Solving Real-World Problems with Trigonometry: Streetlight Height Calculation
Trigonometry is a powerful mathematical tool that can help us solve real-world problems, such as determining the height of a streetlight based on the shadows cast by objects. This article will walk you through the process using a practical example and provide step-by-step solutions and visual aids to enhance your understanding.
Example Problem
A 5 feet student casts a shadow that is 40 feet long while standing 200 feet from the base of a streetlight. We need to find the height of the streetlight.
Mathematical Setup and Solution
Step 1: Analyze the Problem
We have a student with a known height and a known shadow length. We also know the distance from the student to the base of the streetlight. We need to use these measurements to determine the height of the streetlight.
Step 2: Utilize Similar Triangles
The student and their shadow form one triangle, and the streetlight and its shadow form another triangle. Since the angles in these triangles are the same, the triangles are similar.
Step 3: Calculate Total Distance
To calculate the total distance from the streetlight to the end of the student's shadow, we add the distance from the student to the streetlight to the length of the student's shadow.
Total distance 200 feet (student to streetlight) 40 feet (student's shadow) 240 feet
Step 4: Set Up and Solve the Proportion
We can set up a proportion using the properties of similar triangles:
(frac{text{Height of the student}}{text{Length of the student's shadow}} frac{text{Height of the streetlight}}{text{Total distance to the end of the student's shadow}})
In numerical form this looks like:
(frac{5 text{ ft}}{40 text{ ft}} frac{h}{240 text{ ft}})
Step 5: Cross-Multiply and Solve for (h)
Cross-multiply to solve for the height of the streetlight, (h):
(5 text{ ft} cdot 240 text{ ft} 40 text{ ft} cdot h)
1200 40h)
(h frac{1200}{40} 30 text{ ft})
Conclusion
The height of the streetlight is 30 feet.
Other Methodologies
Tan Function Approach
Using the tangent function, we can also solve the problem by setting up the following equation:
(tan theta frac{5}{40} frac{h}{240})
Solving for (h), we get:
(h 30 text{ ft})
Ratio Method
The ratio of the height of the student to the length of the shadow is the same as the ratio of the height of the streetlight to the total distance from the streetlight to the end of the student's shadow:
(frac{5}{40} frac{h}{240})
Since the grade-school equivalent of 1/8, we can directly solve:
(h 30 text{ ft})
Visual Representation
Imagine a diagram with the student standing at point A, the base of the streetlight at point B, and the end of the student's shadow at point C. The streetlight is at point D.
The triangles ABC and AED are similar:
(frac{AE}{DE} frac{AC}{BC})
(frac{40}{5} frac{40 200}{h})
Solving this, we get (h 30 text{ ft}).
Visual Aid
Conclusion
Using either the properties of similar triangles, the tangent function, or the ratio of heights to shadows, we can accurately determine the height of a streetlight. Whether you use the traditional geometry methods or trigonometric functions, the key is to understand the relationship between the objects and their shadows.
Solving Similar Problems
Understanding how to solve this problem is a stepping stone to solving more complex trigonometric challenges. Practice similar problems with different measurements to solidify your understanding. Always remember the principles of similarity and the trigonometric functions to derive accurate results.