A flagpole that is 18 feet tall casts a 6-foot shadow on level ground at a certain time of day. At the same time, a nearby signpost casts a 1.5-foot shadow. How tall, in feet, is the signpost?
Shadow questions at the same time use similar triangles: upright objects and their shadows make the same shape because the sunlight meets them at the same angle. Match each height with its own shadow, then use a Desmos regression to find the unknown height. Keep height over shadow in both ratios. Reversing one ratio gives the wrong size.
Hints
- Hint 1
Each upright object and its shadow form a right triangle. Because the sunlight arrives at the same angle at the same time, the triangles are similar: they have the same shape, though they differ in size.
- Hint 2
In similar triangles, matching sides keep the same ratio. Pair each object's height with its own shadow. How would you write height divided by shadow for the flagpole and for the signpost?
Step-by-step
Match heights with shadows
Step 1Identify the similar triangles
Each upright object makes a right angle with its shadow on level ground. At the same time, the sun's rays give the triangles another matching angle. By angle-angle similarity, two matching angles make the triangles the same shape, so their matching sides have the same ratio.
- Step 2
Write height over shadow
Let be the signpost's height. The flagpole's -foot height matches , while its -foot shadow matches the signpost's -foot shadow. Keep height over shadow in both ratios:
Putting over on only one side would reverse one comparison.
- Step 3
Find the signpost's height
Type in Desmos. Use to ask Desmos to fit the unknown, and instead of so it solves rather than graphs. Under PARAMETERS, Desmos shows . Since represents the signpost's height, the signpost is feet tall. Choice C.