A car is traveling at a constant speed until the car’s brakes are applied. The car’s speed changes at a rate given by $0.08t \, \text{m/s}^2$ after the brakes are applied, where $t$ is the time since the brakes were applied. Three seconds after the brakes are applied, the speed of the car is $15 \, \text{m/s}$. How far will the car travel with the brakes applied before it stops?

The subject of the question is Physics, and the grade level is High School. The student wants to know how far a car will travel before it comes to a stop after the brakes are applied. In order to answer this question, we must first determine how long it will take for the car to come to a stop.

In this particular case, the car's speed change is given by the equation 0.08t m/s², where t is the time in seconds since the brakes are applied. This gives us an acceleration (which in this case is deceleration because the car is slowing down) of -0.08t. After three seconds, the car has a speed of 15 m/s, so the time when the car stops will be when its speed is zero.

Using the formula vf = vi + a*t, where vf is the final velocity (which we know is 0 m/s when the car stops), vi is the initial velocity (15 m/s in this case), a is acceleration (-0.08t in this case), and t is time, we solve for t and find that t = 187.5 seconds.

Now, to find the distance travelled, we'll use the formula d = vi*t + 0.5*a*t². Substituting our known values into this equation, we find that the car will travel 1406.25 meters before it stops, after the brakes are applied.

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