Exponential Functions Assignment : Renzoku Is Logic Puzzle

Assignment Task

Question

1. Renzoku is a logic puzzle also known as Neighbours, where:

• The numbers to fill in are from 1 to 4
• Each row and column must contain only one instance of each number
• A dot between two cells indicates that those two numbers should be consecutive (for example 1× 2 or 2 ×1 are consecutive). If there is no dot then the numbers should NOT be consecutive.

2. Complete the following chart to analyze the characteristics of each exponential function

3. Consider an exponential equation y = ab ^x . Describe the effects the following changes will have on the graph of y = ab ^x .

• The value of a is changed from 3 to 1
• The value of b is changed from 1/2 to 2 

4. Complete the table below by solving each equation algebraically by expressing both sides with the same base, and graphically using technology. If necessary, round your answer to the nearest hundredth.

5. Write an exponential equation, in the form y = ab ^x , to model each of the following situations

• The cost of a home is $385 000 and it increases at a rate of 4.5%/a (per annum). Represent the cost of the home, C ( t ), after t
• A car is valued at $38 000 when it is first purchased, and it depreciates by 14?ch year after Represent the value of the car, V ( n ), after n years. 
• There are 450 bacteria at the start of a science experiment, and this amount triples every Represent the total number of bacteria, T ( h ), after h hour.
• The population of fish in a lake is 4000 and it decreases by 7?ch year. Represent the population of fish, P ( t ), after t

6. A student deposited money into a savings The following equation models the amount of money in the account, A ( t ), after t years. A ( t ) = 1400 (1.035) t

• State the initial amount of money deposited into the 
• Determine the annual interest rate being paid on the
• Use the equation to find the amount of money, to the nearest dollar, in the account after 10 
• How many years, to the nearest whole year, will it take for the account to have at least$2300?

7. The table below shows Canmore’s population from 2010 to 2017. The data is taken from

• Assuming the population of a Canmore has been growing exponentially, find the regression equation in the form P ( t ) = ab ^t , where P ( t ) is the population and t is the number of years since 2010 ( 2010 is year 0 in your calculator ). Express the values of a to the nearest whole number and b to the nearest thousandth. 
• Using your regression equation, calculate what the population of Canmore should have been, to the nearest whole number, in 2020.
• If the actual population of Canmore in 2020 was 14 798 people, why might the predicted population be different than the actual population?
• Assuming that the growth rate of Canmore follows your regression equation, during what year will the population reach 18 000?

8. The energy released during an earthquake can be calculated using the formula  E ( M ) = (10 ^1.5 ) M , where E is the energy in kWh (kilowatt hours) and M is the magnitude of the earthquake. Data is taken from Natural Resources Canada .

• The magnitude 9.0 earthquake off British Columbia on January 26, 1700 is thought to be the strongest earthquake that has occurred in Canada. Use the formula above to calculate the energy released during this earthquake. Express the answer in scientific notation, to the nearest 
• On December 23, 1985, the Nahanni Region in the Northwest Territories experienced an earthquake of magnitude 6.9. This earthquake caused widespread alarm as most people in the Northwest Territories had never felt an earthquake. Calculate the energy that was released during this Express the answer in scientific notation, to the nearest tenth.
• Comparing the British Columbia and Northwest Territories earthquakes, how many times greater was the energy that was released in British Columbia than the Northwest Territories? Round your answer to the nearest whole
• An earthquake occurred 5 km from Banff, AB on February 4, 2021. It released approximately 3 981 000 kWh of energy. Find the magnitude, to the nearest tenth, associated with this earthquake.

9. Sucralose, or Splenda, is made during a chemical process in which three hydrogen-oxygen groups are replaced with chlorine atoms. The result is a sweetener that tastes like sugar but is about 600 times sweeter. Splenda does not occur in nature, so it is not a "natural" product. The body does not metabolize sucralose, so it has no calories).

a. Assume

• the initial amount of sucralose in a can of Diet Coke with Splenda is 40 mg
• the amount of sucralose in a person’s body decreases by 50% every 2 days

b. Determine an exponential regression equation, y = a × b^x , for your table of values. Round a to the nearest whole number and b to the nearest hundredth. Describe the rate of decay of sucralose, as a percentage, based on your regression equation.

c. Sketch your exponential function and state an appropriate range. 

d. The decay of sucralose can be written using the exponential equation S (t ) =40 æ1 ö2 , where S is the amount of sucralose and t is the number of days that have passed. Algebraically determine how many days it takes for only 5 mg of sucralose to remain. Round your answer to the nearest tenth.

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