The distance from city A to city B is approx. 2160 miles. A plane flying directly to city B passes over A at noon. If the plane travels at 500 mph, find the rule of the function f(t) that gives the distance of the plane from city B at time t hours (with t=0 corresponding to noon)
f(t)=?
f(t) = 2160 - 500 t
(or (t - 12) if using a 24 hour clock)
To find the rule of the function f(t) that gives the distance of the plane from city B at time t hours, we need to take into account the speed of the plane and the initial distance between city A and city B.
First, let's determine how long it will take the plane to reach city B. We know that the plane's speed is 500 mph, and the distance from city A to city B is 2160 miles. We can use the formula:
Time = Distance / Speed
Plugging in the values, we get:
Time = 2160 miles / 500 mph
Simplifying, we find:
Time = 4.32 hours
This means that it will take the plane 4.32 hours to reach city B.
Now, at noon (t = 0 hours), the plane is passing over city A. As time passes, the distance from city B will decrease. We can represent this relationship using a linear function, where the distance is a linear function of time.
The general equation for a linear function is:
y = mx + b
In this case, the distance (y) is a function of time (t). We can write:
f(t) = mt + b
To find the specific equation for our problem, we need to determine the slope (m) and the y-intercept (b).
The slope of the line represents the rate at which the distance from city B changes with time. In our case, the plane is traveling at a constant speed of 500 mph. Therefore, the slope (m) is -500 (negative because the distance from city B decreases).
The y-intercept (b) represents the initial distance at time t = 0. At t = 0, the plane has not yet started its journey, so the distance from city B is 2160 miles. Therefore, the y-intercept (b) is 2160.
Plugging the values into the equation, we get:
f(t) = -500t + 2160
So, the rule of the function f(t) that gives the distance of the plane from city B at time t hours is:
f(t) = -500t + 2160