Question

4 Suppose you throw a dart at a circular target of radius 10 inches. Assuming that you hit the target and that the coordinates of the outcomes are chosen at random, find the probability that the dart falls (a) within 2 inches of the center. (b) within 2 inches of the rim. (c) within the first quadrant of the target. (d) within the first quadrant and within 2 inches of the rim.

Answer 1

Answer:

0.04,0.25.0.52

Step-by-step explanation:

Given that you throw a dart at a circular target of radius 10 inches.

Assuming that you hit the target and that the coordinates of the outcomes are chosen at random,

probability that the dart falls

(a) within 2 inches of the center

Here favourable region has area of a circle with radius 2 inches and sample space has area of 10 inches

Prob = $\frac{\pi *2^2}{\pi *10^2} \\\\=\frac{1}{25}$

(b) within 2 inches of the rim.

For within two inches from the rim we have to select area of the ring i.e. area of big circle with 10 inches - area of smaller circle with 10-2 inches

Prob= $\frac{\pi * *8^2}{\pi*10^2} \\=0.64$

c) within I quadrant

area of I quadrant / area of circle=0.25

d) within I quadrant and within 2 inches of the rim

= I quadrant area + 2 inches ring area - common area

= $\frac{\pi*10^2/4 + pi*(10^2-8^2) - \pi(10^2-8^2)/4}{\pi*10^2} \\=\frac{25+36-9}{100} \\=0.52$