Rate of Shadow Movement: A Similar Triangles and Related Rates Problem
Imagine a scenario where a six-foot-tall man is walking away from a 15-foot light pole. This interesting problem involves understanding shadow movement using principles of similar triangles and related rates. In this article, we'll explore the mathematics behind the movement of the man's shadow and provide a detailed step-by-step solution.
Setting Up the Problem
Let's establish the initial conditions and variables. Our six-foot-tall man is walking away from a 15-foot light pole at a rate of 13 feet per second. Let's denote the height of the man as hm and the height of the light pole as hl. The man is initially 5 feet away from the base of the light when we start observing his shadow's movement.
Key Variables
hm 6 feet (man's height) hl 15 feet (light pole's height) dx/dt 13 feet/second (rate at which the man is walking away) x 5 feet (current distance from the man to the base of the light) s length of the shadowUsing Similar Triangles
To understand the relationship between the man, the light, and his shadow, we can use the principle of similar triangles. We have two right triangles here:
A larger triangle formed by the light, the tip of the shadow, and the ground. A smaller triangle formed by the man, the tip of his shadow, and the ground.These triangles are similar, meaning their corresponding sides are proportional. We can set up the following proportion based on their heights and bases:
[ frac{hl}{x s} frac{hm}{s} ]Substituting the known heights:
[ frac{15}{x s} frac{6}{s} ]Solving the Proportion
Cross-multiplying gives:
[ 15s 6(x s) ]Expanding and rearranging the equation:
[ 15s 6x 6s ] [ 9s 6x ] [ s frac{2}{3}x ]To find the rate at which the shadow is moving, we differentiate both sides of the equation with respect to time t:
[ frac{ds}{dt} frac{2}{3} frac{dx}{dt} ]Substituting dx/dt 13 feet/second:
[ frac{ds}{dt} frac{2}{3} times 13 frac{26}{3} text{ feet/second} ]Calculating the Rate at Which the Tip of the Shadow Moves
The total rate at which the tip of the shadow is moving is the sum of the man's walking rate and the shadow's expanding rate:
[ frac{dx s}{dt} frac{dx}{dt} frac{ds}{dt} ]Substituting the known values:
[ frac{dx s}{dt} 13 frac{26}{3} frac{39 26}{3} frac{65}{3} text{ feet/second} ]Conclusion
Therefore, the tip of the shadow is moving away from the light at a rate of (frac{65}{3}) feet per second, or approximately 21.67 feet per second.
Graphical representation
The graphical representation would show the relationship between the distance the man moves and the length of his shadow.
Related Keywords
similar triangles related rates geometryExplanatory Diagram
Here is a diagram to visualise the triangles:
Utilizing Principles of Right Triangles and Proportions
The key is understanding that the two triangles are similar and using this property to set up the proportion. The man's shadow is a direct result of the height of the light pole and the distance the man is from the pole. As the man walks, his shadow grows in proportion to the distance he moves away from the light pole.
Conclusion
Through this problem, we have demonstrated how principles of similarity and related rates can be applied to real-world scenarios involving shadow movement. Understanding these concepts is crucial for a wide range of mathematical and physical applications.