SOLUTION: Hi! This homework problem I have found to be very difficult, if you could help, that'd be great! I've tried all the ways I can think of to try and come up with even the equation, b

Algebra ->  Rational-functions -> SOLUTION: Hi! This homework problem I have found to be very difficult, if you could help, that'd be great! I've tried all the ways I can think of to try and come up with even the equation, b      Log On


   

Question 72373: Hi! This homework problem I have found to be very difficult, if you could help, that'd be great! I've tried all the ways I can think of to try and come up with even the equation, but haven't been successful.. I think part of my problem was that there aren't any variables.. I tried making up my own, but kept coming up with a different constant each time..
Grapefruit problem:
Suppose that you are shopping at a supermarket and find that the Texas grapefruits have twice the diameter of the Florida ones, but cost seven times as much. Recalling that the volumes of similarly shaped solids vary directly with the cube of a linear dimension such as a diameter, you instantly determine which kind of grapefruit gives you more for your money. Which one does, and why?
Thank-you!
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
Let the diameter of grapefruits of Texas be D%5BT%5D and that of Florida be D%5BF%5D.
Let the volume of grapefruits of Texas be V%5BT%5D and that of Florida be V%5BF%5D.
Therefore, V%5BT%5D+=+k%2AD%5BT%5D%5E3 and V%5BF%5D+=+k%2AD%5BF%5D%5E3 where k = constant of proportionality.
So, V%5BF%5D%2FV%5BT%5D+=+%28k%2AD%5BF%5D%5E3%29%2F%28k%2AD%5BT%5D%5E3%29
or V%5BF%5D%2FV%5BT%5D+=+D%5BF%5D%5E3%2FD%5BT%5D%5E3
Given, D%5BT%5D+=+2D%5BF%5D.
So V%5BF%5D%2FV%5BT%5D+=+D%5BF%5D%5E3%2F%282D%5BF%5D%29%5E3+=+1%2F8

Let the cost of each grapefruit of Texas be C%5BT%5D and that of Florida be C%5BF%5D.
Given that C%5BT%5D+=+7C%5BF%5D
Then cost per unit volume of grapefruit from Texas is C%5BT%5D%2FV%5BT%5D and that of grapefruit from Florida is C%5BF%5D%2FV%5BF%5D.

Now,

Hence, it is wiser to buy the grapefruits from Texas.