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 tenth. using this equation, find the height, to the nearest foot, at a time of 12.8 seconds.

time in seconds (x) | height in feet (y)
1.1 | 252
2.2 | 469
3.3 | 645
4.1 | 734
4.7 | 797
5.8 | 900

copy values for calculator
open statistics calculator

answer attempt 1 out of 3
regression equation:
final answer:

Explanation:

Step1: Perform Quadratic Regression

We use a calculator or software to perform quadratic regression on the data points \((x,y)\) where \(x\) is time (in seconds) and \(y\) is height (in feet). The data points are \((1.1, 252)\), \((2.2, 469)\), \((3.3, 645)\), \((4.1, 734)\), \((4.7, 797)\), \((5.8, 900)\). Using a quadratic regression tool (like a graphing calculator or statistical software), we find the quadratic equation of the form \(y = ax^{2}+bx + c\). After performing the regression, we get \(a\approx - 15.7\), \(b\approx223.6\), \(c\approx74.2\). So the quadratic regression equation is \(y=-15.7x^{2}+223.6x + 74.2\) (rounded to the nearest tenth).

Step2: Substitute \(x = 12.8\) into the Equation

Now we substitute \(x = 12.8\) into the quadratic equation \(y=-15.7x^{2}+223.6x + 74.2\).
First, calculate \(x^{2}=(12.8)^{2}=163.84\).
Then, calculate \(-15.7x^{2}=-15.7\times163.84=-15.7\times163.84\approx - 2572.288\).
Next, calculate \(223.6x=223.6\times12.8 = 223.6\times12.8=2862.08\).
Now, add the three terms together: \(y=-2572.288 + 2862.08+74.2\).
First, \(-2572.288+2862.08 = 289.792\).
Then, \(289.792 + 74.2=363.992\approx364\) (rounded to the nearest foot).

Answer:

The quadratic regression equation is \(y = - 15.7x^{2}+223.6x + 74.2\) and the height at \(x = 12.8\) seconds is \(\boxed{364}\) feet.