Question

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. Eight minutes since the experiment started the gas had a mass of 302.4 grams. Seventeen minutes since the experiment started the gas had a mass of 226.8 gramsLet 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 lines slope-intercept equation is [ ] b) 39 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.

Answer 1

Here, we want to model an experiment linearlly

From the question, we have it that;

The coordinates are written as;

(number of minutes, mass of gas)

So, what we have to do know is to set up the two given points

These are the points;

(8,302.4) and (17,226.8)

Now, using these two points, we can model the equation

We start by getting the slope of the line that passes through these two points

To do this, we shall use the slope equation

We have this as;

`\begin{gathered} \text{slope m = }(y_2-y_1)/(x_2-x_1) \\ \\ (x_1,y_1)\text{ = (8,302.4)} \\ (x_2,y_2)\text{ = (17,226.8)} \\ \text{substituting these values;} \\ m\text{ = }(226.8-302.4)/(17-8)\text{ = }(-75.6)/(9)\text{ = -8.4} \end{gathered}`

The general equation representing a linear model is ;

`\begin{gathered} y\text{ = mx + b} \\ m\text{ is slope} \\ b\text{ is y-intercept} \\ y\text{ = -8.4x + b} \end{gathered}`

To get the y-intercept so as to write the complete equation, we use any of the two points and substitute its coordinates

Let us substitute the coordinates of the first point

`\begin{gathered} 302.4\text{ = -8.4(8) + b} \\ 302.4\text{ = -67.2 + b} \\ b\text{ = 67.2 + 302.4} \\ b\text{ = 369.6 } \end{gathered}`

a) Thus, we have the complete linear model as;

`y\text{ = -8.4x + 369.6}`

b) To get this, we simply substitute the value of x given into the linear model

`\begin{gathered} y\text{ = -8.4(39) + 369.6} \\ y\text{ = -327.6 + 369.6} \\ y\text{ = 42} \end{gathered}`

39 minutes after the experiment started, there would be 42 grams

c) If all the gas is gone, then the value of y will br zero at this point

To get the corresponding x-value which is the time, we have it that;

`\begin{gathered} 0\text{ = -8.4x +369.6} \\ 8.4x=\text{ 369.6} \\ x\text{ = }(369.6)/(8.4) \\ x\text{ = 44} \end{gathered}`

In 44 minutes, all the gas in the container will be gone