A person who is standing 15 ft away from a lamp post is casting a shadow that is 6 ft long. The angle of elevation from the tip of the person's shadow on the ground to the top of the lamp post is $27^{\circ}$. What is the height of the lamp post? (Picture is not drawn to scale.)

A$21 \cdot \tan \left(27^{\circ}\right)$
B$21 \cdot \cot \left(27^{\circ}\right)$
C$15 \cdot \tan \left(27^{\circ}\right)$
D$21 \cdot \sin \left(27^{\circ}\right)$
E$6 \cdot \tan \left(27^{\circ}\right)$

F. None of the above.

Question

04-11-2026
Mathematics

Answered

A person who is standing 15 ft away from a lamp post is casting a shadow that is 6 ft long. The angle of elevation from the tip of the person's shadow on the ground to the top of the lamp post is $27^{\circ}$. What is the height of the lamp post? (Picture is not drawn to scale.)

A$21 \cdot \tan \left(27^{\circ}\right)$
B$21 \cdot \cot \left(27^{\circ}\right)$
C$15 \cdot \tan \left(27^{\circ}\right)$
D$21 \cdot \sin \left(27^{\circ}\right)$
E$6 \cdot \tan \left(27^{\circ}\right)$

F. None of the above.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T23:55:39-05:00

Identify the right-angled triangle formed by the lamp post, the ground, and the tip of the shadow.
Determine the lengths: base length $AC = 21$ ft and height $H$.
Use the tangent function $\tan(27^\circ) = \frac{H}{21}$ to express the height.
Conclude that the height is $21 \cdot \tan(27^\circ)$.
Final Answer: $\boxed{21 \cdot \tan(27^\circ)}$.

Explanation

Analysis of the geometric setup

The problem describes a person standing near a lamp post, casting a shadow. We can represent the geometry of this situation as a right-angled triangle, where:

Let $T$ be the top of the lamp post, $A$ be its base on the ground, and $C$ be the tip of the shadow on the ground.
The lamp post stands vertically, so the line segment $TA$ is perpendicular to the ground line $AC$, making $\triangle TAC$ a right-angled triangle with the right angle at $A$.
The person is at a distance of $15$ ft from the lamp post, so the base of the person is at a point $B$ such that $AB = 15$ ft.
The shadow length cast by the person is $BC = 6$ ft.
The total distance from the base of the lamp post $A$ to the tip of the shadow $C$ is the sum of these segments: $AC = AB + BC = 15 \text{ ft} + 6 \text{ ft} = 21 \text{ ft}$.
The angle of elevation from the tip of the shadow (point $C$) to the top of the lamp post (point $T$) is given as $\angle TCA = 27^\circ$.

Calculating the height of the lamp post

In the right-angled triangle $\triangle TAC$, we have:

The angle $\angle TCA = 27^\circ$.
The side adjacent to the angle $\angle TCA$ is the ground segment $AC = 21$ ft.
The side opposite to the angle $\angle TCA$ is the height of the lamp post $AT = H$.
Using the trigonometric definition of the tangent function:

\[\tan(\angle TCA) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AT}{AC}\]

Substituting the known values:

\[\tan(27^\circ) = \frac{H}{21}\]

Solving for the height $H$:

\[H = 21 \cdot \tan(27^\circ)\]

Conclusion and final answer selection

Comparing the calculated height $H = 21 \cdot \tan(27^\circ)$ with the given options, we find it matches the first option. The final answer is $\boxed{21 \cdot \tan(27^\circ)}$.

Examples

This type of geometric problem is a classic application of trigonometry. In real life, land surveyors, architects, and engineers use similar principles to measure the heights of tall structures—like towers, buildings, or trees—without needing to climb them. By measuring the distance to the base of the object and the angle of elevation to its top, they can accurately calculate its height using the tangent function: $\text{height} = \text{distance} \times \tan(\theta)$.