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.