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Advanced Math / Nonlinear functions Difficulty: Hard

The total distance d, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms of t, where t is the time in seconds. At a time of 10.0 seconds, the total distance traveled by the object is 50.0 meters, and at a time of 20.0 seconds, the total distance traveled by the object is 200.0 meters. If the object was at a distance of 0 meters when $t equals 0$ , then what is the total distance traveled, in meters, by the object after 30.0 seconds?

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Explanation

The correct answer is 450. The quadratic equation that models this situation can be written in the form $d equals, a, t squared, plus b t, plus c$ , where a, b, and c are constants. It’s given that the distance, d, the object traveled was 0 meters when $t equals 0$ seconds. These values can be substituted into the equation to solve for a, b, and c: $0 equals, a, times 0 squared, plus, b times 0, plus c$ . Therefore, $c equals 0$ , and it follows that $d equals, a, t squared, plus b t$ . Since it’s also given that d is 50 when t is 10 and d is 200 when t is 20, these values for d and t can be substituted to create a system of two linear equations: $50 equals, a, times 10 squared, plus, b times 10$ and $200 equals, a, times 20 squared, plus, b times 20$ , or $10 a, plus b, equals 5$ and $20 a, plus b, equals 10$ . Subtracting the first equation from the second equation yields $10 a, equals 5$ , or $a, equals one half$ . Substituting $one half$ for a in the first equation and solving for b yields $b equals 0$ . Therefore, the equation that represents this situation is $d equals, one half t squared$ . Evaluating this function when $t equals 30$ seconds yields $d, equals, one half open parenthesis, 30, close parenthesis, squared, equals 450$ , or $d equals 450$ meters.