Scientists are conducting an experiment with a gas in a sealed container. The mass of the gas is measured, and the scientists realize that the gas is leaking over time in a linear way.
Six minutes since the experiment started, the gas had a mass of 144 grams.
Fourteen minutes since the experiment started, the gas had a mass of 115.2 grams.
Let x be the number of minutes that have passed since the experiment started, and let y be the mass of the gas in grams at that moment. Use a linear equation to model the weight of the gas over time.
(a)This line’s slope-intercept equation is:
(b)34 minutes after the experiment started, there would be____grams of gas left.
(c)If a linear model continues to be accurate,____minutes since the experiment started, all gas in the container will be gone.
(a) Well, if we have a linear equation, it will have the form y = mx + b, where m is the slope and b is the y-intercept. To find the slope, we can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are points on the line. Let's choose the points (6, 144) and (14, 115.2):
m = (115.2 - 144) / (14 - 6)
m = -28.8 / 8
m = -3.6
So, our equation will be y = -3.6x + b. To find the y-intercept, we can use one of the given points, let's use (6, 144):
144 = -3.6(6) + b
144 = -21.6 + b
b = 165.6
Therefore, the slope-intercept equation is y = -3.6x + 165.6.
(b) To find the grams of gas left 34 minutes after the experiment started, we substitute x = 34 into the equation we found:
y = -3.6(34) + 165.6
y = -122.4 + 165.6
y ≈ 43.2 grams
So, there would be approximately 43.2 grams of gas left.
(c) To find the minutes when all the gas will be gone, we want to find when y = 0. We can set the equation equal to zero and solve for x:
0 = -3.6x + 165.6
3.6x = 165.6
x = 165.6 / 3.6
x ≈ 46 minutes
If the linear model continues to be accurate, it would take approximately 46 minutes since the experiment started for all the gas in the container to be gone.
To find the slope-intercept equation of the line representing the weight of the gas over time, we first need to determine the slope.
The slope of a line is calculated using the formula:
slope = (change in y) / (change in x)
Given that the gas is leaking over time in a linear way, we can calculate the change in mass as:
(change in mass) = y₂ - y₁
Where y₂ represents the mass at one time and y₁ represents the mass at another time.
Using the given data:
(change in mass) = 115.2 - 144 = -28.8 grams
(change in time) = 14 - 6 = 8 minutes
Therefore, the slope is:
slope = (-28.8) / 8 = -3.6 grams per minute
The slope-intercept equation of the line is of the form:
y = mx + b
Where m represents the slope, and b represents the y-intercept.
From the given data, we can substitute one point (6, 144) into the equation to find the value of b:
144 = (-3.6)(6) + b
Simplifying:
144 = -21.6 + b
b = 144 + 21.6 = 165.6
So, the slope-intercept equation is:
(a) y = -3.6x + 165.6
To find the amount of gas left after 34 minutes:
Substitute x = 34 into the equation:
y = -3.6(34) + 165.6
Simplifying:
y = -122.4 + 165.6 = 43.2 grams
Therefore, there would be 43.2 grams of gas left.
To determine when all the gas in the container will be gone:
Substitute y = 0 into the equation:
0 = -3.6x + 165.6
Simplifying:
3.6x = 165.6
x = 165.6 / 3.6
x = 46
Therefore, if the linear model continues to be accurate, 46 minutes since the experiment started, all gas in the container will be gone.
In summary:
(a) The line's slope-intercept equation is y = -3.6x + 165.6.
(b) 34 minutes after the experiment started, there would be 43.2 grams of gas left.
(c) If a linear model continues to be accurate, 46 minutes since the experiment started, all gas in the container will be gone.
To find the slope-intercept equation for the line that models the weight of the gas over time, we need to determine the slope (m) and the y-intercept (b).
(a) Slope-intercept form equation: y = mx + b
To find the slope (m), we use the formula:
m = (change in y) / (change in x)
Given points:
Point 1: (x1, y1) = (6, 144)
Point 2: (x2, y2) = (14, 115.2)
Change in x = x2 - x1 = 14 - 6 = 8
Change in y = y2 - y1 = 115.2 - 144 = -28.8
Now we can substitute these values into the slope formula:
m = (-28.8) / 8 = -3.6
The slope (m) of the line is -3.6.
To find the y-intercept (b), we can substitute one of the given points into the slope-intercept form equation and solve for b:
Using point (x1, y1) = (6, 144):
144 = (-3.6)(6) + b
144 = -21.6 + b
b = 144 + 21.6
b = 165.6
So, the slope-intercept equation for the weight of the gas over time is:
y = -3.6x + 165.6
(b) To find the grams of gas left after 34 minutes, substitute x = 34 into the equation:
y = -3.6(34) + 165.6
y = -122.4 + 165.6
y = 43.2
Therefore, there would be 43.2 grams of gas left after 34 minutes.
(c) To find the minutes since the experiment started when all the gas in the container will be gone, we need to find the x-intercept of the equation (where y = 0):
0 = -3.6x + 165.6
3.6x = 165.6
x = 165.6 / 3.6
x ≈ 46
Therefore, if the linear model continues to be accurate, it will take approximately 46 minutes since the experiment started for all the gas in the container to be gone.
y = 144+(115.2-144)/(14) x
= 144-2.057x
Now just plug in values for x.