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In the previous part, you learned that the relationship between perimeter and area is dynamic; namely, the amount of area or perimeter of a shape is not fixed in relation to the measure of the other variable.
How about the relationship between surface area and volume? Do prisms with the same volume have the same surface area? Let’s explore this relationship.
Take 24 multilink cubes or building blocks and imagine that each cube represents a fancy chocolate truffle. For shipping purposes, these truffles need to be packaged into boxes in the shape of rectangular prisms. Knowing that you must always package 24 truffles (i.e., your volume is set at 24 cubic units), what are the possible dimensions for the boxes? Record your information in the table below:
The dimensions must all be factors of 24 — 1, 2, 3, 4, 6, 8, 12, 24.
Which of your packaging arrangements requires a box with the least amount of material? The greatest amount of material? Why is the amount of material needed for packaging important?
What do you notice about the shape of the package that has the smallest surface area? How about the package with the greatest surface area?
When the volume is constant (as in the truffles problem), the surface area depends on the shape of the solid. But what happens to the surface area of a solid as its volume increases — does surface area increase at the same rate as volume?
Use multilink cubes or building blocks to create different-sized cubes from the table below. Calculate the volumes and surface areas of the cubes. Examine the proportional relationship between surface area (SA) and volume (V) by creating a surface area-to-volume ratio (SA:V). As volume increases, what happens to the ratio of surface area to volume?
You may find it helpful to build the different-sized cubes first and then to use the models to confirm your calculations of the volumes and surface areas. If cubes aren’t available, you could make a sketch of the cubes on graph paper to help you visualize the surface area of each face. Write the surface area-to-volume ratios and look for patterns.
Size of Cube |
Surface Area |
Volume |
Ratio SA:V |
1 by 1 by 1 | |||
2 by 2 by 2 | |||
3 by 3 by 3 | |||
4 by 4 by 4 | |||
5 by 5 by 5 | |||
6 by 6 by 6 | |||
7 by 7 by 7 | |||
8 by 8 by 8 |
Video Segment Watch this video segment after you’ve completed Problem B4 to see what Lori and Jayne found out about the relationship between the volume and surface area of solid objects.Did you express this relationship differently?You can find this segment on the session video approximately 11 minutes and 49 seconds after the Annenberg Media logo. |
Problem B5
Over the last decade, we have seen the genesis of the giant “superstore.” What kind of surface area-to-volume ratio do you think superstores have? Why do companies build such large stores?
Your body’s surface area is a measurement of the skin that covers your body. You may have noticed that adults and children (and babies in particular) have very different reactions to heat and cold. This happens because the body cools down by sweating at a rate proportional to the area of its skin, but warms up in proportion to its mass (volume). The ratio of surface area to mass is much larger for babies, so they cool down faster than adults. As a result, babies can catch a chill even when adults feel warm.
Problem B6
Use multilink cubes to build a model of a baby and of an adult. You can use a simple, trimmed-down model, or you can build a more realistic one. Once you’ve completed your models, calculate the surface area-to-volume ratio for the baby and the adult.
Problem B7
In the summer, we’re warned not to leave babies or pets in cars. Yet on a hot day, an adult can sit in a car for a short period of time without harm. Use your models and mathematics to explain what is occurring in these situations, and why babies dehydrate so much more quickly than adults.
In Session 6, you learned that about 100 handprints will cover the body. You used your estimate of the area of your handprint to approximate your surface area. Another approach to estimating surface area is to see how much of you fits into a square.
The picture below is based on a famous drawing by Leonardo da Vinci. As shown in the picture, the person more or less fits in the square.
A person’s height is approximately equal to his or her arm span (from fingertip to fingertip).
Problem B9
Another way to approximate a person’s surface area is to use a simple formula:
Height (cm) • Thigh Circumference (cm) • 2 = Body Surface Area (cm^{2})
Since surface area is also related to weight, health care workers usually use a chart called a nomograph to estimate a person’s surface area. To use the nomograph, a person’s height (in centimeters) is located in the left-hand column, and a person’s weight (in kilograms) is located in the right-hand column. These points are connected with a straight line. The surface area of a person’s body is shown where the line crosses the middle scale. Note 2
Problem B10
Use the nomograph to estimate your own body’s surface area. Note that 1 kg = 2.2 lb.
