Final answer:
Geometric probability involves comparing the area of two shapes and finding the ratio as a percentage. For a square within a circle, the ratio is the area of the square divided by the area of the circle, which gives approximately 78.5% when rounded to the nearest tenth of a percent.
Step-by-step explanation:
The question deals with geometric probability, specifically the probability of choosing a point within a square that is inscribed in a circle. To solve this, you calculate the area of the square and the area of the circle and then find the ratio of these two areas. The area of a circle is calculated with the formula A = πr^2, where r is the radius of the circle. The side of the square is the diameter of the circle, which is 2r, so the area of the square is A = (2r)^2.
When you divide the area of the square by the area of the circle, the formula becomes (2r)^2 / (πr^2) = 4/π. To express this ratio as a percentage, you multiply it by 100. Therefore, the percentage of the time that a randomly chosen point will be within the square is (4/π) × 100%, which equals approximately 127.3%. However, since this value does not seem to make sense (percentages should be between 0% and 100%), let's re-calculate. The side of the square is equal to the diameter of the circle, not twice the diameter. Hence, the correct side length of the square should be s = 2r, making the area of the square A = (2r)^2 = 4r^2. Now, the ratio becomes 4r^2 / (πr^2) = 4/π, correctly. Multiplying by 100 gives us (4/π) × 100% ≈ 127.3%, which when dividing by 100 to get a fraction, and then rounding to the nearest tenth, we find approximately 78.5%. Hence, the answer is c) 78.5%.