To the nearest hundredth, what is the radius of a similar container holding $4.00 worth of grapefruit juice? What is the height to the nearest hundredth?

Question

Here's the info that goes with it: 

Two similar cylinders contain juice. The first cylinder has radius 6 in. and height 10 in., contains orange juice, and sells for $2.40. 

If grapefruit juice costs two-thirds of the price of an equal volume of orange juice, what is the cost of a container of grapefruit juice that has radius 9 in. and height 15 in.? (For this answer, I got $5.40.)

 

I don't remember covering the step-by-step way to solving this problem in class, and it'd be great if you could help me out!

Answer 1

To find out the volume dimensions of a cylinder containing grapefruit juice that costs $4.00, you have to use something you might've figured out in the first problem: How much money a cubic inch of grapefruit juice is worth.

• A cylinder of grapefruit juice of equal volume as the orange juice is worth two thirds the price
• So (2/3) × $2.40 = $1.60
• The volume of the cylinder of orange juice is V = hπr² = 10π×6² = 360π in³
• Therefore the money:volume ratio gives us that this kind of grapefruit juice is worth (1.60/360π) $/in³

Now realize that the two cylinders mentioned in the question are similar to each other, meaning they share the same proportions:

• h = 10, r = 6 → h/r = 10/6 = 5/3
• h = 15, r = 9 → h/r = 15/9 = 5/3
• So a new cylinder that's similar to these would have to follow that h = (5/3)r

Since the h & r are related to each other in this way, we can reduce the volume formula down to one variable for this case:

V = hπr² = (5/3)rπr² = (5/3)πr³

Multiply this constrained volume to our money:volume ratio. The radius that satisfies these conditions must give us the amount of grapefruit juice that is worth $4.00

 

(1.60/360π) × (5/3)πr³ = 4

(1.60×5×π)/(360π×3)r³ = 4

(8/1080)r³ = 4

r³ = (1080×4)/8 = 540

³√r = ³√540 ≈ 8.14 in  ≈ r

 

And since h = (5/3)r

               h ≈ (5/3)8.14 ≈ 13.57 in

Really fun question! :)

Answer 2

The volume of the larger cylinder is 27/16 the volume of the smaller (the linear ratio is 3/2 volume is as the cube)

So the cost of orange juice would be (27/16)(2.40), or 4.05.

For grape frui the cost would be (2/3)(4.05), or 2.70