Lesson 4-1: Functions
Worksheet 7-5: Modelling with Exponential and Logarithmic Equations
Key Concepts of Modelling with Exponential and Logarithmic Equations:
• Different technology tools and strategies can be used to construct mathematical models that describe real situations.
• A good mathematical model:
- is useful for both understanding and extrapolating from given data in order to make predictions.
- can be used, in conjunction with other considerations, to aid in decision making.
• Exponential and logarithmic equations often appear in contexts that involve continuous growth or decay
Practice 1: Modelling Population
The population of Decimal Point has been steadily growing for several decades. The table gives the population at 5-year intervals, beginning in 1920, the year the town’s population reached 1015.
a) Is a quadratic or an exponential model a better fit to the given data? Justify your reasoning.
b) Suppose that it is decided that a recreation centre should be built once the town’s population reaches 5000. When should the recreation centre be built based on the given data?
Practice 2: Investment Optimization
Decimal Point has a surplus of $50 000 to invest to build a recreation centre. The two best investment options are described in the table.
a) Complete the table and construct an algebraic model that gives the amount, A, as a function of time, t, in years, for each investment.
b) Which of these investment options will allow the town to double its money faster?
c) If the town needs $80 000 to begin building the recreation centre, how soon can work begin, and which investment option should be chosen?
Practice 3: Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a heated object will decrease exponentially over time towards the temperature of the surrounding medium. That is, the temperature, u, of a heated object at a given time, t, obeys the law [pic], where T is the constant temperature of the surrounding medium, [pic] is the initial temperature of the heated object, and k is a negative number.
a) An object is heated to 100[pic] and is then allowed to cool in a room that has an air temperature of 21[pic], If after 5 min the temperature of the object is 80[pic], when will its temperature be 50[pic]?
b) A thermometer reading [pic] is brought into a room with a constant temperature of 21[pic]. If the thermometer reads 15[pic] after 3 min, what will it read after being in the room for 5 min?
Practice 4: Sound Levels
The difference in two sound levels, [pic] and [pic], in decibels (dB), is given by the logarithmic equation [pic], where [pic] is the ratio of their intensities.
a) The sound level of a jet at take-off is 140 dB, while the level of a normal conversation is 50 dB. What is the ratio of the intensities of the sound level of the jet versus the level of normal conversation?
b) What is the loudness of a jackhammer (in use) if it is known that this sound has an intensity 10 times that of the sound due to heavy city traffic (90dB)?
Practice 5: Quality Assurance Training
On average, the number of items, N, per day, on an assembly line, that a quality assurance trainee can inspect is [pic], where t is the number of days worked. After how many days of training will the employee be able to inspect 43 items?
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Assigned work: P. 404-407 #5, #7-8, #10
WS 7-5
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WS 7-5
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