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Reptiles need the sun to get them going. Many large dinosaurs had to have large fins or plates down their back. How does this fact relate to what you have learned about the proportion of surface area to volume?

A manufacturer is producing half-liter aluminum cans in a cylindrical shape. The volume of the can is 500 cm^{3}.

- Find the radius and height for the can that will use the least aluminum and therefore be the cheapest to manufacture. In other words, minimize the surface area of the can.
- What shape is your can? Do you know of any cans that are made in this shape? Can you think of any practical reasons why more cans are not made in this shape?

Some aspirin-like tablets are said to work “two and a half times faster” than their competitors. What is an obvious way in which this could be accomplished?

Mr. Hobbs had an ugly blob in the middle of his wall. The paint on the rest of the wall looked fresh, so Mr. Hobbs asked the painter to come and paint only the blob. The painter said he would charge according to the area that needed to be painted. To figure out how much the job would cost, Mr. Hobbs ran a string around the edge of the blob so that it covered the border perfectly. Then, to figure out the area, he removed the string, shaped it into a rectangle, and figured out the area of the rectangle.

How close did Mr. Hobbs’s estimate come to the painter’s bill for painting the blob?

In cubes, you found a proportional relationship between volume and surface area. Does the proportional relationship between volume and surface area also exist for spheres?

The volume of a sphere is (4/3)πr^{3}, and the surface area of a sphere is 4πr^{2}.

**Problem H1**

Dinosaurs’ plates were likely the result of their very low surface area-to-volume ratio: The plates served to increase the dinosaurs’ surface area without increasing their volume very much.

Animals with a great deal of surface area but little volume cool down and heat up faster than animals with larger volumes. This is important in reptiles since they obtain their body heat from the sun.

**Problem H2**

- The minimum surface area occurs when the height is exactly twice the radius. If the volume is 500 cm3, the radius is approximately 4.30 cm, while the height is approximately 8.61 cm.
- In this shape, the can fits perfectly inside a cube, since the diameter of the can is the same as the height. Some cans are made in this shape. When cans are displayed in stores or placed on shelves, there is often more of a premium on radius (shelf space), so the radius is often smaller than our “ideal” can.

**Problem H3**

One obvious way is to increase the surface area-to-volume ratio of the tablet. It would dissolve more quickly. A flat caplet dissolves more quickly than a spherical pill.

**Problem H4**

This is the same type of problem as the “area of the hand” problem in Problem A7. The answer depends on the shape of the rectangle Mr. Hobbs used, since different rectangles with the same perimeter will have different areas. There is no way of guaranteeing that, using this method, the area of the two figures would be the same.

**Problem H5**

Yes, a similar ratio exists. The ratio for spheres is r:3 rather than s:6 for cubes and can most easily be found by using a table or by dividing the two formulas into one another.