Question

50 POINTS A point in the figure is selected at random. Find the probability that the point will be in the shaded region.

Answer 1

The first step that we need to take before attempting to solve the problem is to understand what the problem is asking us to do and what they are giving us to help solve the problem.  Looking at the problem statement they are asking for us to determine the probability that a point will randomly be plotted in the shaded region.  We are not given much of anything else which means that we will need to use our own numbers.

The picture that was provided has a square with four equal circles inside right next to each other.  Therefore, we can say that each side of the square is going to be 2 units which causes the diameter of the circle to be half that or 1 unit.  We can go even further and determine that the radius is going to be 0.5 units for each circle.  Let's determine the area of all the shapes.

Area of the square

• $A_{square} = s^2$

• $A_{square} = (2\ units)^2$

• $A_{square} = (2)^2 * (units)^2$

• $A_{square} = 4\ units^2$

Area of a circle

• $A_{circle} = \pi * r^2$

• $A_{circle} = \pi * (0.5\ units)^2$

• $A_{circle} = \pi * (0.5)^2 * (units)^2$

• $A_{circle} = \pi * (0.25\ units)^2$

• $A_{circle} = 0.7854\ units^2$

The area that we got from the circle only gives us the area for one of the circles so we need to multiply the number by four to give us the total area of the circles.

Total area of the circles

• $A_{circles} = 0.7854\ units^2 * 4$

• $A_{circles} = 3.1416\ units^2$

Now that we determined the area of both the square and the circles we can move onto the part of finding the probability of a point randomly landing on a circle.

Determine the probability

• $\textsf{probability = circles / square}$

• $\textsf{probability = } \frac{3.1416\ units^2}{4\ units^2}$

• $\textsf{probability = } \frac{3.1416}{4}$

• $\textsf{probability = } 0.785$

However, now that we have determined what the probability, looking at the answer options we can see that all of the are in percentages.  So let's convert our probability into a percentage.

Convert to percentage

• $\textsf{probability = } 0.785 * 100$

• $\textsf{probability = } 78.5\%$

Therefore, looking at the options given, the option that would best fit this choice would be option B, about 80%.