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Solving Proportions: More Examples (page 7 of 7)

Sections: Ratios, Proportions, Checking proportionality, Solving proportions

A very common class of exercises is finding the height of something very tall by using the daytime shadow length of that same thing, its shadow being down along the ground, and thus easily accessible and measurable. You use the known height of something shorter, along with the length of its daytime shadow as measured at the same time.

This process will of course only work if the ground is perfectly flat but, under that assumption, the reasoning is valid. The sun is far enough away that the rays of light that reach one general area on the planet (say, a particular parking lot) may safely be regarded as being parallel. This obviously would not be the case for a nearby light-source, such as a helicoptor hovering overhead:

As you can see, the light rays from the chopper's spotlight, aimed at the edge of the topmost part of the building and the top of the flagpole in the parking lot, are not even close to being parallel. On the other hand, the sun's rays in the same general area will be close enough to being parallel as makes no difference: Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

To extract the intended picture from the above, you would draw an horizontal line for the ground, vertical lines for the heights of the building and the flagpole, and slanty lines indicating the sun's rays. (An animation in the next exercise illustrates this process.) Because the slanty lines are assumed to be at the same angle from the horizontal, then these two triangles will be similar. Note that, because the height-lines and the ground are (assumed to be) perpendicular, the similar triangles are also right-angled triangles.

Exercises of this sort commonly ask for the heights of buildings, very tall trees, or oversized flag poles, based on known information from short trees, short poles, or simply a measuring stick stood on end, perfectly vertically, on the pavement.

• A building casts a 103-foot shadow at the same time that a 32-foot flagpole casts as 34.5-foot shadow. How tall is the building? (Round your answer to the nearest tenth.)
   

To set up this exercise, I first draw the building, the flagpole, a line for the (flat) ground, and the lines indicating the sunlight's path: This gives me two similar (and right) triangles.

Since the triangles are similar, I can set up a proportion and solve:

34.5 × h = 103 × 32  34.5h = 3296 h = 95.5362318841...

Re-checking the original exercise and how I defined "h", I see that I need to put units on my answer, and round my numerical value to one decimal place:

The building is about 95.5 feet tall.

There is one last type of problem that you may not even think of as being a "ratios and proportions" kind of problem, but it arises often in "real life". Sometimes when you are mixing something (such as "mixed drinks", animal feeds, children's play-clay, potting soil, or color dyes), the measurements are given in terms of "parts", rather than in terms of so many cups or gallons or milliliters. For instance:

• The instructions for mixing a certain type of concrete call for 1 part cement, 2 parts sand, and 3 parts gravel. (The amount of water to add will vary, of course, with the wetness of the sand used.) You have four cubic feet of sand. How much cement and gravel should you mix with this sand?
   

Since the sand is measured in cubic feet and the "recipe" is given in terms of "parts", I will let "one cubic foot" be "one part". The ratio of cement to sand is 1 : 2, and I have four cubic feet of sand. I will define "c" to stand for the amount of cement that I need, and I will set up and solve my proportion.

1 × 4  =  c × 2 4 = 2c 2 = c

I'd better not forget my units! The answer here is not "2", but the statement that "I need two cubic feet of cement".

Now I'll solve for the amount of gravel to add. The ratio of sand to gravel is 2 : 3, and I have four cubic feet of sand. I will define "g" to stand for the amount of gravel that I need, and I will set up and solve my proportion:

2 × g  =  4 × 3 2g = 12 g = 6

Keeping the units in mind, my answer is not "g = 6", but the statement that "I need six cubic feet of gravel". Then my complete answer is:

I need two cubic feet of cement and six cubic feet of gravel.

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