A circular pool has a footpath around the circumference. The equation [tex]x^2+y^2=2,500[/tex], with units in feet, models the outside edge of the pool. The equation [tex]x^2+y^2=3,422.25[/tex], with units in feet, models the outside edge of the footpath. What is the width of the footpath?
Answer (2)
The radius of the pool is found by taking the square root of 2500: 2500 = 50 feet.
The radius of the outside edge of the footpath is found by taking the square root of 3422.25: 3422.25 = 58.5 feet.
The width of the footpath is the difference between the two radii: 58.5 − 50 = 8.5 feet.
The width of the footpath is 8.5 ft .
Explanation
Analyze the problem We are given the equation of the outside edge of the pool as x 2 + y 2 = 2500 and the equation of the outside edge of the footpath as x 2 + y 2 = 3422.25 . Both are in feet. We need to find the width of the footpath.
Find the radii The equation of a circle is given by x 2 + y 2 = r 2 , where r is the radius of the circle. Therefore, the radius of the pool is 2500 and the radius of the outside edge of the footpath is 3422.25 .
Calculate the radii Let's calculate the radius of the pool: r p oo l = 2500 = 50 feet Now, let's calculate the radius of the outside edge of the footpath: r p a t h = 3422.25 = 58.5 feet
Calculate the width The width of the footpath is the difference between the radius of the outside edge of the footpath and the radius of the pool: w = r p a t h − r p oo l = 58.5 − 50 = 8.5 feet
State the answer Therefore, the width of the footpath is 8.5 feet.
Examples
Imagine you're designing a circular garden with a path around it. Knowing the area you want for the garden and the total area including the path, you can use these equations to determine how wide the path should be. This ensures you have enough space for both the garden and a comfortable walkway around it, making your garden both beautiful and functional.
The width of the footpath around the circular pool is 8.5 feet. This is found by calculating the radii of both the pool and the footpath and taking the difference. The calculations are based on the equations provided for both shapes.
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