b Model the situation using a coordinate grid, and use geometric probability.
C
c Model the situation using a coordinate grid, and use geometric probability. In which cases should Meleah wait for the van for company A?
A
a 2584, approximately 0.30, or approximately 30 %
B
b 7084, approximately 0.83, or approximately 83 %
C
c See solution.
Practice makes perfect
a To find the probability that Meleah will have to wait 5 minutes or less to see each van, we will use geometric probability. Let's model this situation using a coordinate grid. The x-axis will represent the time until Meleah sees the van for company B, and the y-axis will represent the time until she sees the van for company A.
Since the van for company A arrives every 7 minutes, it will arrive in at most 7 minutes. Likewise, since the van for company B arrives every 12 minutes, it will arrive in at most 12 minutes.
The sample space in our model is represented by a rectangle with width 12 and length 7. Let's calculate its area A.
The outcomes where both vans appear in 5 minutes or less are represented by points with both coordinates less than or equal to 5.
The event that Meleah will have to wait for 5 minutes or less to see each van is represented by a square with side length 5. Let's calculate the area A_S of this square.
A_S = 5^2 ⇒ A_S = 25
The probability that Meleah will have to wait 5 minutes or less is the ratio of the area A_S of the region representing the outcomes where Meleah has to wait 5 minutes or less to see each van to the area A of the region representing the sample space.
The probability that Meleah will have to wait for 5 minutes or less to see each van is 2584, which can be written as a decimal as approximately 0.30, or approximately 30 %.
b To find the probability that Meleah will have to wait 5 minutes or less to see one of the vans, we will use geometric probability. Let's model this situation using a coordinate grid, like we did in Part A.
The outcomes where at least one of the vans appear in 5 minutes or less are represented by points with either coordinate less than or equal to 5.
The event that Meleah will have to wait 5 minutes or less for either van is represented by a region that consists of two rectangles and a square. The square has side length 5, one of the rectangles has length 5 and width 2, and the other rectangle has length 7 and width 5. Let's calculate the areas of these figures.
Length
Width
Area
Square
5
5
25
Rectangle 1
5
2
10
Rectangle 2
7
5
35
The area A_R of the entire region is the sum of the areas of the square and the two rectangles.
A_R = 25+10+35 ⇒ A_R = 70
Now, let's use geometric probability. The probability that Meleah will have to wait 5 minutes or less to see one of the vans is the ratio of the area A_R of the region representing this event to the area A of the region representing the sample space. We know from Part A that A is equal to 84.
The probability that Meleah will have to wait 5 minutes or less for one of the vans is 7084, which can be written as a decimal as approximately 0.83, or approximately 83 %.
c We want to consider a situation where the van for company B arrives before the van for company A and decide whether Meleah should wait for the van for company A. Let's model our situation using a coordinate grid, like we did in Part A.
Since in this case the van for company B arrives before the van for company A, we will only consider the points where the x-coordinate is less than the y-coordinate.
Here, the sample space is represented by a right triangle with legs of length a = 7 and b = 7. Let's calculate the area A of this region.
Now let's consider the situations where Meleah should wait for the van for company A. If the van for company A will arrive in less than 5 minutes Meleah will not be late, so she should wait. This situation is represented by the points with the y-coordinate less than or equal to 5.
If the van for company B arrives in more than 5 minutes, Meleah runs the risk of being late even if she does not wait, so she should wait as well. This situation is represented by the points with the x-coordinate greater than 5.
Now we will find the area of the shaded region. Note that the shaded region consists of two right triangles: one with both legs of length 5, and another with both legs of length 2. Let's calculate their areas!
Leg Length
Area
Simplify
5
1/2(5)(5)
12.5
2
1/2(2)(2)
2
The area A_S of the shaded region is the sum of the areas of the triangles.
A_S = 12.5 + 2 ⇒ A_S = 14.5
Now we can use geometric probability. The probability that Meleah should wait for the van for company A is the ratio of the area A_S of the shaded region to the area A of the region that represents the sample space.
The probability that Meleah should wait for the van for company A is approximately 0.59. Since this leaves a large probability that Meleah will risk being late for the competition, she should not wait for the van for company A.
