Tim owns shares in a mutual fund. His investment was worth $4,200 when he first bought the shares. One year later, his shares were worth $4,410.
Let t represent the number of years since Tim first bought the shares. Assuming the value has increased exponentially, which expression best models the value of Tim's shares over time?
If the value of Tim's shares continues to increase at the same rate, what will they be worth 20years after he bought them?
Round your answer to the nearest whole number.
$
The expression that best models the value of Tim's shares over time is $4200 \times (1 + r)^t$, where $r$ represents the rate of increase and $t$ represents the number of years since Tim first bought the shares.
To find $r$, we can use the formula for exponential growth:
$4410 = 4200 \times (1 + r)^1$
Dividing both sides by $4200$:
$1.05 = (1 + r)$
Subtracting $1$ from both sides:
$0.05 = r$
So, the expression that models the value of Tim's shares over time is $4200 \times (1 + 0.05)^t$.
To find the value of Tim's shares 20 years after he bought them, we can substitute $t = 20$ into the expression:
$4200 \times (1 + 0.05)^{20} = 4200 \times (1.05)^{20} \approx \$9,155$.
Therefore, Tim's shares will be worth approximately $9,155 after 20 years.