The design of a digital camera box maximizes the volume whil

The design of a digital camera box maximizes the volume while keeping the sum of the dimensions at 6 inches. If the length must be 1.5 inches times the height, what should each dimension be? Please explain how you find the length width and height!! I solved for the equation and got V(x) = -3.75h^3 + 9h^2, but I dont know how to use the equation to find the dimensions.

Solution

let the length of the camera box is \'l\' inches.

  the breadth of the camera box is \'b\' inches.

   the height of the camera box is \'h\' inches.

Given sum of dimensions is 6 inches for maximum volume and  length must be 1.5 inches times the height.

l+b+h = 6 and l = 1.5*h

1.5h + b+h = 6

2.5h+b = 6

b = 6-2.5h

Volume of box = lbh

= 1.5h*(6-2.5h)*h

= 9h² - 3.75h³

A function is maximum or minimum at it\'s first derivative = 0

Derivate volume with respect to h.

9h² - 3.75h³

9(2h) - 3.75(3h²) = 0

18h - 3.75(3h²) = 0

18h = 3.75(3h²)

6 = 3.75h

3.75h = 6

h = 6/3.75

h = 1.6

Now, l = 1.5*h = 1.5*1.6 = 2.4

b =6-2.5h = 6- 2.5*1.6 = 2

Therefore, dimensions of camera box are

length (l) = 2.4inches, breadth(b) = 2inches and height(h) = 1.6inches