Question

math 131: linear regression activity name:austin for this activity, we will use the technique of linear regression to develop a linear model that represents data from a chemical reaction. consider the data in the table below. this data represents the mass in grams (y) of a chemical at time x (measured in seconds). time in seconds (x) 5 7 12 16 20 mass in grams (y) 40 120 180 210 240 1. using the provided desmos link, perform a linear regression. you should get an equation in the form y = mx + b. round to two decimal places. a. what is the slope of this line? 12.21 b. at what point is the y - intercept? 11.48 c. write the equation of the line. y = 12.21x + 11.48 d. how “good” of a model to you think this equation is? explain. yes it is a good motel because it show what a simple way to do it 2. answer the following questions using the linear regression model. round answers to two decimal places. a. what is the mass of the chemical after eight seconds have passed? include units in your answer. b. by how much does the weight of the chemical change in five seconds? is the weight increasing or decreasing over time? c. how much time does it take for the weight of the chemical to 400 grams?

Explanation:

2a.

Step1: Identify the linear equation

We have the linear equation from part 1c: \( y = 12.21x + 11.49 \)

Step2: Substitute \( x = 8 \) into the equation

Substitute \( x = 8 \) into \( y = 12.21x + 11.49 \):
\( y = 12.21\times8 + 11.49 \)
\( y = 97.68 + 11.49 \)
\( y = 109.17 \)

Step1: Recall the slope's meaning

The slope \( m = 12.21 \) in \( y = mx + b \) represents the change in \( y \) per unit change in \( x \).

Step2: Calculate change in 5 seconds

For a change in \( x = 5 \) seconds, the change in \( y \) (mass) is \( m\times5 \).
\( \Delta y = 12.21\times5 = 61.05 \)
Since the slope is positive, the weight is increasing over time.

Step1: Set \( y = 400 \) in the equation

We have \( y = 12.21x + 11.49 \), set \( y = 400 \):
\( 400 = 12.21x + 11.49 \)

Step2: Solve for \( x \)

Subtract \( 11.49 \) from both sides:
\( 400 - 11.49 = 12.21x \)
\( 388.51 = 12.21x \)
Divide both sides by \( 12.21 \):
\( x = \frac{388.51}{12.21} \approx 31.82 \)

Answer:

The mass of the chemical after eight seconds is \( 109.17 \) grams.

2b.