Question

You are planning to make an open rectangular box from a 33​-in.-by-65​-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this​ way, and what is its​ volume?

Answer 1

Answer:

Volume = 6783.27inch³

H = 6.951inch

L = 51.098inch

W = 19.098inch

Step-by-step explanation:

From the given information:

Let h = side length and height of the box

L = length of the box

W = width of the box

V = volume of the box

We have that:

L = 65-2h

W = 33-2h

V = L×W×h

Therefore we have

V = (65-2h)(33-2h)×h

V = (2145-130h-66h+4h²)×h

V = 2145h-196h²+4h³

By differentiating V w.r.t h, we have

V' = 2145-392h+12h²

V = 12h²-392h+2145

Using Almighty formula we have

h = 392+/-√392²-4(12)(2145)/2(12)

h = 6.951 or 25.716

Thus, we find L,W and V.

We use the least value of h in order not to get a negative value of volume, length and width

L = 65-2h = 65-2(6.951) = 51.098

W = 33-2h = 33-2(6.951) = 19.098

V = LWH = 51.098×19.098×6.951 = 6783.2696inches