Your friend shoots a model rocket in the air and you take pictures of it, but you missed the picture at its highest point. The chart below shows the times and heights you recorded on your camera.

Question

Time (s)	0.5	1.0	1.5	2.0
Height (ft)	41	69	89	101

Fill in the blanks to write the equation: y= x² + x +

After finishing the equation,  find the height of the rocket at 3.4 seconds. Round to the nearest hundredth of a foot.

Answer 1

Ignoring air resistance and continued thrust, the rocket follows the path of a parabola when you plot height against time (also when plotting height against horizontal distance traveled). Since time is the independent variable and height is the dependent variable, you can plug in time for x and height for y into the standard equation for a parabola (y = ax² + bx + c). Using the first three pairs of values from the dataset, you get the following equations:

41 = .25a + .5b + c

69 = a + b + c

89 = 2.25a + 1.5b + c

Since we have a system of three equations and three variables, we can solve for the values of the variables a, b, and c. This can be done through substitution, equation subtraction, or matrix algebra. The result is a = -16, b = 80, and c = 5. This gives us the following equation as our result for the first part of the problem:

y = -16x² + 80x + 5

We can check out work by plugging in the fourth set of data, which comes out correctly (101 = -64 + 160 + 5). Also, note that instead of the above process, you can solve for the equation using quadratic regression function on a TI-83 or TI-84 calculator.

The second part of the problem is as simple as plugging 3.4 in for x in our equation, which gives us a result of 92.04 feet.

Interestingly, despite the first sentence of the problem, you were not required to find the time or time or height of the maximum point on the parabola. This can be done by using the maximum function on a TI calculator or by using the first derivative test from calculus.