Question

A box with a square base and open top must have a volume of 296352 cm. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. A(x) = Next, find the derivative, A'(x). A'(x) = The critical value is 3 = The function is decreasing ✓ until the critical value, and increasing value corresponds to a local maximum v after, so the critical

Answer 1

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Soli Given Volume = 296352 cm² dimensions base are xxxx hl of box with Square Volume = xe ² h 296352 → h=296352 x2 Surface of the box Copon top) is A(x) = x²+ ach tachtachtach Arze), érezh =x* ++(216352) :: A(X) = be²t 1185408 2 A cm) = 22+ 1185408 (-2) A' (t) - 123- 20- 1185408 x2 Alcacoa 27- 118 5408 x² = 1185408 A = x=84 2 x2 3. The critical Value is 2= 84 here A"(x) 2 + 2 (1185408) >o at x=84 So a3 Concave Opward. so we have local which means minimum at =84 The function dereasing until the critical Value & increasing after, so the Critical Value Corresponds to local (minimum