Question

At the band's peak of popularity, a personally signed Black Diamond poster sold for $500. Three years later the band was almost forgotten and the poster was worth only $10. What were the annual multiplier and annual percent of decrease?

Answer 1

Answer:

The annual multiplier is 0.27 and Annual Percent in decrease is 73%.

Step-by-step explanation:

Given:

Initial Value $a_i = \ 500$

Elapsed time $n= 3\ years$

Final Value $a_3 = \ 10$

We need to find the annual multiplier and annual percent of decrease.

Solution:

Now we know that;

The Final value after n years is equal to Initial value multiplied by the multiplier raise to number of elapsed years.

framing in equation form we get;

$a_n=a_i\times (m)^n$

m⇒ annual multiplier

Substituting the values we get;

$10=500\times (m)^3\\\\m^3=\frac{10}{500}\\\\m^3=\frac{1}{50}$

Taking cube root we get;

$\sqrt[3]{m^3}=\sqrt[3]{\frac{1}{50}} \\\\m=0.27$

Hence the annual multiplier is 0.27.

Now We will find the annual percent of decrease.

Now we know that;

annual multiplier is equal to 1 minus the depreciation rate.

$m=1-r$

r ⇒ annual percent in decrease.

$0.27 =1-r\\\\r = 0.73\ \ Or \ \ 73\%$

Hence Annual Percent in decrease is 73%.