A dog runs north at a speed of four meters per second for 25 seconds and then runs west at a speed of 6 meters per second for 10 seconds. What is the total distance the dog travels?
In this question, we have a dog that is running, and its journey is comprised of two parts. First the dog runs north, then it runs west. So, let’s use the information from the question to draw a quick sketch showing these two parts of the dog’s journey. The dog starts at some position that we’ve labeled with an 𝑥 and initially runs in the north direction. During this part of the journey, we’re told that the dog runs with a speed of four meters per second for a time of 25 seconds. We’ll label this speed 𝑠 one. So, we have that 𝑠 one is equal to four meters per second. And we’ll label this time 𝑡 one so that we have 𝑡 one is equal to 25 seconds. During this part of the journey, the dog will cover some distance that we’ll label 𝑑 one.
In the second part of the journey, the dog runs to the west. We’ll label the distance that the dog runs in the west direction as 𝑑 two. We know that during the second part of the journey, the dog runs at a speed of six meters per second for a time of 10 seconds. We’ll label this speed 𝑠 two so that 𝑠 two is equal to six meters per second and the time as 𝑡 two so that 𝑡 two is equal to 10 seconds. Then, the dog ends its journey at this position here.
The question is asking us to work out the total distance that the dog travels. So, that’s the total distance between the start position and the end position. Recall that distance is defined as the length of the path between two positions. If we look at our diagram, we can see that the path taken by the dog consists of two straight line segments. The first of these is in the north direction and has a length that we’ve labeled 𝑑 one. The second is in the west direction with a length 𝑑 two.
We know that the total distance traveled by the dog which we’ve labeled 𝑑 is equal to the length of the path that it takes between its start position and its end position. So, that’s the distance 𝑑 one from the first segment of its journey plus the distance 𝑑 two from the second segment. In order to calculate the total distance traveled, we therefore need to find the values of 𝑑 one and 𝑑 two.
For each of the two parts of the dog’s journey, we know its speed and we know the time that it travels for. We can recall that there is a formula that relates the three quantities speed, distance, and time. Specifically, for a speed 𝑠, a distance 𝑑, and a time 𝑡, we have that 𝑠 is equal to 𝑑 divided by 𝑡. In our case, for both parts of the journey, we have a value for speed and for time, and we’re looking to calculate a value of distance. So, let’s rearrange this formula to make 𝑑 the subject. If we multiply both sides by 𝑡, then on the right-hand side, the 𝑡’s in the numerator and denominator cancel each other out. Then swapping the left- and right-hand sides of this equation over, we have that distance 𝑑 is equal to speed 𝑠 multiplied by time 𝑡.
Now, let’s take this equation and apply it to each part of the dog’s journey. In the first part of the journey, the dog travels some distance 𝑑 one equal to the speed 𝑠 one multiplied by the time 𝑡 one. We know that 𝑠 one is equal to four meters per second and 𝑡 one is equal to 25 seconds. Substituting in these values, we get that 𝑑 one is equal to four meters per second multiplied by 25 seconds. When we do this multiplication, we get a result of 100 meters. So, we have that the distance 𝑑 one traveled by the dog during the first northward part of its journey is equal to 100 meters.
Now, we’ll do the same thing for the second part of the journey. This time, the dog travels some distance 𝑑 two that’s equal to its speed 𝑠 two multiplied by the time 𝑡 two. We know that 𝑠 two is equal to six meters per second and 𝑡 two is equal to 10 seconds. Substituting these values in gives us that 𝑑 two is equal to six meters per second multiplied by 10 seconds. Doing the multiplication gives a result for 𝑑 two, the distance traveled by the dog during the second westward part of its journey, of 60 meters.
So, we know that 𝑑 one is equal to 100 meters and 𝑑 two is equal to 60 meters. And we also know that the total distance traveled by the dog is equal to 𝑑 one plus 𝑑 two. When we substitute in these values for 𝑑 one and 𝑑 two, we have that the total distance 𝑑 traveled by the dog is given by 100 meters plus 60 meters. Adding together 100 meters and 60 meters gives us our final result that the total distance traveled by the dog is equal to 160 meters.