Chapter 1.3, Problem 35E

a.

To determine

To find: the slope of line containing the given points.

Answer to Problem 35E

  $0.25$

Explanation of Solution

Given information:

The given variables are as follows:

Pressure (in Pascal) = $P$

Temperature (in $°\text{C}$ ) = $T$

According to Charles’ Law, $P$ is linearly related to $T$ at a fixed volume of gas.

  $\begin{array}{l}\left(T_1,P_1\right)=\left(40,90\right)\\ \left(T_2,P_2\right)=\left(80,100\right)\end{array}$

Calculation:

Put the values of $\left(T_1,P_1\right)\text{ and }\left(T_2,P_2\right)$ in the slope formula and evaluate the value of m as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{100-90}{80-40}\\ =\frac{10}{40}\\ =\frac{1}{4}\\ =0.25\end{array}$

Thus, the slope of line is 0.25.

b.

To determine

To explain: the meaning of slope here.

Answer to Problem 35E

In this context, slope shows that pressure increases at a rate of 0.25 times faster than temperature.

Explanation of Solution

Given information:

The given variables are as follows:

Pressure (in Pascal) = $P$

Temperature (in $°\text{C}$ ) = $T$

According to Charles’ Law, $P$ is linearly related to $T$ at a fixed volume of gas.

  $\begin{array}{l}\left(T_1,P_1\right)=\left(40,90\right)\\ \left(T_2,P_2\right)=\left(80,100\right)\end{array}$

  $m=0.25$

Calculation:

Since the function of the line represented by $P\left(T\right)$ . The slope intercept equation for this function will be as follows:

  $\begin{array}{c}y=mx+b\\ P=0.25T+b\mathrm{...}\left(1\right)\end{array}$

Since the line is linear, so the rate is constant i.e. $b=0$ . So, the line equation (1) can be written as follows:

  $\begin{array}{l}P=0.25T+0\\ P=0.25T\end{array}$

From the above equation, it shows that pressure increases at a rate of 0.25 times faster than temperature.

c.

To determine

To graph: the function.

Answer to Problem 35E

  

Explanation of Solution

Given information:

The given variables are as follows:

Pressure (in Pascal) = $P$

Temperature (in $°\text{C}$ ) = $T$

According to Charles’ Law, $P$ is linearly related to $T$ at a fixed volume of gas.

  $\begin{array}{l}\left(T_1,P_1\right)=\left(40,90\right)\\ \left(T_2,P_2\right)=\left(80,100\right)\end{array}$

  $m=0.25$

Calculation:

Since the function of the line represented by $P\left(T\right)$ . The slope intercept equation for this function will be as follows:

  $\begin{array}{c}y=mx+b\\ P\left(T\right)=0.25T+b\mathrm{...}\left(1\right)\end{array}$

Put the first point coordinates $\left(T_1,P_1\right)$ in equation (1) and evaluate b as follows:

  $\begin{array}{c}90=\left(0.25\right)\left(40\right)+b\\ 90=10+b\\ b=90-10\\ b=80\end{array}$

Now, put the value of b in equation (1) to evaluate the slope intercept equation $P\left(T\right)$ as follows:

  $P\left(T\right)=0.25T+80\mathrm{...}\left(2\right)$

Thus, the y- intercept is $\left(0,80\right)$ .

Put $P=0$ in equation (2) and solve to find the x- intercept as follows:

  $\begin{array}{c}0=0.25T+80\\ 0.25T=-80\\ T=-\frac{80}{0.25}\\ =-\frac{80}{\left(\frac{25}{100}\right)}\\ =-\frac{80}{\left(\frac{1}{4}\right)}\\ =-80\left(4\right)\\ =-320\end{array}$

Thus, the x- intercept is $\left(-320,0\right)$ .

Now, plot the graph for point coordinates $\left(0,80\right)$ and $\left(-320,0\right)$ as shown below.