INTERMEDIATE ALGEBRA

SUMMER TERM 2001

Naturals, Integers, Fractions, Decimals, Rationals, Square Roots, Irrationals, Reals, Scientific Notation, Graphing

**Equations and Formulas -
**Formulas are rules we state in symbolic form, usually as

- determine the perimeter, area and volume of geometric shapes;
- calculate the density of materials;
- calculate the period of a simple pendulum;
- convert between degrees Celsius and degrees Fahrenheit; and vice versa, and
- verify Boyle's Law.

**PERIMETER AND AREA **

**Complete the following problems on a
separate sheet of paper. Be sure to show all of the work necessary to complete the
problems**. (Calculators may be used to check the answer, but should not be used to
solve the problem originally.)

1. Find the perimeter of a square whose sides are each 4.5 cm long.

Step 1 - Choose the appropriate formula for a square.

Step 2 - Substitute the numerical values of the known physical quantities into the formula. Use correct units for the numerical value.

Step 3 - Solve for the unknown quantity, showing all of your work.

* Example*:

P = 4x

P = 4*4.5 cm

P = 18 cm.

**HOMEWORK PROBLEMS** :

2. Find the area of a square whose sides are each 7.25 m long.

3. Find the perimeter of a rectangle whose length is 9.32 cm long and whose width is 3.7 cm long

4. Find the area of a rectangle whose length is 9.32 cm long and whose width is 3.7 cm long

5. Find the circumference of a circle whose diameter is 14 cm long.

6. Find the area of a circle whose diameter is 7.0 cm long.

**VOLUME**

**HOMEWORK PROBLEMS** :

7. Find the volume of a cube whose sides are each 8.12 cm.

8. Find the volume of a rectangular box with a length of 4.32 cm, a width of 3.8 cm and a height of 6.28 cm.

9. Find the volume of a rectangular box with a length of 1.72 m, a width of 78 cm and a height of 528 mm.

10. Find the volume of a sphere whose diameter is 3.14 cm.

**DENSITY OF MATERIALS**- **Density**
is the amount of **mass** in a given unit of **volume**.

**Mass**- The balances that you use in laboratory measure mass, not weight. Mass (**m**) is sometimes defined as the amount of matter in an object. 10.0 grams of gold would contain twice as many gold atoms as 5.0 grams of gold. Students sometimes confuse mass and volume because the term "massive" can mean "large" in English. In science, mass has nothing to do with size. The SI unit for mass is the kilogram (kg).**Volume**- Volume (**V**) is the amount of space that an object takes up. When you buy a 2-liter bottle of soda, the soda takes up 2 liters of space. A 200 cm^{3}sample of gas is twice as large as a 100 cm^{3}sample of gas. You can determine the volume of regularly-shaped objects with a meter stick. Objects that have an irregular shape are often measured through what is called "the water displacement method." This means that you determine the volume of the object by finding out how much water it displaces. In Chemistry we often measure volume in millileters (ml) or cubic centimeters (cm^{3}).**Density**- Density (**r**) is the amount of matter in a given unit of volume. It can be measured in grams per cubic centimeter (g/cm3). It is a measure of how tightly packed the atoms of a substance are. When we say that ice is less dense than water, we mean that the water molecules are more tightly packed when they are in the liquid state. The formula for determining density is:

**HOMEWORK PROBLEMS** :

11. Dried walnut has a density of 0.64g/cm^{3}.
What must the volume of 33.3 g of wood?

Step 1 - Starting with the density formula, isolate the required physical quantity to derive a "new" formula.

Step 2 - Substitute the numerical values of the known physical quantities into the formula. Use correct units for the numerical value.

Step 3 - Solve for the unknown quantity, showing all of your work.

* Example*:

12. What is the density of a piece of iron that has a mass
of 16.4g and a volume of 2.08 cm^{3}?

13. Granite has a density of 2.68 g/cm^{3}. What
is the mass of 46.8 cm^{3} of granite?

14. Olive oil has density of 0.918 g/cm^{3}. What
is the mass of 34.0 cm^{3} of corn oil?

15. Copper has a density of 8.96 g/cm^{3}. What is
the volume of 2.78g of copper?

**PERIOD OF A SIMPLE PENDULUM** - Pendulums
have been used to make accurate clocks since 1656, and they have not changed dramatically
since then. Christiaan Huygens, a Dutch scientist, is credited with making the first
pendulum clock. Pendulums are useful because they have an extremely interesting property.
The period (the amount of time it takes for a pendulum to go back and forth once) of a
pendulum's swing is related only to the length of the pendulum and the acceleration of
gravity. Since gravity is constant at any given spot on the planet, the only thing that
affects the period of a pendulum is the length of the pendulum. The amount of weight does
not matter. Nor does the length of the arc that the pendulum swings through. Only the
length of the pendulum matters. The formula relating the period (**T**) of a
pendulum to its length (**L**) and the acceleration of gravity (**g**)
is:

**HOMEWORK PROBLEMS** :

16. If the acceleration of gravity is 9.81 m/s^{2},
what is the length of a pendulum whose period is 2.00 s?

Step 1 - Starting with the formula for the period of a pendulum, isolate the required physical quantity to derive a "new" formula.

