Question

The manufacturer of a wooden feeding trough for farm animals makes the sides and bottom of the trough 1 foot thick. The outer height and outer width of the trough are the same, and the outer length is twice the outer height and outer width. What should the outer dimensions of the trough be if the trough is to hold 36 cubic feet of feed?

Answer 1

outer width of trough = outer height of trough = x = 4ft

outer length of trough = 2x = 8ft

Explanation:

Let,

x = outer width of trough = outer height of trough

Now, according to given condition:

2x = outer length of trough

Now we calculate the inner dimensions by subtracting the thickness (1 ft)

Therefore,

x - 2(1ft) = x - 2ft = inner width of trough (because 1ft from both sides will be subtracted)

x - 1ft = inner height of trough (because only bottom thickness will be subtracted)

2x - 2ft = inner length of trough (because 1ft from both sides will be subtracted)

Now, for the holding volume or inside volume of trough will be:

`Inner Volume = 36\ ft^3 = inner\ height*inner\ width*inner\ length \\36 = (x - 1)(x - 2)(2x - 2)\\36 = (x - 1)(2x^2 - 6x + 4)\\36 = (2x^3 - 6x^2 + 4x - 2x^2 + 6x - 4)\\36 = (2x^3 -8x^2 + 10x - 4) \\2x^3 -8x^2 + 10x - 40 = 0\\2x^2(x-4)+10(x-4) = 0\\(x-4)(2x^2+10) = 0\\`

So, we will have three roots from this solution. Two, of them will be complex from the second factor. So, we ignore them and take the third one as our answer:

x - 4 = 0

x = 4ft

Now the outer dimensions will be:

outer width of trough = outer height of trough = x = 4ft

outer length of trough = 2x = 8ft