A square has side length of 4 metres. Inside the square is a circle of radius 1.5 metres.If a dart thrown at the square is equally likely to land at any point inside the square,then the probability that it will land outside the circle is closest to单项选择题
A
a. 0.250
B
b. 0.558
C
c. 0.442
D
d. 0.295

Question

A square has side length of 4 metres. Inside the square is a circle of radius 1.5 metres.If a dart thrown at the square is equally likely to land at any point inside the square,then the probability that it will land outside the circle is closest to单项选择题

A

a. 0.250

B

b. 0.558

C

c. 0.442

D

d. 0.295

BlackTom AI · Accepted Answer

Area approach: probability equals the area of the region outside the circle but inside the square divided by the area of the square.

1. Relevant knowledge points and concepts
- For a uniformly random point inside a region, probability of landing in a subregion = (area of subregion)/(area of whole region).
- Area of a square = side^2.
- Area of a circle = πr^2.
- Complement rule: probability outside the circle = 1 − probability inside the circle.

2. Step-by-step reasoning process
- Compute the square’s area.
- Compute the circle’s area.
- Subtract the circle’s area from the square’s area to get the area outside the circle.
- Divide that area by the square’s area (or use 1 − (circle area / square area)) to get the probability.

3. Calculations with intermediate steps
- Square area = side^2 = 4^2 = 16 square metres.
- Circle area = π r^2 = π × (1.5)^2 = π × 2.25 = 2.25π.
- Fraction of square covered by the circle = (2.25π) / 16 = 0.140625π.
- Numerically: 0.140625 × π ≈ 0.140625 × 3.14159265 ≈ 0.441964.
- Probability outside the circle = 1 − 0.441964 ≈ 0.558036.

4. Why the correct answer is correct
Because the dart is equally likely to land anywhere in the square, the chance it lands outside the inscribed circle equals the proportion of the square’s area not occupied by the circle. The arithmetic above gives ≈ 0.558036, which rounds to 0.558.

5. Why the other options are incorrect
- a. 0.250 is far too small; it would imply the circle covers 75% of the square, which is not the case.
- c. 0.442 is the probability of landing inside the circle (the complement), not outside.
- d. 0.295 does not match either the complement or the correct area calculation and is therefore incorrect.

6. Conclusion
The probability that the dart lands outside the circle is approximately 0.558, matching the provided correct answer (b. 0.558).

A square has side length of 4 metres. Inside the square is a circle of radius 1.5 metres.If a dart thrown at the square is equally likely to land at any point inside the square,then the probability that it will land outside the circle is closest to单项选择题

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