# The length of a box is 2 centimeters less than its height. the width of the box is 7 centimeters more than its height. If the box had has a volume of 180 cubic centimeters, what is its surface area?

So the surface area becomes

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Let's denote the height of the box as (h) centimeters.

Given:

- The length of the box is (2) centimeters less than its height, so the length (l) can be expressed as (h - 2) centimeters.
- The width of the box is (7) centimeters more than its height, so the width (w) can be expressed as (h + 7) centimeters.
- The volume of the box is (180) cubic centimeters, so we have the equation (l \times w \times h = 180).

Substituting the expressions for (l) and (w) into the volume equation, we get: [ (h - 2) \times (h + 7) \times h = 180 ]

Expanding and simplifying: [ (h^2 + 7h - 2h - 14) \times h = 180 ] [ (h^2 + 5h - 14) \times h = 180 ] [ h^3 + 5h^2 - 14h - 180 = 0 ]

Now, we need to find the value of (h) that satisfies this equation. By trying different values or using numerical methods, we find that (h = 6) is a solution to this equation.

So, the height of the box is (6) centimeters.

Now, we can find the length and width: [ l = h - 2 = 6 - 2 = 4 \text{ centimeters} ] [ w = h + 7 = 6 + 7 = 13 \text{ centimeters} ]

Now that we have the dimensions of the box, we can find its surface area. The surface area (A) of a rectangular box is given by the formula: [ A = 2lw + 2lh + 2wh ]

Substituting the values we found: [ A = 2(4 \times 13) + 2(4 \times 6) + 2(13 \times 6) ] [ A = 2(52) + 2(24) + 2(78) ] [ A = 104 + 48 + 156 ] [ A = 308 \text{ square centimeters} ]

Therefore, the surface area of the box is (308) square centimeters.

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