A laptop computer is purchased for
$5000
. Each year, its value is
70%
of its value the year before. After how many years will the laptop computer be worth
$300
or less
300 = 5000 (.7)^n
.06 = .7^n
log .06 = n log .7
so
n = log.06/log.7 = 7.89 years
Let's break down the problem step by step:
Step 1: Determine the value of the laptop after the first year.
The value after the first year will be 70% of $5000:
Value after the first year = $5000 * 70% = $5000 * 0.7 = $3500
Step 2: Determine the value of the laptop after the second year.
The value after the second year will be 70% of its value after the first year:
Value after the second year = $3500 * 70% = $3500 * 0.7 = $2450
Step 3: Repeat Step 2 for subsequent years until the value of the laptop is $300 or less.
Value after the third year = $2450 * 70% = $2450 * 0.7 = $1715
Value after the fourth year = $1715 * 70% = $1715 * 0.7 = $1200.50
Based on the calculations above, we can see that the value of the laptop will be $1200.50 or less after the fourth year.
Therefore, it will take 4 years for the laptop computer to be worth $300 or less.
To determine how many years it will take for the laptop computer to be worth $300 or less, we can set up an equation and solve for the number of years.
Let's define the value of the laptop in year n as Vn.
According to the given information, the value of the laptop in year n is 70% of its value in the previous year, so we can express this relationship mathematically as:
Vn = 0.7 * Vn-1
We need to find the number of years it takes for the value of the laptop to be $300 or less, which means we need to find the first value of n for which Vn is less than or equal to $300.
We can start with the initial value of the laptop, which is $5000, and continue calculating the value for subsequent years until it reaches or falls below $300.
Let's tabulate the values of Vn for each year:
Year 1: V1 = 0.7 * $5000 = $3500
Year 2: V2 = 0.7 * $3500 = $2450
Year 3: V3 = 0.7 * $2450 = $1715
Year 4: V4 = 0.7 * $1715 = $1200.50
After 4 years, the value of the laptop falls to $1200.50, which is less than $300. Therefore, it takes 4 years for the laptop computer to be worth $300 or less.