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?

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

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

If a verb in the following sentences is incorrect, cross it out and write the correct form above it. If the verb is already correct, write C above it.

Example 1. The chef $\overset{\textit{\color{#c34632}{set}}}{\sout{\text{sat}}}$ an assortment of ingredients on the counter.

She had set on a high stool at the work counter.

The economic advisor of a large tire store proposes the demand function D(p)=

$$\frac { 1800 } { p - 40 }$$

, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p. For what prices is the demand elastic? Inelastic?

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: $\mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}$, Time: $t_0=\pi$

The profits, in dollars, for $x$ weeks in one year for a small business can be modeled by the function $y=-0.55(x-12)^2+630$. Does the graph of the function have a maximum or a minimum? What is that value and during what week did it occur?

The government spends $\$ 3$ billion to buy police cars. Explain why aggregate demand might increase by more than $\$ 3$ billion. Explain why aggregate demand might increase by less than $\$ 3$ billion.

The demand function for a product is modeled by p=20-0.02x, $0 \leq x \leq 1000$ where p is the price (in dollars) and x is the number of units. (a) Determine when the demand is elastic, inelastic, and of unit elasticity. (b) Use the result of part (a) to describe the behavior of the revenue function.

Label each of the following functions, g(x), as a vertical stretch or a vertical compression of the parent function, f(x), and tell whether it is reflected about the x-axis. $g(x)=-1.2(5)^{x}, f(x)=5^{x}$

Mr.Martinez is comparing the price of oranges from several different markets. Which market's pricing guide is based on a constant unit price? A.

$$\begin{matrix} \text{Number of Oranges} & \text{5} & \text{10} & \text{15} & \text{20}\\ \text{Total Cost (\$)} & \text{3.50} & \text{6.00} & \text{8.50} & \text{11.00}\\ \end{matrix}$$

B. 

$$\begin{matrix} \text{Number of Oranges} & \text{5} & \text{10} & \text{15} & \text{20}\\ \text{Total Cost (\$)} & \text{3.50} & \text{6.50} & \text{9.50} & \text{12.50}\\ \end{matrix}$$

C. 

$$\begin{matrix} \text{Number of Oranges} & \text{5} & \text{10} & \text{15} & \text{20}\\ \text{Total Cost (\$)} & \text{3.00} & \text{5.00} & \text{7.00} & \text{9.00}\\ \end{matrix}$$

D. 

$$\begin{matrix} \text{Number of Oranges} & \text{5} & \text{10} & \text{15} & \text{20}\\ \text{Total Cost (\$)} & \text{3.00} & \text{6.00} & \text{9.00} & \text{12.00}\\ \end{matrix}$$

An object located $100 \mathrm{~cm}$ from a concave mirror forms a real image $75 \mathrm{~cm}$ from the mirror. The mirror is then turned around so that its convex side faces the object. The mirror is moved so that the image is now $35 \mathrm{~cm}$ behind the mirror. How far was the mirror moved? Was it moved toward the object or away from the object?

Find the rank by theorem "Rank in Terms of Determinants " and check by row reduction. (Show details.)

$$\left[\begin{array}{rrr}2 & 1 & 0 \\ 13 & -13 & 12 \\ -3 & 5 & -4\end{array}\right]$$

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

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

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