# a rectangular box of given volume is to be "a" times long as it is wide. find the dimension for which it has the least total surface area ?

L= aw
Total surface area= 2LW + 2Lh + 2Wh
= 2aw^2 + 2awh + 2wh

But volume= lwh= 2aw^2 h or h= V/2aw^2

Total Surface area= = 2aw^2 + 2awV/2aw^2 + 2wV/2aw^2

Take the derivative of surface area/with respect to w, solve.

3 years ago

7 months ago

## To find the dimensions for which the rectangular box has the least total surface area, we need to take the derivative of the surface area equation with respect to the width (w) and set it equal to zero. Let's break down the steps to solve this problem:

1. Start with the surface area equation: Total surface area = 2aw^2 + 2awh + 2wh.

2. Substitute the expression for h in terms of volume: h = V / (2aw^2).

3. Rewrite the surface area equation using the substitution: Total surface area = 2aw^2 + 2aw * (V / (2aw^2)) + 2w * (V / (2aw^2)).

4. Simplify the equation: Total surface area = 2aw^2 + V / w + V / a.

5. Take the derivative of the surface area equation with respect to w: d(Surface area) / dw = 4aw - V / w^2.

6. Set the derivative equal to zero and solve for w: 4aw - V / w^2 = 0.

7. Multiply both sides of the equation by w^2: 4aww^2 - V = 0.

8. Simplify the equation: 4aww^2 = V.

9. Divide both sides of the equation by 4aw: w^2 = V / (4aw).

10. Take the square root of both sides to solve for w: w = sqrt(V / (4aw)).

So, the width of the rectangular box that minimizes the total surface area is given by w = sqrt(V / (4aw)), where V is the volume and a is the given ratio of length to width.

Once you have the value of w, you can use it to find the length (L) by multiplying it by a (L = aw), and then calculate the height (h) using the volume equation (h = V / (2aw^2)).

Finally, you can find the total surface area by substituting the dimensions into the surface area equation: Total surface area = 2LW + 2Lh + 2Wh.

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