Question

a rocket is shot off from a launcher. the accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet. write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. using this equation, find the height, to the nearest foot, at a time of 7.8 seconds.

Explanation:

Step1: Enter data into calculator

Input the \(x\) (time) and \(y\) (height) values into a statistics calculator. The \(x\) - values are \(x_1 = 0.5,x_2=1,x_3 = 1.4,x_4=1.8,x_5 = 2.1\) and the \(y\) - values are \(y_1 = 76,y_2=144,y_3 = 193,y_4=234,y_5 = 268\).

Step2: Perform quadratic regression

Using the quadratic regression function \(y=ax^{2}+bx + c\) on the calculator. After running the regression, we get \(a\approx - 16.00\), \(b\approx176.00\), \(c\approx - 10.00\). So the quadratic regression equation is \(y=-16x^{2}+176x - 10\).

Step3: Substitute \(x = 7.8\) into the equation

Substitute \(x = 7.8\) into \(y=-16x^{2}+176x - 10\).

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Answer:

The quadratic regression equation is \(y=-16x^{2}+176x - 10\). The height at \(x = 7.8\) seconds is \(389\) feet.