Which are the correct answers?
In a certain candy factory, the amount of candy bars produced annually is double the amount from the year before. This relationship can be described by:
Select one:
a. a linear function
b. an exponential function
Question 5
In a certain apple plant, the amount of apples shipped annually is increasing by 15%15% of the amount from the year before. This relationship can be described by:
Select one:
a. a linear function
b. an exponential function
Question 6
In a certain production plant, the annual production amount is decreasing by 10%10% of the amount from the year before. This relationship can be described by:
Select one:
b. an exponential function
c. both a linear and exponential function
No function is both linear and exponential!
If there is a constant amount (say, m) added or subtracted, that is a linear function (with slope m)
If there is a constant multiplication factor, that is an exponential function.
Question 1: In a certain candy factory, the amount of candy bars produced annually is double the amount from the year before. This relationship can be described by:
Answer: b. an exponential function
To explain how to arrive at this answer, we need to understand the characteristics of a linear function and an exponential function.
A linear function is represented by a straight line when graphed, and it has a constant rate of change. This means that the increase or decrease in the value is constant over time. For example, if you produce 100 candy bars in the first year and increase by 100 each subsequent year, the relationship would be linear.
However, in this case, the amount of candy bars produced annually is double the amount from the year before. This means that the rate of increase is not constant. If you produce 100 candy bars in the first year, you would produce 200 in the second year (double the amount), 400 in the third year (double the amount again), and so on. This growth pattern indicates an exponential relationship, where the value is multiplied by a constant factor in each step.
Question 2: In a certain apple plant, the amount of apples shipped annually is increasing by 15% of the amount from the year before. This relationship can be described by:
Answer: b. an exponential function
To arrive at this answer, we need to understand the given information. The amount of apples shipped annually is increasing by 15% of the amount from the year before. This means that each year, the amount of apples shipped is 15% more than the previous year.
If we start with a base value of, let's say, 100 apples shipped in the first year, the second year would be 100 + 15% of 100 = 100 + 15 = 115 apples. In the third year, it would be 115 + 15% of 115 = 115 + 17.25 = 132.25 apples. Each subsequent year, the amount of apples shipped is increasing by a factor of 15%. This type of growth is characteristic of an exponential relationship.
Question 3: In a certain production plant, the annual production amount is decreasing by 10% of the amount from the year before. This relationship can be described by:
Answer: c. both a linear and exponential function
To analyze the given information, we have to understand that the annual production amount is decreasing by 10% of the amount from the year before. This means that each year, the production amount is reduced by 10% of the previous year's value.
If we start with a base value of, let's say, 100 units produced in the first year, the second year's production would be 100 - 10% of 100 = 100 - 10 = 90 units. In the third year, it would be 90 - 10% of 90 = 90 - 9 = 81 units. As we can see, the decrease in production is constant by percentage, indicating that the decrease follows an exponential pattern.
However, this relationship can also be described as linear. If we consider the year-over-year change in the production amount, it is constant at -10 units. So, if we start with 100 units in the first year and decrease by 10 units each subsequent year, the relationship would be linear.
Therefore, the given relationship can be described as both a linear function (constant decrease by units) and an exponential function (decrease by a percentage).