You are under water in a pond and look up through the smooth surface of the water at the Sun in the sky. Is the Sun in fact higher in the sky than it appears to you, or is it lower?

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error.

$$\cos \left(30^{\circ}+20^{\circ}\right)=\cos 30^{\circ}+\cos 20^{\circ}$$

Identify the weaknesses of a partnership.

Decide whether $\frac{24}{75}$ would yield a repeating or a terminating decimal. (Hint: Write in lowest terms before trying to decide.)

Use matrix multiplication to find the image of the vector (- 2, 1, 2) under the stated rotation. (a) Through an angle of $30^{\circ}$ about the positive x-axis. (b) Through an angle of $45^{\circ}$ about the positive y-axis. (c) Through an angle of $-60^{\circ}$ about the positive z-axis.

A partnership owned by 25 partners is worth 5.4 million dollars. The partnership loses a lawsuit worth 6.2 million dollars. How much of the settlement is each partner liable for after the partnership is sold? Explain.

A gas turbine burns octane $\left(\mathrm{C}_8 \mathrm{H}_{\mathrm{18}}\right)$ completely with $400 \%$ of theoretical air. Determine the amount of $\mathrm{N}_2$ in the products, in $\mathrm{kmol}$ per $\mathrm{kmol}$ of fuel.

The value of $K_{s p}$ for cadmium sulfide, $\mathrm{CdS}$, is $8.0 \times 10^{-27}$. Compare this value with the $K_{s p}$ for lead (II) sulfate, $2.5 \times 10^{-8}$.

Which compound will have a higher concentration of ions in solution at equilibrium? Explain your reasoning.

A bond currently sells for $1,050, which gives it a yield to maturity of 6%. Suppose that if the yield increases by 25 basis points, the price of the bond falls to$1,025. What is the duration of this bond?

In this exercise, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth?

$$x=2 \cos \theta$$

$$y=2 \sin \theta$$

Let $T$ be a solid with a piecewise-smooth boundary. Show that if $f$ and $g$ have continuous second partials, then the flux of $\nabla f \times \nabla g$ out of $T$ is zero.

An isothermally transformed eutectoid steel is found to have a yield strength of $410 \mathrm{MPa}$. Estimate (a) the transformation temperature and (b) the interlamellar spacing in the pearlite.

Calculate the $\mathrm{pH}$ and the percent dissociation in $1.5 \mathrm{M} \mathrm{HNO}_2$ $\left(K_{\mathrm{a}}=4.5 \times 10^{-4}\right)$

Compute the assessed value for each property.

$$\begin{aligned} \begin{array}{lccc} \textbf{Market Value} && \textbf{Assessment Rate}\\ \$60,000 && 45\% \end{array} \end{aligned}$$