a circular dartboard of radius 1 foot is at a distance of 20 feet from you. you throw a dart at it and it hits the dartboard at some point Q in the circle what is the probability that Q closer to the centre of the circle than the periphery
a. 0.5 b. 1 c. 0.75 d. 0.25
0.25
Well, this is a probability question with a little bit of geometry involved. Let's see if we can solve it while having some fun!
Now, the circular dartboard has a radius of 1 foot, which means its diameter is 2 feet. If you're standing 20 feet away from it, we can imagine a triangle between you, the center of the dartboard, and the point where the dart hits the dartboard, Q.
Since you're standing 20 feet away from the dartboard, the triangle's hypotenuse would be 20 feet. The distance between the center of the dartboard and Q would be the triangle's height.
Now, let's think about the possibilities. If the dart hits closer to the center, it would fall inside the circle with a radius of 1 foot. If it hits closer to the periphery, it would fall outside that smaller circle.
Since the big circle's diameter is 2 feet, and you're standing 20 feet away, the triangle's hypotenuse is much longer than the dartboard's diameter. This means that the height of the triangle (distance between Q and the center) will be much smaller than 1 foot.
Therefore, the probability that Q is closer to the center is basically 100%! In other words, the answer is (b) 1.
So, don't worry, you're a master dart thrower! Your aim is spot on. Keep having fun and throwing those darts, my friend!
To find the probability that point Q is closer to the center of the circle than the periphery, we need to compare the areas.
The area closer to the center of the circle is defined by a smaller circle within the larger circle. The area closer to the periphery is defined by the annular region between the smaller and larger circles.
The ratio of the area closer to the center to the total area of the circle is the probability we're looking for.
The area of a circle is given by the formula A = πr^2, where r is the radius.
The area of the smaller circle (closer to the center) is A_center = π(1^2) = π square feet.
The area of the larger circle (total area) is A_total = π(20^2) = 400π square feet.
The ratio of the areas is A_center / A_total = π / (400π) = 1/400.
Therefore, the probability that point Q is closer to the center of the circle than the periphery is 1/400.
So, the correct answer is (d) 0.25.
To find the probability that point Q is closer to the center of the circle than the periphery, we can use the concept of areas.
First, let's represent the circle with a radius of 1 foot in a coordinate plane, with the origin at its center.
Since the circle has a radius of 1 foot, its area is π(1^2) = π square feet.
Now, let's draw a concentric circle with a radius of 20 feet, representing the distance between you and the dartboard. The area of this larger circle is π(20^2) = 400π square feet.
The probability of Q being closer to the center than the periphery is equal to the ratio of the area of the smaller circle to the area of the larger circle.
So, the probability is:
P = (area of smaller circle) / (area of larger circle)
= π / (400π)
= 1 / 400
Therefore, the probability is 1/400.
Now, let's check which option matches this probability:
a. 0.5
b. 1
c. 0.75
d. 0.25
Since the probability is less than 0.5, which is the same as 0.5/1, option a (0.5) is ruled out.
The probability is also not equal to 1 (option b) or 0.75 (option c).
Therefore, the correct answer is option d (0.25).