CONSTANT RATE OF CHANGE & THE POINT-SLOPE FORMULA
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1 CONSTANT RATE OF CHANGE & THE POINT-SLOPE FORMULA 1. In Worksheet 3 we defined the meaning of constant rate of change. a. Explain what it means for two quantities to be related by a constant rate of change. If two quantities are related by a constant rate of change, then the corresponding changes in the quantities are proportional In Worksheet 3 we defined the meaning of constant rate of change. b. The cost of covering an area with a certain type of concrete paver increases at a constant rate of $4.50 per square foot with respect to the size of the area covered. What does this mean? The change in the cost of covering the area (in dollars) is always 4.50 times as large as the change in the size of the space covered (in square feet). That is, if a represents the change in the size of the area covered (in square feet) and c represents the change in the total cost of covering the area, then c/ a = 4.50 and c = 4.50 a
2 1. In Worksheet 3 we defined the meaning of constant rate of change. c. On a job application for an administrative assistant position at a law firm, an applicant listed that he is able to type 90 words per minute. What does this mean? The change in the number of words typed is 90 times as large as the change in the time spent typing (in minutes). Put another way, if t represents the change in the time spent typing (in minutes) and w the change in the number of words typed, then w/ t = 90 and w = 90 t for all corresponding values of t and w In Worksheet 3 we defined the meaning of constant rate of change. d. Let x be the value of one quantity and let y be the value of another quantity. Suppose y changes at a constant rate of change of 8.2 with respect to x. What does this mean? y/ x = 8.2 and y = 8.2 x e. Let x be the value of one quantity and let y be the value of another quantity. Suppose y changes at a constant rate of change of 4.95 with respect to x. What does this mean? y/ x = 4.95 and y = 4.95 x 112 2
3 113 a. Are the quantities hours candle has been burning and candle length (in inches) related by a constant rate of change? Explain. Yes the quantities are related by a constant rate of change: the change in the candle length is always 1.6 times the change in the hours the candle has been burning. b. Represent an increase in the time spent burning of 2 hours from the given point on the graph
4 an increase of 2 hours from 3.5 hours since the candle began burning 115 c. By how much will the length of the candle change when the time spent burning increases by 2 hours? Represent this on the graph. It will change by 1.6(2), or 3.2 inches
5 a change of 3.2 inches of length from 8.3 inches 117 d. What is the length of the candle 5.5 hours since it began burning? Explain how you determined your answer. The length of the candle at 3.5 hours was 8.3 inches. We know that over the next two hours the length of the candle changed by 3.2 inches. This means that it is now 5.1 inches long. e. Represent a decrease in the time spent burning of 1.8 hours from the given point on the graph
6 a decrease of 1.8 hours (a change of 1.8 hours) from 3.5 hours since the candle began burning 119 f. By how much will the length of the candle change when the time spent burning is decreased by 1.8 hours? Represent this on the graph. It will change by 1.6( 1.8), or 2.88 inches. e. Represent a decrease in the time spent burning of 1.8 hours from the given point on the graph
7 a change of 2.88 inches of length from 8.3 inches 121 g. What is the length of the candle 1.7 hours since it began burning? Explain how you determined this. The length of the candle at 3.5 hours was 8.3 inches. We know that for a change in 1.8 hours of burning the length of the candle changes by 2.88 inches. This means that 1.8 hours earlier, the candle was inches long. h. What was the original length of the candle before it started burning? Explain how you determined this value and represent your reasoning on the graph. When the candle had burned for 3.5 hours it was 8.3 inches long. If the change in time is 3.5 hours, the change in the length of the candle is 3.5 times 1.6, or 5.6 inches from our reference length of 8.3 inches. The original length of the candle was 13.9 inches
8 (0, 13.9) a change of 5.6 inches of length from 8.3 inches A change of 3.5 hours from 3.5 hours since the candle began burning 123 i. Draw the graph that represents the length of the candle in inches with respect to the number of hours spent burning
9 length of the candle in inches hours candle has been burning Refer to Exercise #2 to answer the following questions. Let t be the amount of time elapsed since the candle began burning (in hours). a. Suppose the candle has been burning for 4 hours. Write the expression that calculates the change in t from t = 3.5 to t = b. Suppose the candle has been burning for 5.1 hours. Write the expression that calculates the change in t from t = 3.5 to t = c. Suppose the candle has been burning for 1 hour. Write the expression that calculates the change in t from t = 3.5 to t =
10 3. Refer to Exercise #2 to answer the following questions. Let t be the amount of time elapsed since the candle began burning (in hours). d. Write an expression that calculates the change in the length of the candle for each of the changes in time spent burning from parts (a) through (c). a. 1.6(4 3.5) b. 1.6( ) c. 1.6(1 3.5) Refer to Exercise #2 to answer the following questions. Let t be the amount of time elapsed since the candle began burning (in hours). e. Suppose the candle has been burning for x hours. i. What expression represents the change in t from t = 3.5 to t = x? x 3.5 ii. What expression represents the change in the length of the candle from t = 3.5 to t = x? 1.6(x 3.5) iii. What expression represents the length of the candle after burning for x hours? 1.6(x 3.5) or ( 1.6)(x 3.5)
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