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 5.7 seconds.

time in seconds (x) | height in feet (y)
0.3 | 40
0.8 | 101
1.1 | 131
1.5 | 169
1.8 | 198

copy values for calculator
open statistics calculator

answer attempt 1 out of 2

regression equation:
final answer:

Explanation:

Step 1: Find Quadratic Regression Equation

We use the given data points to perform quadratic regression. Using a calculator or software, we determine the coefficients \( a \), \( b \), and \( c \) for the quadratic equation \( y = ax^2 + bx + c \). After calculation, we get \( a \approx -5.68 \), \( b \approx 106.53 \), and \( c \approx 13.27 \), so the equation is \( y = -5.68x^2 + 106.53x + 13.27 \).

Step 2: Substitute \( x = 5.7 \) into the Equation

We substitute \( x = 5.7 \) into the quadratic equation. First, calculate \( (5.7)^2 = 32.49 \). Then, calculate each term:

• \( -5.68\times32.49 \approx -184.54 \)
• \( 106.53\times5.7 \approx 607.22 \)
• The constant term is \( 13.27 \)

Step 3: Sum the Terms

We sum the three terms: \( -184.54 + 607.22 + 13.27 \approx 435.95 \), which rounds to 436.

Answer:

First, we use a quadratic regression calculator (or software) to find the quadratic equation \( y = ax^2 + bx + c \) for the given data points \((0.3, 40)\), \((0.8, 101)\), \((1.1, 131)\), \((1.5, 169)\), \((1.8, 198)\).

Using a calculator, we find the coefficients:

• \( a \approx -5.68 \)
• \( b \approx 106.53 \)
• \( c \approx 13.27 \)

So the quadratic regression equation is \( y = -5.68x^2 + 106.53x + 13.27 \)

Now, to find the height at \( x = 5.7 \) seconds, we substitute \( x = 5.7 \) into the equation:

\( y = -5.68(5.7)^2 + 106.53(5.7) + 13.27 \)

First, calculate \( (5.7)^2 = 32.49 \)

Then, \( -5.68\times32.49 \approx -184.54 \)

\( 106.53\times5.7 \approx 607.22 \)

Now, sum these values: \( -184.54 + 607.22 + 13.27 \approx 435.95 \approx 436 \)

The final answer for the height at 5.7 seconds is \(\boxed{436}\)