Two storage bins are built in the form of rectangular prisms, and the two bins are similar.

Question

One stores wheat at a cost of $.15 per bushel, and the other stores corn at a cost of $.20 per bushel. The bin storing the wheat has a square base 80 ft on a side and is 120 ft tall. If the cost of storing the wheat is $8000, and the cost of storing the corn is $36,000, find the height and the length to the nearest whole number of a side of the base of the bin storing corn.

Answer 1

Well, Mark M has a point there... however the Wikipedia page for bushel says:

"...The old bushel was equal to 4 pecks or 8 gallons and was used mostly for agricultural products such as wheat. At present, the volume is usually only nominal, with bushels referring to standard quantities of mass instead..."

So there is some historic use of bushel to be a volume measure.  

Given that this is a math problem we might go ahead and treat "bushel" here as a volume unit.

 

They have disguised the volumes by the prices, so first uncover the volumes...

The $8000 storage cost of the wheat at $0.15 per bushel gives the wheat volume:

  8000/0.15 = 53,333. bushels

For the corn the volume is:

  36,000/0.20 = 180,000 bushels

 

So the ratio:  corn volume / wheat volume = 180,000 / 53,333 ~ 3.375

 

The next ingredient is recognizing that the word "similar" is a mathematical term that has a precise meaning:

two geometric objects are similar if all of their corresponding lengths are in the same proportion.

For example a 4"x6" rectangle is similar to a 12"x18" rectangle because the lengths and widths are both scaled by the same factor of x3.

 

For a rectangular prism:

  V₁ = L * W * H

A similar prisim has all dimensions scaled by a factor, f:

  V_f = f*L * f*W * f*H  = f*f*f * L * w * H 

       = f³ * V₁ 

This is a general result for any 3D object: when the dimensions are scaled by f the volume changes by f³.

 

So the scale factor is related to the volume ratio we calculated above by:

  f³ = 3.375

Taking the cube-root of each side gives:

  f = cuberoot(3.375) ~ 1.500

 

So, the corn dimensions are all x1.5 of the wheat ones:

   80 * 1.5 = 120'  <-- sides of the square base

 120 * 1.5 = 180'  <-- height 

 

Answer 2

 

1 bushel = 1.24446 cubic feet

This conversion rate is needed to solve this problem.

 

 

    To determine the number of bushels of corn stored. we take $36000 and divide that by the cost per bushel (.20)

    36000/.20 = 180,000 bushels of corn

 

    To determine the number of cubic feet of corn stored, we multiply by 1.24446

    180,000 x 1.24446 = 224,002.8 cubic feet of corn stored

 

    The bins for the corn and the wheat are similar ... (given)

    Note that the height of the wheat bin is 120 ... that is 1.5 times 80, (the length of one of the square base sides)

    

    If we let b represent one of the sides of the square base of the corn's bin, then the height of the corns bin is 1.5x

 

    So now we can construct an equation 

 

    The volume of the corn's bin (computed earlier) is 224,002.8 cubic feet

   

       Volume = length times width times height

       b x b x 1.5b = 224,002.8

       1.5b³ = 224,002.8

       b³  = 224,002.8/1.5

 

    Taking the cube root of both sides of the equation

 

      b = 60.7320 ft  = 61

 

    Since the height is 1.5 times the base

 

      the height of the corn's bin is 91.098 ft = 91