is,

[tex]A=2l^2+4lh[/tex]

Put the value of height,

[tex]A=2l^2+4l\dfrac{29}{l^2}\\A=2l^2+\dfrac{116}{l}\\[/tex]

Hence, the surface area of the closed box in terms of l only, which has the square base is,

[tex]2l^2+\dfrac{116}{l}\;\rm ft^2\\[/tex]

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