SOLUTION: An exponential decay graph shows the expected depreciation for a new boat, selling for $3,500, over 10 years.
a. Write an exponential function for the graph.
b. Use the fun
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Question 628699: An exponential decay graph shows the expected depreciation for a new boat, selling for $3,500, over 10 years.
a. Write an exponential function for the graph.
b. Use the function in part a to find the value of the boat after 9.5 years.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
First, this problem has nothing to do with Trigonometry. It belongs in one of the categories related to exponents. Please post using a possibly relevant category.
Second, without the graph you were given it will be impossible for a tutor to solve this problem. I will, nevertheless, try to help as much as I can.
The general equation for exponential growth and decay is:
where
t is a number of units of time. (Often the units are years so t would be a number of years.)
A = the amount after "t" units of time.
is the starting amount (IOW the amount at t = 0. This is why the A has a subscript of zero.)
r is a decimal (or fraction) between 0 and 1 which represents the rate of change. A positive r means growth. A negative r means decay. (Note: Some books use , merging the 1 and r. In this case if r > 1 then the equation is for growth and if r is between 0 and 1 then the equation is for decay. I prefer the first form I gave you because that percent growths or decays are easier to use with that form.)
You have been given the original amount, . So your equation so far is:
From this point on I can only tell you what you should do to finish:- Find the value for r:
- Look at the graph and find the coordinates of any point (other than where t = 0) on the graph. (If the axes are not labeled t and A, then the "t" in the equation corresponds to "x" and the "A" corresponds to the "y".)
- Replace the t and A in your equation with the coordinates you found in step 1.
- Solve this equation for r.
- Rewrite your working equation:
substituting in the value for r you just found. This (or the equation you get if you add the 1 to r) is the answer to part a. - Using the equation from part a, replace the t with 9.5 and solve for A. This will be the answer for part b.
Here's an example:
1. Solve for r
1.1 Find a point. Suppose that your graph shows that at t (or x) = 10, the A (or y) is 700.
1.2 Insert the coordinates into your working equation:
1.3. Solve for r.
Dividing both sides by 3500:
Find the 10th root of each side. (On your calculator use 0.2^(1/10) or 0.2^0.1)
Subtract 1 from each side:
(BTW: As a percent, r is an approximately 15% rate of decay.)
2. Replace the r in your working equation:
This, or , would be the answer to part a.
3. To find the value of the car after 9.5, replace the t in you equation with 9.5 and solve for A:
Simplifying...
Using 0.85133992^(9.5) on our calculators:
So after 9.5 years the car would be worth approximately $758.69.
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