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Sample Question 7:
Step 1: Calculate the area of the square. Since the side length of the square is 10, the area is \(10^2 = 100\).
Step 2: Calculate the area of the circle. The formula for the area of a circle is \(\pi r^2\), where \(r\) is the radius, so the area is \(\pi \times (10^2) = 100\pi\).
Step 3: Realize that exactly 1/4 of the circle lies inside the square.
Step 4: The correct area of the union of the regions enclosed by the square and the circle excludes the quarter of the circle that is inside the square, so we subtract 1/4 of the circle's area from the total circle's area, resulting in \(\frac{3}{4} \times 100\pi = 75\pi\).
Step 5: Add the area of the square and the adjusted area of the circle together to get the total area, \(100 + 75\pi\). This corresponds to answer choice B.
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
\((\mathrm {A}) 200+25\pi \quad (\mathrm {B}) 100+75\pi \quad (\mathrm {C}) 75+100\pi \quad (\mathrm {D}) 100+100\pi \quad (\mathrm {E}) 100+125\pi\)
Answer Keys
Question 7: BSolutions
Question 7Step 1: Calculate the area of the square. Since the side length of the square is 10, the area is \(10^2 = 100\).
Step 2: Calculate the area of the circle. The formula for the area of a circle is \(\pi r^2\), where \(r\) is the radius, so the area is \(\pi \times (10^2) = 100\pi\).
Step 3: Realize that exactly 1/4 of the circle lies inside the square.
Step 4: The correct area of the union of the regions enclosed by the square and the circle excludes the quarter of the circle that is inside the square, so we subtract 1/4 of the circle's area from the total circle's area, resulting in \(\frac{3}{4} \times 100\pi = 75\pi\).
Step 5: Add the area of the square and the adjusted area of the circle together to get the total area, \(100 + 75\pi\). This corresponds to answer choice B.