We are given a square with a side length of 74, and a circle inscribed in the square. We are asked to find the probability that a randomly chosen point within the square lies inside the circle. The key to solving this problem is understanding the ratio of the areas of the circle and the square.

Step-by-step solution:

1. Area of the square: The area of a square is given by: $\text{Area of the square} = \text{side length}^2 = 74^2 = 5476$

2. Radius of the inscribed circle: Since the circle is inscribed in the square, the diameter of the circle is equal to the side length of the square. Therefore, the radius of the circle is half the side length of the square: $\text{Radius of the circle} = \frac{74}{2} = 37$

3. Area of the circle: The area of a circle is given by: $\text{Area of the circle} = \pi r^2 = \pi (37)^2 = \pi \times 1369$ $\text{Area of the circle} \approx 3.1416 \times 1369 = 4300.84$

4. Probability calculation: The probability that a randomly chosen point is inside the circle is the ratio of the area of the circle to the area of the square: $\text{Probability} = \frac{\text{Area of the circle}}{\text{Area of the square}} = \frac{4300.84}{5476}$ $\text{Probability} \approx 0.7852$ Converting this to a percentage: $\text{Probability} \approx 0.7852 \times 100 = 78.52\%$

Final answer:

The probability that a randomly chosen point is inside the circle is approximately 78.5% (rounded to the nearest tenth of a percent).

Would you like further details or have any questions?

Related questions:

1. What would the probability be if the side length of the square was doubled?
2. How does the probability change if the circle is not inscribed, but placed randomly within the square?
3. What is the area of the square that lies outside the circle?
4. How does the probability change if the circle is replaced with an ellipse inscribed in the square?
5. How would you find the probability for a random point inside an inscribed triangle?

Tip:

When solving geometry-based probability problems, always start by finding the areas of both shapes and compare them for ratio-based solutions.