2 Geometry Qs: Who is eating the largest mass of freeze pop?

[Ficle]

1. Three friends are eating enormous freeze pops on a hot summer day. Benjamin has a strawberry freeze pop that has a density of 0.8 pounds per cubic inch. His freeze pop is shaped like a square pyramid with a 1.5 inch side and a height of 2 inches. Kendra has a giant grape freeze pop which has a density of 0.2 pounds per cubic inch. The freeze pop is a cylinder which has a radius of 1 inch and a height of 4 inches. Luke has a giant cherry freeze pop, which has a density of 0.05 pounds per cubic inch and is also shaped like a cylinder which has a radius of 1.5 inches and a height of 3 inches. Answer the following questions.

• Who is eating the largest mass of freeze pop? Round to the nearest tenth of a pound.
• How many more pounds of freeze pop does Kendra have compared to Luke? Round your answer to the nearest tenth.
• If all three freeze pops were the same size, which flavor would you prefer, based on the densities? Explain why.

2. Find the largest possible rectangular area you can enclose with 420 meters of fencing. What is the significance of the dimensions of this enclosure, in relation to geometric shapes?

 

[ksdhart2]

Okay, so, for the first problem, you're told the shapes of the various freeze pops, and their densities. The problem was even nice enough to make them constant densities, rather than varying. As you know, the formula for mass is density times volume. So, what are the volumes of the various freeze pops? Do you know the formulas for a square pyramid and a cylinder?

For the second problem, let's pretend that the length and width of the rectangle are restricted to integers (whole numbers), to make calculations easier, and to get a feel for what's going on. If you have a length of 1 meter, how many meters must the width be in order to use all 420 meters? What would the area of that rectangle be? If you have a length of 2 meters, what must be the width? What must be the area? If the length is 3 meters? Are you seeing a pattern? Now, what would happen if the length were 209 meters? If the length were 208 meters? Does this data change your pattern? If so, can you find a new pattern? What does that suggest the maximum area might be?

Now that you're armed with an intuition for the process, let's check your answer for part 2. Is your proposed solution really the maximum area? How would you know? Well, one such method is outlined here.

If you get stuck again, that's okay, but please share with us all of your work, even if you know it's wrong. Thank you.