b Calculate the travel time on each route and find the shortest one.
A
a County Line Road, see solution.
B
b Highway 4 and Route 6, see solution.
Practice makes perfect
a Let's consider the given diagram.
We can see that County Line Road, Highway 4, and Route 6 are the sides of the triangle. Let's now review what the Triangle Inequality Theorem states.
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Using this theorem, we can conclude that the length of the County Line Road is less than the sum of the lengths of Highway 4 and Route 6. Therefore, Takoda should take the Country Line Road.
Let's check this answer by comparing the lengths of the possible routes. We are given that Highway 4 is 22 miles long and Route 6 is 25 miles long. Adding these two values, we can find the length of the first route.
22+25=47miles
We also know that the County Line Road is 30 miles long. This is the length of the second possible route.
47>30
The second route, which goes along the County Line Road, is shorter than the first route. Our answer is correct.
b From the diagram we know that County Line Road is 30 miles long. It is also given that the speed limit on this road is 30 miles per hour. If Takoda drives at this speed, it takes him an hour to reach the Sportsplex. However, since he always drives below the speed limit, it will take him longer than an hour.
County Line Road: t>1hour
To find the time Takoda will travel along Highway 4 and Route 6, we can use the following formula.
t=d/s
Here t is the time, d is the distance, and r is the rate. In Part A we have calculated that the length of this route is 47 miles. We are also told that the speed limit on this route is 55 miles per hour. Let's substitute these values into the formula and find t.
Next, let's multiply 0.85 by 60, to convert this time from hours to minutes.
0.85 * 60=51min
Thus, it will take at least 51 minutes for Takoda to reach the Sportsplex driving the second route. This is less than an hour, so the second route that consists of Highway 4 and Route 6 is faster.
