Solving Proportions: The Sun's Rays & "Parts"

This obviously would not be the case for a nearby light-source, such as a helicoptor hovering overhead:

As you can see above, the light rays from the chopper's spotlight, with respect to 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, as you can see below, the sun's rays in the same general area will be close enough to being parallel as makes no difference:

To extract the intended and necessary information from the above picture, we 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 exercise below illustrates this process.)

Because the slanty lines are assumed to be at the same angle from the horizontal, then these two triangles must be similar. Then we can use what you know about proportions and similar figures to solve the exercise.

Note that, because the height-lines and the ground are (assumed to be) perpendicular, the similar triangles are also right-angled triangles. This doesn't usually matter, but try to make the triangles that you draw look at least roughly right-angled.

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Exercises involving the sun's rays commonly ask for the heights of buildings, very tall trees, or oversized flag poles; and you're expected to determine these heights based on known information from short trees, short poles, or something as simple as a measuring stick stood on end, perfectly vertically, on the pavement, or maybe your own shadow.

• 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.)

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• You're concerned that your neighbor's backyard flagpole may fall over in heavy winds (it looked really wobbly during the last thunderstorm), and you're hoping it doesn't fall over and hit your house. One afternoon, you notice that the four-foot high chain-link fence is casting a shadow seven feet long, while the neighbor's pole is casting a shadow 26 feet long. The base of the pole is 18 feet from your house. Is your house safe?

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• Will is heading out to his car. He notices a tall pine tree whose shadow, from base to tip (and parallel to the parking lot), is currently the length of two parking spaces, each of which is 2.75 meters wide. Will is 178 cm tall, and he notices that, with his back to the sun, his shadow extends exactly one shoe-length length past the end of his feet. When he gets home, he finds that his shoes are 33 cm long. How tall is the tree? (Round to the nearest half-meter.)

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"Parts"

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 being used, and will be determined on-site.) You have four cubic feet of sand. How much cement and gravel should you mix with this sand?

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• A particular liquid-cleanser concentrate says to dilute it by combining one part concentrate with 32 parts water. Your spray bottle's top "fill" line says "946 mL". How much concentrate should you pour into the bottle? (You'll then add water up to the fill line.) Round to the nearest whole number.