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 9 seconds.
Time in Seconds (x)	Height in Feet (y)
0, point, 70.7	100100
1, point, 21.2	160160
1, point, 71.7	213213
2, point, 12.1	249249
2, point, 72.7	298298
3, point, 13.1	319319


import numpy as np
import pandas as pd
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression

# Given data
time = np.array([0.7, 1.2, 1.7, 2.1, 2.7, 3.1])
height = np.array([100, 160, 213, 249, 298, 319])

# Reshape the time data for regression model
time_reshaped = time.reshape(-1, 1)

# Apply quadratic regression
poly = PolynomialFeatures(degree=2)
time_poly = poly.fit_transform(time_reshaped)

model = LinearRegression()
model.fit(time_poly, height)

# Get the quadratic equation coefficients
coefficients = model.coef_
intercept = model.intercept_

# Generate the equation
a, b, c = coefficients[2], coefficients[1], intercept

# Predict height at time = 9 seconds
time_9 = poly.transform([[9]])
predicted_height_9 = model.predict(time_9)

a, b, c, predicted_height_9[0](-15.434908733877872,
 150.53353862632224,
 1.8630785022538987,
 106.43731869504646)(-15.434908733877872,
 150.53353862632224,
 1.8630785022538987,
 106.43731869504646)successThe quadratic regression equation for the given data is:

\[
y = -15.4x^2 + 150.5x + 1.9
\]

where \(x\) is time (in seconds) and \(y\) is height (in feet).

Using this equation, the height of the rocket at \(x = 9\) seconds is approximately \(106\) feet (rounded to the nearest foot).

Would you like details on how this equation was derived, or do you have any other questions?

### Related Questions:
1. How does quadratic regression work to fit a curve to the data?
2. Can this equation be used to predict the rocket's height for values of \(x\) beyond the provided data range?
3. How does rounding coefficients affect the accuracy of predictions?
4. What does each term in the quadratic equation represent in the context of the rocket's motion?
5. Can we determine the maximum height the rocket reaches from this equation?

### Tip:
When using a quadratic regression equation for prediction, ensure the range of the data supports the reliability of extrapolated values.

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 9 seconds. Time in Seconds (x): [0.7, 1.2, 1.7, 2.1, 2.7, 3.1], Height in Feet (y): [100, 160, 213, 249, 298, 319].

Learn how to calculate a quadratic regression equation for a rocket's height based on time and predict its height at 9 seconds. Explore polynomial fitting and regression techniques using data points like time (0.7, 1.2, 1.7, 2.1, 2.7, 3.1 seconds) and height (100, 160, 213, 249, 298, 319 feet). Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation