A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100 , the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:

Step-by-step solution

​ Let l = 8 x , b = 15 x ∴ Volume = 8 x - 2 a 15 x - 2 a a = 4 a 3 - 46 a 2 x + 120 a x 2 d V d a = 6 a 2 - 46 a x + 60 x 2 d V d a a t a = 5 = 0 ∴ x = 3 and 5 6 d 2 V d a 2 = 6 a - 23 x d 2 V d a 2 a t a = 5 & x = 3 0 So, at x = 3 , it gives maxima: d 2 V d a 2 a t a = 5 & x = 5 6 > 0 So, at x = 5 6 , it gives minima. So by x = 3 (for max volume): 8 x = 24 , 15 x = 45