Question

You are planning to make an open rectangular box from a 24​-in.-by-47​-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:

The dimensions are  13.92inches by 36.92inches by 5.04inches

Volume is 2590.18 inches³

Step-by-step explanation:

see attachment for the figure

supposing 'h' is side length of the square and also the height of the box

then, According to the question

Length of the box 'L'= 47 - 2h

Width of the box 'w'= 24 - 2h

Volume of the box 'V'= hLw

Substituting the values of 'L' and 'w' in above equation

V= h(47 - 2h)(24 - 2h)

V=(47h - 2h²)(24 - 2h) => 1128h - 94h²-48h²+4h³

V= 4h³ -142h²+1128h -->eq(1)

Taking derivative on both sides.

V' = 12h² - 284h + 1128

Setting the equation to zero, we will have

0=12h² - 284h + 1128

0= 3h² - 71h + 282

Using Quadratic formula to find h

h=  (-b+-√b²-4ac) / 2a

  = 71+-√71² - (4 x 3 x 282) / (2 x 3)=> 71+-√5041- 3384 / 6

h  =(71+- 40.706)/ 6

EITHER h= 18.6

OR h=5.04

By substituting the value of 'h' in eq(1)

(1)=>V(18.6)= 4(18.6)³ -142(18.6)²+1128(18.6) => -2406.096

(1)=>V(5.04)= 4(5.04)³ -142(5.04)²+1128(5.04) => 2590.18

By ignoring the negative value, we will have h=5.04

Therefore,

L= 47 - 2h => 47 -2(5.04) =>36.92

w= 24 - 2h=> 24- 2(5.04)=> 13.92

The dimensions are  13.92inches x 36.92inches x 5.04inches

Volume is 2590.18 inches³

Answer 2

We  are planning to make an open rectangular box from a 24​-in.-by-47​-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. The dimensions of the box of largest volume we can make are

Length of the box =  47-2(5.04) =  36.92 inch

Width of the box = 24-2(5.04) =  13.92 inch

Height of box = 5.04 inch

Length of the cardboard  = 47 inch

Width of the cardboard = 24 inch

Let the height of box be h inch

Length of side of each square cut = h inch

Then from figure attached we can write that the

$\rm Length \; of \; the \; box = 47 -2h\\Width \;of \;the \; box = 24-2h$

So Volume of the box   can be written as

As box is cuboidal in shape

$\rm Volume \; of \; box \; = Length \times Width \times Height$

$\rm V_{box}= (47-2h)(24-2h)(h)\\On \solving \; we\; get \\V = 4 h^3 -142h^2 +1128h ............(1)$

On differentiating equation (1) with respect to height h

We can get

$\rm V' = 12h^2 -284\; h +1128....(2)$

For finding out maximum volume   on putting equation (2) equal to zero we get

$\rm 12h^2 -284\; h +1128 =0 .....(3)$

On solving equation(3) we get

h = 18.61 and h = 5.04

on putting these h values in equation (1) we get

V = -2406.12 cubic inch  for h = 18.61  (unacceptable, since it is negative)

V = 2590.19 cubic inch  for h = 5.04 inch

So the length  and width of the box

Length of the box =  47-2(5.04) =  36.92 inch

Width of the box  = 24-2(5.04) =  13.92 inch

For more information please refer to the link below

https://brainly.com/question/22105103