Chapter 1.4, Problem 50E

To determine

To graph: The volume as a function of mass from the data provided between measurement of a growing plant. Also evaluate the equation of line in both point-slope and slope intercept form and volume when mass reaches $30.0\text{ g}$ . Compare the density at this with density when $a=0.5$ .

$\begin{array}{cccc}\begin{array}{l}\text{Age, a}\\ \left(\text{days}\right)\end{array}& \begin{array}{l}\text{Mass, M}\\ \left(g\right)\end{array}& \begin{array}{l}\text{Volume, V}\\ \left(c{m}^3\right)\end{array}& \begin{array}{l}\text{Glucose production, G}\\ \left(mg\right)\end{array}\\ 0.5& 2.5& 5.1& 0.0\\ 1.0& 4.0& 6.2& 3.4\\ 1.5& 5.5& 7.3& 6.8\\ 2.0& 7.0& 8.4& 10.2\\ 2.5& 8.5& 9.5& 13.6\\ 3.0& 10.0& 10.6& 17.0\end{array}$

Explanation of Solution

Given information:

The table that provides relation between measurement of a growing plant.

$\begin{array}{cccc}\begin{array}{l}\text{Age, a}\\ \left(\text{days}\right)\end{array}& \begin{array}{l}\text{Mass, M}\\ \left(g\right)\end{array}& \begin{array}{l}\text{Volume, V}\\ \left(c{m}^3\right)\end{array}& \begin{array}{l}\text{Glucose production, G}\\ \left(mg\right)\end{array}\\ 0.5& 2.5& 5.1& 0.0\\ 1.0& 4.0& 6.2& 3.4\\ 1.5& 5.5& 7.3& 6.8\\ 2.0& 7.0& 8.4& 10.2\\ 2.5& 8.5& 9.5& 13.6\\ 3.0& 10.0& 10.6& 17.0\end{array}$

Graph:

Consider the table that provides relation between measurement of a growing plant.

$\begin{array}{cccc}\begin{array}{l}\text{Age, a}\\ \left(\text{days}\right)\end{array}& \begin{array}{l}\text{Mass, M}\\ \left(g\right)\end{array}& \begin{array}{l}\text{Volume, V}\\ \left(c{m}^3\right)\end{array}& \begin{array}{l}\text{Glucose production, G}\\ \left(mg\right)\end{array}\\ 0.5& 2.5& 5.1& 0.0\\ 1.0& 4.0& 6.2& 3.4\\ 1.5& 5.5& 7.3& 6.8\\ 2.0& 7.0& 8.4& 10.2\\ 2.5& 8.5& 9.5& 13.6\\ 3.0& 10.0& 10.6& 17.0\end{array}$

To graph the volume as a function of mass, on the horizontal axis that is x- axis, plot the mass and on vertical axis that is y -axis plot the volume.

Now plot the points $\left(2.5,5.1\right),\left(4.0,6.2\right),\left(5.5,7.3\right),\left(7.0,8.4\right),\left(8.5,9.5\right),\left(10.0,10.6\right)$ on the graph and connect the dots through line.

  

Interpretation:

The point slope form of a line is $y=mx+\left(y_0-m{x}_0\right)$ where m is the slope and $\left(x_0,y_0\right)$ is the point through line the line passes.

The slope-intercept form of a line is $y=mx+c$ where m is the slope and c is the intercept of the line.

The slope is $m=\frac{y-y_0}{x-x_0}$ where $\left(x,y\right)$ is an arbitrary point and $\left(x_0,y_0\right)$ is the base point.

Here independent variable is x and dependent variable is y.

According to the question, volume V is dependent variable and massVis independent variable.

Slope of the line is evaluate when first two points are considered that is, $\left(2.5,5.1\right)\text{ and }\left(4.9,6.2\right)$ ,

$\begin{array}{c}m=\frac{6.2-5.1}{4-2.5}\\ =0.73\end{array}$

Slope of the line is $0.73$ .

Equation of line in point slope form is, consider $\left(2.5,5.1\right)$ as the base point.

$\begin{array}{c}V=mx+\left(y_0-m{x}_0\right)\\ =0.73M+\left(5.1-0.73\left(5.5\right)\right)\\ =0.73M+1.085\end{array}$

Therefore, the equation in point-slope form is $V=0.73M+1.085$ .

Intercept on y -axis is $1.085$ 4 when compared with the equation in slope-intercept form $y=mx+c$ .

The above statement is interpreted as volume of the seed when mass reaches 0 gis $1.085{\text{ cm}}^3$ .

Biologically this could be the volume of a new seedling.

Next, volume of seed when mass reaches $30.0\text{ g}$ , substitute M as $30.0$ in the equation of line.

$\begin{array}{c}V=0.73\left(30.0\right)+1.085\\ =22.985\end{array}$

Therefore, volume is $22.985{\text{ cm}}^3$ when mass reaches $30.0\text{ g}$ .

Density is defined as mass per volume.

Density when volume is $22.985{\text{ cm}}^3$ and mass is $30.0\text{ g}$ ,

$\begin{array}{c}\text{Density}=\frac{30.0g}{22.985{\text{ cm}}^3}\\ =1.31{\text{ g/cm}}^3\end{array}$

When age is $a=0.5$ days then volume is $5.1{\text{ cm}}^3$ and mass is $2.5\text{ g}$ , so density is

$\begin{array}{c}\text{Density}=\frac{2.5g}{5.1{\text{ cm}}^3}\\ =0.5{\text{ g/cm}}^3\end{array}$

Density is less when age of the plant is $0.5$ days.

Thus, equation in slope-intercept and point-slope form is $V=0.73M+1.085$ . All points lie on the line and volume is $22.985{\text{ cm}}^3$ when mass reaches $30.0\text{ g}$ .