Word problems in calculus

Question

The base and sides of a container is made of wood panels. The container does not have a lid. The base and sides are rectangular. The width of the container is   . The length is three times the width. The volume of the container is . 

1.Determine the minimum surface area, in cm², that this container will have. Round to two decimal places. Please show all steps. I need this for finals! Thank you!

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Answer 1

SA of a box is 2lw + 2lh + 2wh. Without a lid, we lose an lw, and SA = lw + 2lh+2wh. length is 3 times width, so l = 3w. Then, we haveSA = 3w(w) + 2(3w)h + 2wh = 3w^2 + 8wh

Volume = lwh = 735; (3w)wh = 735; 3w^2h = 735

minimize 3w^2 + 8wh subject to 3w^2h = 735. substitute h = 735/3w^2 = 245/w^2.

minimize 3w^2 +8w(245/w^2) = 3w^2 + 1960/w

We do this by taking the derivative.

6w - 1960/w^2 = 0

w^3 = 1960/6

w = (1960/6)^(1/3) = 6.88707703165523

SA = 3(1960/6)^(2/3) + 1960/(1960/6)^(1/3) = 9(1960/6)^(2/3) = 426.886469862238 = 426.89

Answer 2

The volume of the container is l*w*h, so we have:

x(3x)h = 3x²h = 735

The surface area is just the sum of the areas of each side, which is:

S = 3x² + 2xh + 2(3xh) = 3x² + 8xh

Since h = 245/x² by the volume formula, we have:

S(x) = 3x² + 8x(245/x²) = 3x² + 1960/x

Now, to find the minimum value of S, we need to take the derivative and set it equal to 0. We have:

S'(x) = 6x - 1960/x² = 0

6x³ = 1960

x³ = 326.667

x = 6.887

Therefore, the minimum surface area is:

S(6.887) = 3(6.887)² + 1960/6.887 = 426.8865 cm²

Answer 3

3x²h = 735

h = 735 / 3x²

SA = 3x² + 2xh + 6xh

SA = 3x² + 8xh

SA = 3x² + 8x(735 / 3x²)

SA = 3x² + 1960 / x

Now differentiate for the minimum.