exponential equation

Question

exponential equation
The retail value of a 2011 15.4" Apple Macbook Pro laptop was $2199.99. The resale value of the Macbook Pro decreases by 13.8% each year. Use this information to complete the following problems.
a. Write the equation for the exponential function that could model this situation, where t is the number of years after 2011. Call it V(t).

V(t)=2199.99(.862)^x is this correct?
c. Use your model to estimate GRAPHICALLY when the value of the laptop will depreciate to $1,000.
e. Suppose it would not be worth your time to sell the laptop once the value dropped below $500. Determine graphically AND algebraically in which year this will happen.

Answer 1

Hi Steve:

 

  Evaluation of exponential functions can be done by using either log table of both base 10 and base e

 

  For this case:

 

     2199.99( 1 - 0.138) ^4 = 2199.99( 0.862) ^4 = 1214. 646128

 

 

      2199. 99 * e^{-0.138(4) = 1266 .747783}

   

        The answer come out different with 2%

      However:

 

          Consider:

 

          2199.99* ( 0.862) ^t  = 2199.99 e ^kt

 

             t ( ln 0.862) = kt

 

             k = ln (0.862) 

 

           2199. 99* e^((ln0.862) * 4) = 1214.646128

 

           Read exactly the same value, but textbook don't consider this fact, and use both bases as valid calculation.

   Actually they are valid, but ln table do not use the same method as log tables . 

 

 

                       

                       

               

                            

Answer 2

V(t)=2199.99(.862)^x is NOT correct.

 

Original question:

 

The retail value of a 2011 15.4" Apple Macbook Pro laptop was $2199.99. The resale value of the Macbook Pro decreases by 13.8% each year. Use this information to complete the following problems.

a. Write the equation for the exponential function that could model this situation, where t is the number of years after 2011. Call it V(t).

b. Use your model to predict the resale value of the Macbook Pro laptop in 2015.

c. Use your model to estimate GRAPHICALLY when the value of the laptop will depreciate to $1,000.

d. Use your model to estimate ALGEBRAICALLY when the value of the laptop will depreciate to $1,000;

e. Suppose it would not be worth your time to sell the laptop once the value dropped below $500. Determine graphically AND algebraically in which year this will happen.

 

My answers:

a.)

V(t) = 2199.99 e^(-0.138 t)

b.)

V(2015-2011) = 2199.99 e^(-0.138 * 4)
≈ 1266.74778258838322 ≈ $1,266.75

c.)

See http://www.wyzant.com/resources/files/263768/depreciation_of_mac for graphs.
Graphically V(5.71343) = 1000.

d.)

V(c) = 1000 = 2199.99 e^(-0.138 c)

100000/219999 = e^(-0.138 c)

-0.138 c = ln(100000/219999)

c = ( ln(100000/219999))/(-0.138) ≈ 5.71342619492312 years from 2011

e.)

Graphically V(10.73623) = 500.

V(c) = 500 = 2199.99 e^(-0.138 c)

50000/219999 = e^(-0.138 c)

-0.138 c = ln(50000/219999)

c = ( ln(50000/219999))/(-0.138) ≈ 10.73623185115477 years from 2011

[“Half Life” is 10.73623185115477 - 5.71342619492312 = 5.02280565623165 years;
i.e., about every 5 years the value halves.]