Math Problem Statement

The accompanying table shows eleven altitudes​ (in thousands of​ feet) and the speeds of sound​ (in feet per​ second) at these altitudes. Complete parts​ (a) through​ (d) below.

Altitude, x

Speed of​ sound, y

00

1116.61116.6

55

1095.61095.6

1010

1077.41077.4

1515

1057.21057.2

2020

1037.11037.1

2525

1014.91014.9

3030

994.3994.3

3535

968.3968.3

4040

968.5968.5

4545

968.5968.5

5050

968.5968.5

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​(a) Display the data in a scatter plot. Choose the correct graph below.

Solution

matplotlib.pyplot as plt

Data for the scatter plot

altitude = [0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50] speed_of_sound = [1116.6, 1095.6, 1077.4, 1057.2, 1037.1, 1014.9, 994.3, 968.3, 968.5, 968.5, 968.5]

Creating the scatter plot

plt.figure(figsize=(8, 6)) plt.scatter(altitude, speed_of_sound, c='blue', label="Speed of Sound") plt.title("Scatter Plot of Altitude vs Speed of Sound") plt.xlabel("Altitude (thousands of feet)") plt.ylabel("Speed of Sound (feet per second)") plt.grid(True) plt.legend() plt.show()The scatter plot displays the relationship between altitude (in thousands of feet) and the speed of sound (in feet per second). The data points show a general decrease in speed with increasing altitude, leveling off around 968.5 feet/second for altitudes above 35,000 feet.

Would you like details on interpreting the plot or have further questions? Here are five related questions:

  1. How does the speed of sound generally vary with altitude?
  2. Can we fit a trendline or regression model to this data?
  3. At what altitude does the speed of sound stabilize?
  4. Is there a theoretical explanation for the leveling off of the speed of sound?
  5. How does temperature or pressure affect the speed of sound at these altitudes?

Tip: In physics, the speed of sound decreases with altitude primarily due to lower temperatures, which affect air density.

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Math Problem Analysis

Mathematical Concepts

Data Visualization
Scatter Plots
Relationship Analysis

Formulas

None (focus on data representation)

Theorems

None (conceptual understanding of data trends)

Suitable Grade Level

Grades 9-12