A tourist being chased by an angry bear is running in a straight line toward his car at a speed of 4.38 m/s. The car is a distance d away. The bear is 25.4 m behind the tourist and running at 5.88 m/s. The tourist reaches the car safely. What is the maximum possible value for d?
I know the equations I need are...
d + x = v(bear) * t
d = v(person) * t
I just don't know what to do with them.
The distance between tourist and bear "closes" at a rate of 1.5 m/s. It becomes zero at time T = 25.4/(1.5 m/s). In that time, the tourist runs
4.38 m/s * T = (4.38/1.5)*25.4 = 74.2 m. That is the longset possible distance without being caught.
To find the maximum possible value for d, we need to consider the scenario where the bear is closest to catching the tourist.
Let's start by calculating the time it takes for the bear to catch up with the tourist.
The equation for the distance traveled by the bear is:
25.4 m + x = (5.88 m/s) * t
The equation for the distance traveled by the tourist is:
d = (4.38 m/s) * t
Since the bear and the tourist are moving towards each other, the total distance traveled by both of them is equal to the distance between them.
25.4 m + x + d = total distance
We can substitute the equations for both the bear and the tourist's distances into the total distance equation:
25.4 m + x + (4.38 m/s) * t = (5.88 m/s) * t
Now, we can solve this equation for t:
25.4 m + x = (5.88 m/s - 4.38 m/s) * t
25.4 m + x = 1.5 m/s * t
Since we are looking for the maximum possible value of d, we want to find the scenario where the bear catches up with the tourist in the shortest amount of time. In other words, we want to minimize the time t.
To minimize t, we want to maximize x. This is because when the bear catches up with the tourist, the total distance traveled by both of them would be minimized.
So, we can set x = 0 to find the shortest possible time for the bear to catch up with the tourist:
25.4 m + 0 = 1.5 m/s * t
25.4 m = 1.5 m/s * t
Solving for t:
t = 25.4 m / (1.5 m/s)
t ≈ 16.93 s
Now, we can find the maximum possible value for d by substituting this value of t into the equation for the tourist's distance:
d = (4.38 m/s) * t
d = (4.38 m/s) * 16.93 s
d ≈ 74.18 m
Therefore, the maximum possible value for d is approximately 74.18 meters.
To find the maximum possible value for d, we need to use the given information about the distances and speeds of the tourist and the bear.
Let's assume that the time it takes for the tourist to reach his car is 't'. During this time, the bear will also cover a certain distance based on its speed.
From the given information, we can write the following equations:
Distance covered by the tourist: d = v(person) * t
Distance covered by the bear: d + 25.4 = v(bear) * t
Since the tourist reaches the car safely, it means that at some point, the bear must be stopped or caught up by the tourist. This implies that the distances covered by the tourist and the bear are equal. So we can set the two equations equal to each other:
v(person) * t = v(bear) * t - 25.4
Now, let's substitute the known values into the equation. The speed of the tourist is 4.38 m/s, and the speed of the bear is 5.88 m/s:
4.38 * t = 5.88 * t - 25.4
Simplifying the equation:
25.4 = 1.5 * t
Now, solve for t:
t = 25.4 / 1.5
t ≈ 16.93 seconds
Now that we have the time it takes for the tourist to reach the car, we can substitute this value back into one of the original equations to find the maximum possible value for d:
d = v(person) * t = 4.38 * 16.93
d ≈ 74.17 meters
Therefore, the maximum possible value for d is approximately 74.17 meters.