So, the car will be 8 years old when it is worth $2400.
If the value of a car is inversely proportional to its age, we can express this relationship using the formula:
![\[V = (k)/(A),\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/hp6pyamsmnts3nzxqk7laqs01pxltf7nwa.png)
where:
is the value of the car,
is the age of the car, and
is the constant of proportionality.
Given that the car is worth $6400 when it is 3 years old, we can use this information to find the value of
:
![\[6400 = (k)/(3).\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/9ct56j83ahafgu3ohbiuj5ni0n5h4o4le5.png)
To find
, multiply both sides by 3:
![\[k = 6400 * 3 = 19200.\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/42xzi4ddayvgwymodpqgvtw2ss4lreh4nj.png)
Now that we have the constant of proportionality
, we can use it to determine the age of the car when it is worth $2400. Let
:
![\[2400 = (19200)/(A).\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/zpnblniq5j3qmwcoqnyr5t8a9u5x63sphl.png)
To find
, divide both sides by 2400:
![\[A = (19200)/(2400) = 8.\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/q5h8sf78qtmnv7dfju4091x8wo1lz2jgrf.png)