Explore IDNLearn.com's extensive Q&A database and find the answers you're looking for. Join our community to receive prompt, thorough responses from knowledgeable experts.
Sagot :
Givn:
Value of the car in 1995 = $32,000
Value of the car in 2001 = $14,000
Let's solve for the following:
• (A). What was the annual rate of change between 1995 and 2001?
Apply the exponential decay formula:
[tex]f(t)=a(1-r)^t[/tex]Where:
• t is the number of years between 2001 and 1995 = 2001 - 1995 = 6
,• a is the initial value = $32000
,• r is the rate of decay.
,• f(t) is the present value
Thus, we have
[tex]\begin{gathered} 14000=32000(1-r)^6 \\ \end{gathered}[/tex]Divide both sides by 32000:
[tex]\begin{gathered} \frac{14000}{32000}=\frac{32000(1-r)^6}{32000} \\ \\ 0.4375=(1-r)^6 \end{gathered}[/tex]Take the 6th root of both sides:
[tex]\begin{gathered} \sqrt[6]{0.4375}=\sqrt[6]{(1-r)^6} \\ \\ 0.87129=1-r \end{gathered}[/tex]Solving further:
[tex]\begin{gathered} r=1-0.87129 \\ \\ r=0.1287 \\ \\ r=0.1287*100=12.87\text{ \%} \end{gathered}[/tex]Therefore, the rate of change between 1995 and 2001 is 0.1287
• (B). What is the correct answer to part A written in percentage form?
In percentage form, the rate of change is 12.87 %
• (C),. Assume that the car value continues to drop by the same percentage. What will the value be in the year 2005?
We have the equation which represents this situation below:
[tex]f(t)=32000(1-0.1287)^t[/tex]Here, the value of t will be the number of years between 1995 and 2005.
t = 2005 - 1995 = 10
Now, substitute 10 for t and solve for f(10):
[tex]\begin{gathered} f(10)=32000(1-0.1287)^{10} \\ \\ f(10)=32000(0.8713)^{10} \\ \\ f(10)=32000(0.25216) \\ \\ f(10)=8069.14\approx8100 \end{gathered}[/tex]Therefore, the value in the year 2005 rounded to the nearest 50 dollars is $8100
ANSWER:
• (a). 0.1287
,• (b). 12.87%
,• (c). $8100
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.
