How much larger is the area of the circle enclosed by x^2 + y^2 = 100 than the area of the circle enclosed by x^2 + y^2 = 25?
My answer: The area of the circle enclosed by x^2 + y^2 = 100 is the times larger than the area of the circle enclosed by x^2 + y^2 = 25.
Hmmm. pretty coy there. Not sure what to make of "the times larger."
The ratio of areas is 100/25 = 4
To find the difference in the areas of the two circles, we need to calculate the areas of each circle separately and then subtract the smaller area from the larger area.
The equation x^2 + y^2 = r^2 represents a circle with radius r. Given that the first circle has r = 10 (r^2 = 100) and the second circle has r = 5 (r^2 = 25), we can calculate their respective areas:
Area of the first circle = π * (10^2)
Area of the first circle = π * 100
Area of the second circle = π * (5^2)
Area of the second circle = π * 25
To find the difference in the areas, we subtract the area of the smaller circle from the area of the larger circle:
Difference in areas = (π * 100) - (π * 25)
Difference in areas = π * (100 - 25)
Difference in areas = π * 75
Therefore, the area of the circle enclosed by x^2 + y^2 = 100 is 75π (approximately 235.62) larger than the area of the circle enclosed by x^2 + y^2 = 25.