Problem B11
The density of the human body is a little greater than the density of water because of our bones and organs. One kilogram of body mass occupies a volume of about 0.9 L. Determine the volume of your body by multiplying your weight in kilograms by 0.9.
Problem B12
This problem will be easier to solve if you convert to metric measures.
“Human Measurements” adapted from Romberg, T., et al. Made to Measure. Mathematics in Context. © 1998 by Encyclopedia Britannica Educational Corporation. Used with permission. All rights reserved.
Note 2
The problems in this section require some conversions between the U.S. customary system and the metric system. You can use the Web site www.onlineconversion.com to find the proper conversions.
Problem B1
Here is the completed table:
Length |
Width |
Height |
Volume |
Surface Area |
1 | 1 | 24 | 24 | 98 |
1 | 2 | 12 | 24 | 76 |
1 | 3 | 8 | 24 | 70 |
1 | 4 | 6 | 24 | 68 |
2 | 2 | 6 | 24 | 56 |
2 | 3 | 4 | 24 | 52 |
Problem B2
The least amount of material is required by the 2-by-3-by-4 box. The greatest amount of material is required by the 1-by-1-by-24 box. The amount of packaging material needed is important, since charges for such material are figured in square inches or square feet, and reducing the amount of needed material will reduce the cost of shipping. Also, the cost of making the box itself will be lower if there is a smaller surface area.
In practical terms, however, you may want to use more packaging material so that the truffles don’t get damaged. Also, customers may feel that they’re getting more for their money if there is more packaging and the box is larger.
Problem B3
The rectangular prisms with the greatest surface area have dimensions that are far apart (1 by 1 by 24), whereas the prisms with smaller surface area have dimensions close to each other (e.g., 2 by 3 by 4).
Problem B4
Here is the completed table:
Size of Cube |
Surface Area |
Volume |
Ratio SA:V |
1 by 1 by 1 | 6 • 1^{2} = 6 | 1^{3} = 1 | 6:1 |
2 by 2 by 2 | 6 • 2^{2} = 24 | 2^{3} = 8 | 6:2 (or 3:1) |
3 by 3 by 3 | 6 • 3^{2} = 54 | 3^{3} = 27 | 6:3 (or 2:1) |
4 by 4 by 4 | 6 • 4^{2} = 96 | 4^{3} = 64 | 6:4 (or 3:2) |
5 by 5 by 5 | 6 • 5^{2} = 150 | 5^{3} = 125 | 6:5 |
6 by 6 by 6 | 6 • 6^{2} = 216 | 6^{3} = 216 | 6:6 (or 1:1) |
7 by 7 by 7 | 6 • 7^{2} = 294 | 7^{3} = 343 | 6:7 |
8 by 8 by 8 | 6 • 8^{2} = 384 | 8^{3} = 512 | 6:8 (or 3:4) |
In general, the ratio is 6:s, since the surface area formula is 6s^{2} and the volume formula is s^{3}.
Here is the graph that shows what happens to the ratio of surface area to volume (expressed as a decimal) as the volume increases:
Problem B5
The superstore’s surface area-to-volume ratio is as small as possible, compared to smaller stores. This gives a company the ability to put more products inside the store (since volume is relatively large), in comparison to how much it may cost to build the store (since surface area is relatively small).
Problem B6
Answers will vary, but the baby has a higher surface area-to-volume ratio than the adult.
Problem B7
Since babies have a higher surface area-to-volume ratio, and water is a volume-based measurement, babies lose proportionately more water per minute than adults do. As a result, babies will dehydrate more quickly.
Problem B8
Problem B9
Problem B10
Answers will vary. One example is someone with the height of 170 cm and weight of 65 kg, who has the approximate surface area of 17,600 cm^{2}.
Problem B11
Answers will vary. For example, a person with the weight of 65 kg will have the volume of 58.5 L.
Problem B12