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A rectangular box with no top (pictured above) is to contain 2250 cubic inches of volume.
Find the dimensions of the box that will minimize the surface area.
The length (l) of the base is three times the width (w).
But what about the height of the box? Is it the same as the width,
or do we have to find it? (h), assuming this is the case:
l * w * h = 2250 cu/inches
Let w = x, then we can write it:
3x * x * h = 2250
3x^2 * h = 2250
is the height of the box
Surface area for box without a top
S.A. = (l*w) + 2(l*h) + 2(w*h)
Substitute 3x for l and x for w
S.A. = 3x*x + 2(3x*h) + 2(x*h)
S.A = 3x^2 + 6hx + 2hx
S.A = 3x^2 + 8hx
S.A = 3x^2 + 8(x*
S.A = 3x^2 + 8(
S.A = 3x^2 +
Graph this equation and find the min surface area:
On a Ti-83, it says, min S.A. at x = 10, which is 900 sq/in, as indicated here.
h = 7.5
Then l= 30, w=10, h=7.5; the dimensions of the box
Check volume: 30 * 10 * 7.5 = 2250
Did this make sense?