The total distanced, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms oft, wheretis the time in seconds. At a time of10.0seconds, the total distance traveled by the object is50.0meters, and at a time of20.0seconds, the total distance traveled by the object is200.0meters. If the object was at a distance of0meters whent=0, then what is the total distance traveled, in meters, by the object after30.0seconds?
Question
The total distanced, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms oft, wheretis the time in seconds. At a time of10.0seconds, the total distance traveled by the object is50.0meters, and at a time of20.0seconds, the total distance traveled by the object is200.0meters. If the object was at a distance of0meters whent=0, then what is the total distance traveled, in meters, by the object after30.0seconds?
Solution
The problem states that the total distance traveled by an object can be modeled by a quadratic function. A quadratic function has the form f(t) = at^2 + bt + c, where t is the time, a, b, and c are constants, and f(t) is the total distance traveled at time t.
We know three points on this curve:
(0,0) because the object was at a distance of 0 meters when t=0. (10,50) because the object was at a distance of 50 meters when t=10. (20,200) because the object was at a distance of 200 meters when t=20.
We can use these points to form a system of equations to solve for a, b, and c.
From the point (0,0), we know that c = 0.
Substituting the points (10,50) and (20,200) into the equation f(t) = at^2 + bt + c gives us the following two equations:
50 = 100a + 10b 200 = 400a + 20b
Solving this system of equations gives a = 1 and b = 0.
So, the function that models the total distance traveled by the object is f(t) = t^2.
To find the total distance traveled by the object after 30 seconds, we substitute t = 30 into the function:
f(30) = (30)^2 = 900 meters.
So, the object will have traveled a total of 900 meters after 30 seconds.
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