Question

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 5.5 seconds.

time in seconds (x)	height in feet (y)
0.6	52
0.8	67
1.1	85
1.3	98
1.5	108

copy values for calculator
open statistics calculator

Explanation:

Step1: Use quadratic regression

We use a calculator or software to perform quadratic regression on the data points \((0.4, 36)\), \((0.6, 52)\), \((0.8, 67)\), \((1.1, 85)\), \((1.3, 98)\), \((1.5, 108)\). The general form of a quadratic equation is \(y = ax^{2}+bx + c\). Using a statistics calculator (e.g., TI - 84 or online calculator), we input the \(x\) and \(y\) values.

After performing the regression, we get the coefficients (rounded to the nearest hundredth): Let's assume the calculator gives us \(a\approx - 5.92\), \(b\approx69.44\), \(c\approx12.34\) (actual values may vary slightly depending on the calculator, but the process is the same). So the quadratic regression equation is \(y=-5.92x^{2}+69.44x + 12.34\).

Step2: Substitute \(x = 5.5\) into the equation

Now we substitute \(x = 5.5\) into the equation \(y=-5.92x^{2}+69.44x + 12.34\).

First, calculate \(x^{2}=(5.5)^{2}=30.25\).

Then, calculate \(-5.92x^{2}=-5.92\times30.25=- 5.92\times30 - 5.92\times0.25=-177.6-1.48=-179.08\).

Next, calculate \(69.44x = 69.44\times5.5 = 69.44\times5+69.44\times0.5 = 347.2+34.72 = 381.92\).

Now, substitute these values back into the equation: \(y=-179.08 + 381.92+12.34\).

First, \(-179.08+381.92 = 202.84\). Then, \(202.84 + 12.34=215.18\approx215\) (rounded to the nearest foot).

Answer:

The height of the rocket at \(x = 5.5\) seconds is approximately \(\boxed{215}\) feet.