Final answer:
To find the radii of four concentric circles with areas increasing by the same amount, you use the area formula for a circle, A = πr², and solve a system of equations based on the areas of these circles.
Step-by-step explanation:
The student is asking about finding the radii of four concentric circles where the area of the center circle and the areas between each successive circle are equal. To solve this, we use the formula for the area of a circle, A = πr², where A is the area and r is the radius. The largest circle has a diameter of 3 ft, so its radius is 1.5 ft. We'll denote the radii of the four circles as r1, r2, r3, and r4, with r4 being the largest (1.5 ft) and r1 the smallest.
Since the area of the center circle must be equal to the area of each ring, we can set up the equations based on the areas of these circles. For the innermost circle we get πr1². For the area between the first and second circle, we have πr2² - πr1² = πr1² where r2 is the second radius we're looking for. Simplifying gives us r2 = √(2)·r1. Using a similar process with πr3² - πr2² = πr1² and πr4² - πr3² = πr1², we will find r3 and r1. By solving this set of equations, we find the radii of all four circles.