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?

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

the final price would be 29.6925

step-by-step explanation:

39.59 minus 25% equals 29.6925

step-by-step explanation:

given that there are two triangles j'k'l' and jkl>

the former is got by dilating the second one.

when drawn on a grid, (graph paper) coordinates of vertices of first triangle is

(6,12)j, (9,6)k and (12,9)l

side jk has length = distance between j and k

=

side j'k'=distance between j' and k'

=

ie j'k'l' has sides 1/3 of that of corresponding sides of jkl.

hence scale factor of 1/3 is used to get j'k'l'from jkl