Some organisms exhibit a density-dependent mortality from one generation to the next. Let R > 1 be the net reproductive rate (that is, the number of surviving offspring per parent), let x > 0 be the density of parents, and y be the density of surviving offspring. The Beverton-Holt recruitment curve is $y=\frac{R x}{1+\left(\frac{R-1}{K}\right) x}$ where K > 0 is the carrying capacity of the organism’s environment. Show that $\frac{d y}{d x}>0$, and interpret this as a statement about the parents and the offspring.

Acetic acid, $\mathrm{HC}_2 \mathrm{H}_3 \mathrm{O}_2$, is responsible for the sour taste of vinegar. Combustion of acetic acid gives off $14.52 \mathrm{~kJ} / \mathrm{g}$ of heat. When $15.00 \mathrm{~g}$ of acetic acid is burned in a bomb calorimeter (heat capacity $=2.166 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}$ ) containing $0.800 \mathrm{~kg}$ of water, the final temperature in the calorimeter is $64.14^{\circ} \mathrm{C}$. What was the initial temperature of the calorimeter?

A solar cooker consists of a curved reflecting surface that concentrates sunlight onto the object to be warmed . The solar power per unit area reaching the Earth's surface at the location is $600 \mathrm{~W} / \mathrm{m}^2$. The cooker faces the Sun and has a diameter of $0.600 \mathrm{~m}$. Assume that $40.0 \%$ of the incident energy is transferred to $0.500 \mathrm{~L}$ of water in an open container, initially at $20.0^{\circ} \mathrm{C}$. How long does it take to completely boil away the water? (Ignore the heat capacity of the container.)

In the following exercise, sketch the graph of the plane curve given by the vector-valued function and, at the point on the curve determined by $r\left(t_0\right)$, sketch the vectors $T$ and $N$. Note that $\mathbf{N}$ points toward the concave side of the curve.

Function: $\mathbf{r}(t)=t \mathbf{i}+\frac{1}{t} \mathbf{j}$, Time: $t_0=2$

The recommended calcium intake for teenagers is 1300 mg per day. A glass of milk contains 305 mg of calcium. One glass contains a volume of 8 fluid ounces. How many liters of milk should a teenager drink per day to get the recommended amount of calcium? One fluid ounce equals 29.6 mL.

The chief executive officer (CEO) of your company recently returned from a luncheon meeting where activity-based costing was presented and discussed. Though her background is not in accounting, she has worked for the company for 15 years and is thoroughly familiar with its operations. Her impression of the presentation about ABC was that it was just another way of dividing up total overhead cost and that the total would still be the same “no matter how you sliced it.” Write a memorandum to the CEO, no more than one page, explaining how ABC is different from traditional volume-based costing methods. Also, identify its advantages and disadvantages vis-a-vis traditional methods. Be sure it is written to be understandable to someone who is not an accountant.

a. Solve the system by linear combination:\

$$\ \begin{aligned} &5 x-8 y=47 \\ &2 x+7 y=-22 \end{aligned}$$

b. Solve the system by determinants:\

$$\begin{array}{lr} 4 x-3 y= & 5 \\ 7 x+2 y= & -1 \end{array}$$

Let $I _ { m , n } = \int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } d x$ for constant m, n. Show that $I _ { m , n } = I _ { n , m }.$

The heat capacity C(T) of a substance is the amount of energy (in joules) required to raise the temperature of 1 g by

$$1 ^ { \circ } \mathrm { C }$$

at temperature T. (a) Explain why the energy required to raise the temperature from

$$T_1$$

to

$$T_2$$

is the area under the graph of C(T) over

$$[T_1, T_2].$$

(b) How much energy is required to raise the temperature from

$$50 ^ { \circ } \mathrm { C }$$

to

$$100 ^ { \circ } \mathrm { C }$$

if

$$C ( T ) = 6 + 0.2 \sqrt { T } ?$$

Use determinants to find the area of a triangle with vertices (3, -1), (7, -1), and (7, 5). Confirm your answer by plotting the points in a coordinate plane and using the formula Area = 1/2(base)(height).

Use properties of determinants to find the value of each determinant if it is known that

$$\left|\begin{array}{lll}{x} & {y} & {z} \\ {u} & {v} & {w} \\ {1} & {2} & {3}\end{array}\right|=4.$$

$$\left|\begin{array}{rrr}{x} & {y} & {z} \\ {-3} & {-6} & {-9} \\ {u} & {v} & {w}\end{array}\right|$$

Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning.

$$\left| \begin{array} { r r r } { 0 } & { 0 } & { 1 } \\ { 0 } & { 5 } & { 2 } \\ { 3 } & { - 1 } & { 4 } \end{array} \right|$$

Compute the determinants using cofactor expansion along any row or column that seems convenient.

$$\left| \begin{array} { r r r } { - 4 } & { 1 } & { 3 } \\ { 2 } & { - 2 } & { 4 } \\ { 1 } & { - 1 } & { 0 } \end{array} \right|$$

A shiny spoon provides both a concave mirror and a convex mirror, one on each side.

Hold a shiny spoon about 25 cm from your face. Observe your image in the concave side. Is your image up-side down or right-side up?