MAT104: Fundamentals of Mathematics II Summary of Section 14-5: Volume, Temperature, and Dimensional Analysis with Area & Volume.

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1 MAT104: Fundamentals of Mathematics II Summary of Section 14-5: Volume, Temperature, and Dimensional Analysis with Area & Volume For prisms, pyramids, cylinders, and cones: Volume is the area of one base is the height of the figure (perpendicular distance from one base to the other base or the apex) For figures with two bases (prisms and cylinders the flat top figures): For figures with one base and an apex (pyramids and cones the pointy figures): For spheres (spheres are in their own little category, so to speak): is the radius of the figure Example: Determine the volume of a right circular cone with a slant height of having a diameter of inches. inches and a base First, note the diameter is inches, so the radius is inches. Next, note the slant height is given, so using the Pythagorean Theorem (or recognizing a Pythagorean Triple, ), the actual height can be found to be inches. The volume is then calculated as follows: 1. The symbol is used to represent the area of ANY shape base; it is expected that the area of any shape given in any exercise can be calculated using techniques and/or formulas presented in class (possible shapes of bases include triangles, squares, rectangles, trapezoids, parallelograms, circles, and any regular polygon, as well as any other shapes we have formally discussed). 2. Slant height is NOT used to calculate the volume of any figure; however, slant height may be provided in the problem description rather than the actual height, in which case the Pythagorean Theorem or some other technique may have to be used to determine the actual height.

2 Temperature Conversions For any temperature conversions: is the temperature in degrees Fahrenheit is the temperature in degrees Celsius is the temperature in degrees Kelvin To convert from Fahrenheit to Celsius (if Fahrenheit is already known): To convert from Celsius to Fahrenheit (if Celsius is already known): To convert from Celsius to Kelvin (if Celsius is already known): To convert from Kelvin to Celsius (if Kelvin is already known): To convert from Fahrenheit to Kelvin or from Kelvin to Fahrenheit: First, convert to Celsius; then, convert to the necessary final temperature scale. Example 1: Convert to Celsius. Example 2: Convert to Fahrenheit. First, convert to Celsius: Then, convert to Fahrenheit: 1. Rounding a final result is often desirable for readability and understandability. 2. Kelvin is NOT discussed in the textbook but is expected to be known.

3 Dimensional Analysis Involving Area and Volume For conversions within the English system of measurement: Use the same base conversion factors to change units (e.g., ), but square the conversion factor for area (e.g., ), or cube the conversion factor for volume (e.g., ). Example 1: Convert to square inches. Example 2: Convert to cubic yards. 1. The given starting value should NOT be squared or cubed; the given starting value is a statement of fact that cannot be altered (note that neither the in the first example above nor the in the second example above is modified). 2. In some cases, it may be most practical to round final answers obtained on a calculator to a reasonable number of decimal places if no specific place value is provided, such as in the second example above, although exact results are generally preferred if possible. Specifically, results containing a large number of decimal places are often too cumbersome to be practical, so round to the nearest hundredth, thousandth, or other specified place value may enhance readability and understandability. 3. Whenever converting units through dimensional analysis, particularly when converting units of volume, results obtained on a calculator may be expressed in scientific notation in which a positive or negative integer follows a capital on the screen (e.g., or ). In such cases, a positive value following the capital indicates the decimal point must be moved to the right that number of places, whereas a negative value indicates the decimal point must be moved to the left that number of places. Using the aforementioned parenthetical examples, values expressed in scientific notation on a calculator display may be written out as follows: becomes becomes

4 For conversions within the metric system of measurement: For area, move the decimal point double the number of places in the same direction as the base conversion factor indicates (e.g., move the decimal point places left to convert from square millimeters to square meters); for volume, move the decimal point triple the number of places in the same direction as the base conversion factor indicates (e.g., move the decimal point places right to convert from cubic kilometers to cubic centimeters). Example 1: Convert to cubic meters. Begin by noting that converting centimeters to meters involves moving the decimal point places to the left. Since the exercise concerns volume (cubic units), the desired result is obtained by moving the decimal point places to the left. Example 2: Convert to square millimeters. Begin by noting that converting kilometers to millimeters involves moving the decimal point places to the right. Since the exercise concerns area (square units), the desired result is obtained by moving the decimal point places to the right. 1. As with conversions within the English system, rounding a final result to a reasonable or specified number of decimal places may be desirable (see preceding discussion for details). 2. Perhaps the most common mistake made when converting units of area or volume within the metric system is squaring or cubing the number of places to move the decimal point rather than doubling or tripling the number of places, respectively. In the first example above, this type of error results in moving the decimal point, or places to the left; in the second example above, this type of error results in moving the decimal point, or places to the right.

5 For conversion between the English and metric systems: As with linear measure, the only correct technique is to use the conversion factor, but be sure to square or cube that conversion factor for area or volume, respectively. If starting in the English system, first convert the given value to square or cubic inches using the appropriate conversion factor(s). Next, convert that result to square or cubic centimeters by multiplying by or, respectively. Finally, convert that result to the necessary metric units by moving the decimal point the appropriate number of places in the proper direction. If starting in the metric system, first convert the given value to square or cubic centimeters by moving the decimal point the appropriate number of places in the proper direction. Next, convert that result to square or cubic inches by dividing by or, respectively. Finally, convert that result to the necessary English units using the appropriate conversion factor(s). Example: Convert to square kilometers. First, convert from square miles to square inches: Next, convert from square inches to square centimeters: Finally, convert from square centimeters to square kilometers by moving the decimal point, or places to the left: 1. As with conversions within the English system, the starting value should NOT be squared or cubed (see preceding discussion for details). 2. As with conversions within the English system, rounding a final result to a reasonable or specified number of decimal places may be desirable (see preceding discussion for details). 3. As with conversions within the English system, a result at any step may be displayed in scientific notation on a calculator (see preceding discussion for details).