Step 2 - Substitute the numerical values of the known physical quantities into the formula. Use correct units for the numerical value.

Step 3 - Solve for the unknown quantity, showing all of your work.

* Example*:

17. If the acceleration of gravity is 9.81 m/s^{2},
what is the length of a pendulum whose period is 4.00 s?

18. If the acceleration of gravity is 1.635 m/s^{2},
what is the length of a pendulum whose period is 2.00 s?

19. If the acceleration of gravity is 9.81 m/s^{2},
what is the period of a pendulum whose length is 1.00 m?

20. If the acceleration of gravity is 9.81 m/s^{2},
what is the period of a pendulum whose length is 4.00 m?

**CELSIUS AND FAHRENHEIT DEGREES** - Both the
Fahrenheit and Celsius temperature scales are based on the freezing conditions of water, a
very common and available liquid. Since water freezes and boils at temperatures that are
rather easy to generate (even before modern refrigeration), it is the most likely
substance on which to base a temperature scale.

On the **Fahrenheit** scale, the freezing
point of water is 32 degrees and the boiling point is 212 degrees. Zero Fahrenheit was the
coldest temperature that the German-born scientist Gabriel Daniel Fahrenheit could create
with a mixture of ice and ordinary salt. He invented the mercury thermometer and
introduced it and his scale in 1714 in Holland, where he lived most of his life.

Anders Celsius, a Swedish astronomer, introduced his scale
is 1742. For it, he used the freezing point of water as zero and the boiling point as 100.
For a long time, the **Celsius** scale was called "**centigrade**."
The Greek prefix "centi" means one-hundredth and each degree Celsius is
one-hundredth of the way between the temperatures of freezing and boiling for water. The
Celsius temperature scale is part of the "metric system" of measurement (SI) and
is used throughout the world, though not yet embraced by the American public.

**To convert Fahrenheit temperatures into Celsius**:

- Begin by subtracting 32 from the Fahrenheit number.
- Divide the answer by 9.
- Then multiply that answer by 5.
- These steps can be expressed by the following formula.

**To convert Celsius temperatures into Fahrenheit**:

- Begin by multiplying the Celsius temperature by 9.
- Divide the answer by 5.
- Now add 32.
- These steps can be expressed by the following formula.

**HOMEWORK PROBLEMS** :

21. Change 98.6 degrees Fahrenheit to Celsius.

22. Change -100 degrees Fahrenheit to Celsius.

23. Change 520 degrees Celsius to Fahrenheit.

24. Change -15.2 degrees Celsius to Fahrenheit.

**BOYLE'S LAW** - Boyle's Law involves a
relationship between the pressure (**p**) and volume (**V**) of
a gas in a container.

- The
**pressure**of the gas is measured in units of force per area -- for example, newtons per square meter. A pressure of 1.013 x 10^{5}newtons/m^{2}is called**one atmosphere**(atm). - The
**volume**of the container is measured in units of length cubed -- for example, cubic meters.

Robert Boyle, in 1660, found that if the temperature of a fixed mass of gas was held constant while its volume was varied, the pressure exerted by the gas varied also, and in such a way that the product of pressure and volume remained constant. This can be expressed by the following formula:

p_{1}V_{1} = p_{2}V_{2}

where the variables with the 1 subscript mean initial values before the manipulation and the variables with the 2 subscript mean final values after the manipulation.

A practical application illustrating Boyles Law would be the action of a syringe. When we draw fluids into a syringe, we increase the volume inside the syringe, this correspondingly decreases the pressure on the inside where the pressure on the outside of the syringe is greater and forces fluid into the syringe. If we reverse the actin and push the plunger in on the syringe we are decreasing the volume on the inside which will increase the pressure inside making the pressure greater than on the outside and fluids are forced out.

**HOMEWORK PROBLEMS** :

25. If 50 cm^{3} of oxygen gas is compressed from
20 atm of pressure to 40 atm of pressure, what is the new volume at constant temperature?

Step 1 - Starting with the formula for Boyle's Law, isolate the required physical quantity to derive a "new" formula.

Step 3 - Solve for the unknown quantity, showing all of your work.

* Example*:

26. If a gas sample in a balloon had a volume of 100 ml and a pressure of 3 atm, and was compressed to a pressure of 10 atm, what would be its volume? Assume the temperature remains fixed.

27. If a gas sample in a balloon had a volume of 125 ml and a pressure of 5.2atm, and was compressed to a volume of 46 ml, what would be its pressure? Assume the temperature remains fixed.

28. What volume of nitrogen gas is compressed from 1.0 atm of pressure to 12.0 atm of pressure, if the final volume is 4.25 liters? Assume the temperature remains fixed.

29. What was the original pressure of nitrogen gas compressed from 5.26 liters to 1.12 liters, if the final pressure is 8.0 atm? Assume the temperature remains fixed.

30. What is the pressure in Problem 29 in terms of
newtons/m^{2}?