The following data represent the weight (in grams) of a box of raisins and the number of raisins in the box.

$$\begin{matrix} \text{Weight (in grams), w} & \text{Number of Raisins, N}\\ \hline \text{42.3} & \text{87}\\ \text{42.7} & \text{91}\\ \text{42.8} & \text{93}\\ \text{42.4} & \text{87}\\ \text{42.6} & \text{89}\\ \text{42.4} & \text{90}\\ \text{42.3} & \text{82}\\ \text{42.5} & \text{86}\\ \text{42.7} & \text{86}\\ \text{42.5} & \text{86}\\ \end{matrix}$$

Select two points and find a linear model that contains the points.

Find the slope of the line passing through the given points. (4,5) and (6,15)

If 2.00 g of N$_2$ gas has a volume of 0.40 L and a pressure of 6.0 atm, what is its Kelvin temperature?

Write an equation of each line.

slope = 3; through (1, 5)

For each of the following, write the equilibrium-constant expression for $K_{\mathrm{sp}}$ :
(a) $\mathrm{Ca}(\mathrm{OH})_2$
(b) $\mathrm{Ag}_3 \mathrm{PO}_4$
(c) $\mathrm{BaCO}_3$
(d) $\mathrm{Ca}_5\left(\mathrm{PO}_4\right)_3 \mathrm{OH}$

At a pressure of 1.00 atm and a temperature of $20^{\circ} \mathrm{C}$, 1.72 g $CO_2$ will dissolve in 1 L of water. How much $CO_2$ will dissolve if the pressure is raised to 1.35 atm and the temperature stays the same? A. 2.32 g/L, B. 1.27 g/L, C. 0.785 g/L, D. 0.431 g/L.

Which state in each of the following pairs has the higher entropy per mole of substance?
(a) Ice at $-40^{\circ} \mathrm{C}$ or ice at $0^{\circ} \mathrm{C}$
(b) $\mathrm{N}_2$ at $\mathrm{STP}$ or $\mathrm{N}_2$ at $0^{\circ} \mathrm{C}$ and 10 atm
(c) $\mathrm{N}_2$ at STP or $\mathrm{N}_2$ at $0^{\circ} \mathrm{C}$ in a volume of $50 \mathrm{~L}$
(d) Water vapor at $150^{\circ} \mathrm{C}$ and 1 atm or water vapor at $100^{\circ} \mathrm{C}$ and $2 \mathrm{~atm}$

What is the temperature, in degrees Celsius, of 5.23 moles of helium (He) gas confined to a volume of 5.23 L at a pressure of 5.23 atm?

Graph the function. f(x)=4 tan x

Calculate y'. $y=\tan \sqrt{1-x}$

Find the directional derivative of $f$ at $P$ in the direction of a, where

$$f=\ln \left(x^2+y^2\right), P:(4,0), \mathbf{a}=\mathbf{i}-\mathbf{j}$$

Resource Competition Tilman (1982) developed a model for how plants compete for a single resource-for instance, nitrogen. If $B(t)$ denotes the total biomass at time $t$ and $R(t)$ is the amount of the resource available at time $t$, then the dynamics are described by the following system of differential equations:

$$\begin{aligned} & \frac{d B}{d t}=B[F(R)-M] \\ & \frac{d R}{d t}=a(S-R)-\operatorname{CBF}(R) \end{aligned}$$

The first equation describes the rate of change of biomass, where the function $f(R)$ describes how the species growth rate depends on the resource, and $m$ is the per capita plant mortality rate. The second equation describes the resource dynamics; the constant $S$ is the maximal amount of the resource in a given habitat. The rate of resource supply $(d R / d t)$ is assumed to be proportional to the difference between the current resource level and the maximal amount of the resource; the constant $a$ is the constant of proportionality. The term $c B f(R)$ describes the resource uptake by the plants; the constant $c$ can be considered a conversion factor.

$$f(R)=\frac{d R}{k+R}$$

In what follows, we assume that $f(R)$ follows the Monod growth function where $d$ and $k$ are positive constants. (a) Find all equilibria. Show that if $d>m$ and $S>m k /(d-m)$, then there exists a nontrivial equilibrium. (b) Sketch the zero isoclines for the case in which the system admits a nontrivial equilibrium, and sketch the direction of the vector field on the isoclines and in the regions between them.

Determine the domain and range of each relation. Explain why each nonfunction is not a function.

$$\begin{matrix} \text{x} & \text{0} & \text{3} & \text{8} & \text{0} & \text{3} & \text{8}\\ \text{y} & \text{-1} & \text{-2} & \text{-3} & \text{1} & \text{2} & \text{2}\\ \end{matrix}$$

What is an equation of a line passing through (3, 3) and (4, 7)?