A traffic signal that is 24 feet tall casts a 32-foot shadow on level ground. At the same time, a fire hydrant nearby casts a 4-foot shadow.
What is the height, in feet, of the fire hydrant?
For word problems involving an object and its shadow under the same sun, immediately think of similar triangles: height and shadow lengths are proportional. Write a simple proportion using height/shadow for the known object and height/shadow for the unknown object, simplify any fractions to make arithmetic easier, and then solve for the missing value—often in one or two quick steps. Always check that the final ratio for your answer matches the original ratio to avoid common mistakes.
Hints
Think about the shapes formed
Imagine drawing a line from the top of each object (traffic signal and hydrant) to the end of its shadow. What kind of triangles do these form, and why might they be similar?
Compare height to shadow
For the traffic signal, you know both the height and the shadow length. What ratio can you form from these two measurements?
Set up a proportion
If the height-to-shadow ratio is the same for both objects, how can you use the ratio from the traffic signal to write an equation involving the hydrant’s unknown height and its 4-foot shadow?
Solve the equation
Once you have an equation of the form , solve for using either cross-multiplication or by simplifying the fraction first.
Desmos Guide
Compute the height-to-shadow ratio
In Desmos, type 24/32 and note the simplified decimal value; this is the common ratio of height to shadow for both objects.
Use the ratio to find the hydrant’s height
Next, type (24/32)*4 (the ratio multiplied by the hydrant’s 4-foot shadow). The numerical output Desmos gives for this expression is the hydrant’s height in feet.
Step-by-step Explanation
Recognize similar triangles
Both the traffic signal and the fire hydrant stand vertically and cast shadows on level ground at the same time of day. The sun’s rays are effectively parallel, so the triangles formed by each object and its shadow are similar right triangles. That means their height-to-shadow ratios are equal.
Write a proportion using height and shadow
Let be the height of the fire hydrant in feet.
For the traffic signal:
- Height:
- Shadow:
For the hydrant:
- Height:
- Shadow:
Since the triangles are similar, set up the proportion:
Solve the proportion for the hydrant’s height
Simplify the left side:
So we have:
Because the denominators are the same, the numerators must be equal, so .
Therefore, the height of the fire hydrant is 3 feet, which corresponds to answer choice B) 3.