Question

Jason purchased a machine for his factory. The value of the machine was $21,500 in 2010. In 2011, the value of the machine was $18,920. In 2012, the value of the machine was $16,649.60, and in 2013, the value of the machine was $14,651.65. Which statement best describes this situation?
A. An increasing exponential function can be used to describe this situation because it has a constant percentage of change.
B. An increasing linear function can be used to describe this situation because it has a constant rate of change.
C. A decreasing linear function can be used to describe this situation because it has a constant rate of change.
D. A decreasing exponential function can be used to describe this situation because it has a constant percentage of change.

Answer 1

I think it would be D. A decreasing exponential function.

Answer 2

Answer:

D. A decreasing exponential function can be used to describe this situation because it has a constant percentage of change.

Step-by-step explanation:    

We have been given that the value of the machine was $21,500 in 2010. In 2011, the value of the machine was $18,920. In 2012, the value of the machine was $16,649.60, and in 2013, the value of the machine was $14,651.65.

First of all we can see that the value of machine is decreasing as 14,651.65 is less than 21,500.  

Now let us see if the decay rate is linear or exponential.  

$\text{Rate of change}=\frac{21500-18920}{2010-2011}$

$\text{Rate of change}=\frac{2580}{-1}=-2580$

$\text{Rate of change}=\frac{18920-16649.60}{2011-2012}$

$\text{Rate of change}=\frac{2270.4}{-1}=-2270.4$

$\text{Rate of change}=\frac{16649.60-14651.65}{2012-2013}$

$\text{Rate of change}=\frac{1997.95}{-1}=-1997.95$  

We can see that the rate of change between any two consecutive years is not same. Since we know that a linear function has a constant rate of change and rate of change of value of machine is not constant, therefore, our function is not a linear function.

Let us find rate of change of value of machine using percentage change formula.

$\text{Percentage change}=\frac{\text{New-Original}}{\text{Original}}*100$          

$\text{Percentage change}=\frac{18,920-21,500}{21,500}*100$  

$\text{Percentage change}=\frac{-2580}{21,500}*100$  

$\text{Percentage change}=-0.12*100=-12$  

$\text{Percentage change}=\frac{16,649.60-18,920}{18,920}*100$  

$\text{Percentage change}=\frac{-2270.4}{18,920}*100$  

$\text{Percentage change}=-0.12*100=-12$  

We can see that the value of machine is decreasing 12% per year, therefore,  A decreasing exponential function can be used to describe this situation because it has a constant percentage of change (12%).