On a bright clear day, you are at the top of a mountain and looking at a city $12 \mathrm{~km}$ away. There are two tall towers $20.0 \mathrm{~m}$ apart in the city. Can your eye resolve the two towers if the diameter of the pupil is $4.0 \mathrm{~mm}$ ? If not, what should be the minimum magnification power of the telescope needed to resolve the two towers? In your calculations use $550 \mathrm{~nm}$ for the wavelength of the light.

The resolving power of refracting telescope increases with the size of its objective lens. What other advantage is gained with a larger lens?

As the width of the slit producing a single-slit diffraction pattern is reduced, how will the diffraction pattern produced change?

A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Describe how it can sparkle with flashes of brilliant color when illuminated by white light.

Estimate the area under the graph of f(x)=1+x^2 from x=-1 to x=2 using three rectangles and left endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.

A standing wave pattern on a string is described by $y(x, t) = 0.040 (\sin 5 \pi x)(\cos 40 \pi t)$, where $x$ and $y$ are in meters and $t$ is in seconds. For $x \geq 0$, what is the location of the node with the smallest value of x?

If the form of a sound wave traveling through air is s(x, t) = (6.0 nm) cos(kx + (3000 rad/s)t + Φ), how much time does any given air molecule along the path take to move between displacements s = +2.0 nm and s = -2.0 nm?

It is quire common for a solid to change from one structure to another at a temperature below its melting point. For example, sulfur undergoes a phase change from the rhombic crystal structure to the monoclinic crystal form at temperatures above $95 ^ { \circ } \mathrm { C }$. Predict the signs of $\Delta H$ and $\Delta S$ for the process $S _ { \text { rhombic } } \longrightarrow S _ { \text { monoclinic } }$.

In 1967, New Zealander Burt Munro set the world record for an Indian motorcycle, on the Bonneville Salt Flats in Utah, of 295.38 km/h. The one-way course was 8.00 km long. Acceleration rates are often described by the time it takes to reach 96.0 km/h from rest. If this time was 4.00 s and Burt accelerated at this rate until he reached his maximum speed, how long did it take Burt to complete the course?

Given the following equilibrium constants at $427 ^ { \circ } \mathrm { C },$

$$\mathrm { Na } _ { 2 } \mathrm { O } ( s ) \rightleftharpoons 2 \mathrm { Nal } ( l ) + \frac { 1 } { 2 } \mathrm { O } _ { 2 } ( g ) \quad K _ { 1 } = 2 \times 10 ^ { - 25 } \\ \mathrm { NaO } ( g ) \rightleftharpoons \mathrm { Na } ( l ) + \frac { 1 } { 2 } \mathrm { O } _ { 2 } ( g ) \quad K _ { 2 } = 2 \times 10 ^ { - 5 } \\ \mathrm { Na } _ { 2 } \mathrm { O } _ { 2 } ( s ) \rightleftharpoons 2 \mathrm { Nal } ( l ) + \mathrm { O } _ { 2 } ( g ) \quad K _ { 3 } = 5 \times 10 ^ { - 29 } \\ \mathrm { NaO } _ { 2 } ( s ) \rightleftharpoons \mathrm { Na } ( l ) + \mathrm { O } _ { 2 } ( g ) \quad K _ { 4 } = 3 \times 10 ^ { - 14 }$$

determine the values for the equilibrium constants for the following reactions. $\mathrm { a }. \mathrm { Na } _ { 2 } \mathrm { O } ( s ) + \frac { 1 } { 2 } \mathrm { O } _ { 2 } ( g ) \rightleftharpoons \mathrm { Na } _ { 2 } \mathrm { O } _ { 2 } ( s ),$ $\mathrm { b }. \mathrm { NaO } ( g ) + \mathrm { Na } _ { 2 } \mathrm { O } ( s ) \rightleftharpoons \mathrm { Na } _ { 2 } \mathrm { O } _ { 2 } ( s ) + \mathrm { Na } ( l ),$ $\mathrm { c }. 2 \mathrm { NaO } ( g ) \rightleftharpoons \mathrm { Na } _ { 2 } \mathrm { O } _ { 2 } ( s ).$

A room has dimensions 3.00 m (height) × 3.70 m × 4.30 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) Could the length of its path be less than this magnitude? (c) Greater? (d) Equal? (e) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. (f) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

A monument that you are creating is for a city park which will be a rectangular prism with the following measurements: x + 1 feet, x - 5 feet, and x - 6 feet. Make a function f(x) for the volume of the monument.

Use your graph to determine if the city residents would be able to walk across all of the bridges without crossing the same bridge twice. Explain your answer.

The city of McGregor borrowed $7.8 million to build a zoo. The interest rate is 4% compounded annually, and the loan is to be paid in 20 years. Find the annual payments.