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
