High School

Ian currently owns the original Darth Vader helmet from "Star Wars Episode IV: A New Hope," valued at \$60,000. Unfortunately, the value of the helmet depreciates by 8\% every time a new Star Wars movie is released.

**Multiple Choice**

Which equation represents this situation?

A. [tex]y = 20000(1.13)^t[/tex]
B. [tex]y = 60000(0.92)^t[/tex]
C. [tex]y = 60000(0.87)^t[/tex]
D. [tex]y = 60000(1.08)^t[/tex]

**Formula**

How much will the helmet be worth after they release 17 more movies? (Round to the nearest whole number)

Answer: [tex]\square[/tex]

Answer :

We start with an initial helmet value of \[tex]$60,000. Every time a new movie is released, the helmet loses 8% of its value. Losing 8% means that the helmet retains 92% of its value from the previous movie release. In mathematical terms, after each movie, the value is multiplied by $[/tex]0.92[tex]$. This situation is represented by the exponential equation

$[/tex][tex]$
y = 60000 \cdot (0.92)^t,
$[/tex][tex]$

where $[/tex]t[tex]$ is the number of new movies released.

Thus, the correct equation from the multiple choice options is

$[/tex][tex]$
y=60000(0.92)^t.
$[/tex][tex]$

Next, we calculate the value of the helmet after 17 more movies. For $[/tex]t = 17[tex]$, the equation becomes

$[/tex][tex]$
y = 60000 \cdot (0.92)^{17}.
$[/tex][tex]$

Evaluating the expression $[/tex](0.92)^{17}[tex]$, we obtain approximately $[/tex]0.2423221228[tex]$. Multiplying this by \$[/tex]60,000 gives

[tex]$$
y \approx 60000 \cdot 0.2423221228 \approx 14539.3274.
$$[/tex]

Rounding to the nearest whole number, the helmet will be worth about \$14,539 after 17 movies.