A box is to be made from a rectangular piece of cardboard 8cm x 15cm?

Question

An open box is to be made using cardboard 8cm by 15cm by cutting a square from each corner and folding the sides up. Find the length of a side of the square being cut so that the box will have a maximum voulme.

Answer 1

s = length of square
V = L*w*h
h = s
L = 15 - 2s
w = 8 - 2s
V = s(15-2s)(8-2s) = 120s - 46s^2 +4s^3
V' = 120 - 92s + 12s^2
0 = 12s^2 - 92s + 120
0 = 3s^2 - 23s + 30
0 = (s - 6)(3s - 5)
s = 6, 5/3
check these: cutting two 6cm pieces out of a 8cm cardboard won't work, so the answer is 5/3 cm

Answer 2

Let the squares that are cut from the cornaer have sides length x. Then the sides of the box will be 15 - 2x, 8 - 2x and x for a total volume of 4 x^3 - 46 x^2 + 120x. Differentiate this w.r.t. x to get 12 x^2 - 92 x +120. Solve this quadratic equation to get extrema at x = 5/3 and x = 6. Plug these back into the volume function and find that x = 5/3 yields the maximum (clearly x = 6 is nonsensical since 8 - 2x6 is negative) .

Answer: 5/3

Answer 3

volume V = x(8 -- 2x)(15 -- 2x)
V' = 12x^2 -- 92x + 120 = 0 gives x = 6 (not possible with width 8), 5/3
length of side of squares cut from corners = 5/3 cm
V"(5/3) = 24(5/3) -- 92 = -- 52 < 0

Answer 4

properly, I discern i could extra suitable sharpen my math skills contained in direction of the summer so i will help you. First I drew a diagram of the situation with a sq. of "x" * "x" in each and each nook of the field. From this diagram I observed that x=top 7-2x=length 3-2x=width quantity = top * length * width so V = x *(7-2x)*(3-2x) or 4x^3-20x^2+21x to locate the optimal of this function we ought to locate the 1st by-product and carry out the 1st by-product try. so V'=12x^2-40x+21is the 1st by-product. resolve for the zeroes, using the quadratic formula which provide you x, yet of direction examine to be sure your answer is a optimal and does not provide a detrimental quantity. Then as quickly as you're specific you ahve stumbled on the optimal for x, plug it in to the unique quantity equation for C.