Efficiency usually refers to comparing two quantities. In Geometry, when we discuss the efficiency of solids, we want to compare volumes with surface areas of those figures.
Let’s say you own a business selling cereal. You want to make sure that the boxes you make for your cereal don’t waste material to make the box and also hold the most amount of cereal! This ratio comparing how much it holds (in other words, volume) and how much material it takes to make the box (in other words, total surface area), is called efficiency.
Most of the time, you want a box to hold more for the least amount of material to make the box. Efficiency means more volume per less surface area.
Let’s say you own a business selling cereal. You want to make sure that the boxes you make for your cereal don’t waste material to make the box and also hold the most amount of cereal! This ratio comparing how much it holds (in other words, volume) and how much material it takes to make the box (in other words, total surface area), is called efficiency.
Most of the time, you want a box to hold more for the least amount of material to make the box. Efficiency means more volume per less surface area.
Example 1:
Using the efficiency ratio for each box, determine which box is more efficient.
Using the efficiency ratio for each box, determine which box is more efficient.
|
Volume (cubic units) |
Surface Area (square units) |
Box A |
120 |
98 |
Box B |
150 |
124 |
Solution:
At first glance, it looks like Box B holds more, as it has more volume than Box A. However, when we are looking at efficiency, we need to find the ratio between volume and surface area for each box:
Efficiency Ratio for Box A = \(\large\frac{\text{Volume of Box A}}{\text{TSA of Box A}}= \frac{120}{98} = \normalsize 1.2244898….\)
Efficiency Ratio for Box B = \(\large\frac{\text{Volume of Box B}}{\text{TSA of Box B}}= \frac{150}{124} = \normalsize 1.20967742….\)
Since the efficiency ratio for Box A is larger than the efficiency ratio for Box B, Box A is more efficient.
Example 2:
Given the following table, what is a possible value for total surface area of container C that will make container C most efficient?
At first glance, it looks like Box B holds more, as it has more volume than Box A. However, when we are looking at efficiency, we need to find the ratio between volume and surface area for each box:
Efficiency Ratio for Box A = \(\large\frac{\text{Volume of Box A}}{\text{TSA of Box A}}= \frac{120}{98} = \normalsize 1.2244898….\)
Efficiency Ratio for Box B = \(\large\frac{\text{Volume of Box B}}{\text{TSA of Box B}}= \frac{150}{124} = \normalsize 1.20967742….\)
Since the efficiency ratio for Box A is larger than the efficiency ratio for Box B, Box A is more efficient.
Example 2:
Given the following table, what is a possible value for total surface area of container C that will make container C most efficient?
\(\) |
Volume (cubic units) |
Surface Area (square units) |
Container A |
300 |
123 |
Container B |
300 |
400 |
Container C |
300 |
??? |
Container D |
300 |
250 |
Solution:
Notice that the volumes for each container is the same. In order for us to find the largest efficiency ratio, we need to choose the box that holds the most for the least surface area. Each of them hold the same amount, but currently Box A has the least surface area. In order to make container C the most efficient, its total surface area must be smaller than that of Box A.
Therefore, the total surface area of container C must be less than \(123\) square units.
Notice that the volumes for each container is the same. In order for us to find the largest efficiency ratio, we need to choose the box that holds the most for the least surface area. Each of them hold the same amount, but currently Box A has the least surface area. In order to make container C the most efficient, its total surface area must be smaller than that of Box A.
Therefore, the total surface area of container C must be less than \(123\) square units.
Example 3:
Your company manufactures candy bars:
Your company manufactures candy bars:
Your company would like to know how increasing the height of the candy bars by \(10\%\) would affect the volume, surface area and efficiency ratio of the package of candy.
Solution:
Watch the video for the solution.
Watch the video for the solution.
Quick Check
1) Determine which of the following containers is the most efficient.
1) Determine which of the following containers is the most efficient.
\(\) |
Volume (cubic units) |
Surface Area (square units) |
Container L |
123 |
425 |
Container I |
123 |
320 |
Container V |
123 |
404 |
Container E |
123 |
190 |
2) Given the following table, what is a possible value for the volume of container O that will make container O the most efficient?
\(\) |
Volume (cubic units) |
Surface Area (square units) |
Container L |
300 |
347 |
Container O |
??? |
347 |
Container V |
423 |
347 |
Container E |
127 |
347 |
3) Given the following table, which of the following containers is the LEAST efficient.
\(\) |
Volume (cubic units) |
Surface Area (square units) |
Container M |
248 |
259 |
Container A |
540 |
562 |
Container T |
371 |
235 |
Container H |
718 |
820 |