Explain how you could experimentally determine whether the outside temperature is higher or lower than $0^{\circ} \mathrm{C}$ $\left(32^{\circ} \mathrm{F}\right)$ without using a thermometer.

Evaluate each definite integral. $\int_{3}^{5} \frac{x}{x-1} d x$

A heat pump for a 20$^{\circ} \mathrm{C}$ house uses R-410a, and the outside temperature is –5$^{\circ} \mathrm{C}$. What is the minimum high P and the maximum low P it can use?

Suppose that you choose a plant from a large batch of redflowering pea plants and cross it with a white-flowering pea plant. What percentage of the red-flowering parent plants are of genotype Cc if 90% of the offspring have red flowers?

Suppose a vector

$$\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$$

has length 4 and is $70^{\circ}$ counterclockwise from the negative $x_{2}$-axis. Find $x_{1}$ and $x_{2}.$

To prepare for a laboratory period, a student lab assistant needs 125 g of a compound. A bottle containing 1/4 lb is available. Did the student have enough of the compound?

In Problems classify each community matrix at equilibrium according to the five cases considered in Subsection 11.4.3 and determine whether the equilibrium is stable. (Assume in each case that the equilibrium exists.)

Let X be a random variable with distribution function

$$F(x)=\left\{\begin{array}{ll} 0 & x<0 \\ 0.05 & 0 \leq x<1.3 \\ 0.30 & 1.3 \leq x<1.7 \\ 0.85 & 1.7 \leq x<1.9 \\ 0.90 & 1.9 \leq x<2 \\ 1.0 & x \geq 2 \end{array}\right.$$

Determine the probability mass function of X.

Suppose that two species of beetles are reared together. Species 1 wins if there are initially 100 individuals of species 1 and 20 individuals of species 2. But species 2 wins if there are initially 20 individuals of species 1 and 100 individuals of species 2. When the beetles are reared separately, both species seem to reach an equilibrium of about 120. On the basis of this information and assuming that the densities follow the Lotka–Volterra model of interspecific competition, can you give lower bounds on $\alpha_{12}$ and $\alpha_{21}?$

Suppose that

$$L=\left[\begin{array}{ll} 0.5 & 2.3 \\ a & 0 \end{array}\right]$$

is the Leslie matrix of a population with two age classes. For which values of a does this population grow?

A large piece of jewelry has a mass of 132.6 g. A graduated cylinder initially contains 48.6 mL water. When the jewelry is submerged in the graduated cylinder, the total volume increases to 61.2 mL. (a) Determine the density of this piece of jewelry. (b) Assuming that the jewelry is made from only one substance, what substance is it likely to be? Explain.

In Problems we consider communities composed of two species. The abundance of species 1 at time t is given by N1(t), the abundance of species 2 at time t by N2(t). Their dynamics are described by Assume that when both species are at low abundances their abundances increase and that f1 and f2 change sign when crossing their zero isoclines. In each problem, determine the sign structure of the community matrix at the nontrivial equilibrium (indicated by a dot) on the basis of the graph of the zero isoclines. Determine the stability of the equilibria if possible. See Figure 11.69 .

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the x-axis. In each case, sketch the region and a typical disk element. $y=4-x^{2}, y=0, x=0$ (in the first quadrant)

Assume that $P\left(A \cap B^{c}\right)=0.1, P\left(B \cap A^{c}\right)=0.5,$ and $P\left((A \cup B)^{c}\right)=0.2.$ Find $P(A \cap B).$