A business finds that it has to increase its staff by 5% each year due to growth. After how many years would the size of their staff double?
2 = (1 + .05)^y
log(2) = y * log(1.05)
To find out after how many years the size of the staff would double, we can use the concept of exponential growth. Let's assume the initial size of the staff is "S".
According to the problem statement, the business has to increase its staff by 5% each year. This means that the staff size will grow by 5% every year.
To calculate the size of the staff after one year, we can use the following formula:
Staff after 1 year = S + (5/100) * S = S + 0.05S = 1.05S
Similarly, to calculate the size of the staff after two years, we can use the formula:
Staff after 2 years = (1.05S) + (5/100) * (1.05S) = (1.05S) + 0.0525S = 1.1025S
In general, the size of the staff after "n" years can be calculated using the formula:
Staff after "n" years = S * (1 + r)^n
where "r" is the rate of increase, which is 5% expressed as a decimal (0.05), and "n" is the number of years.
We want to find the value of "n" when the staff size doubles. This means we want to find the value of "n" when "Staff after n years" is equal to 2S.
Equating the two, we have:
2S = S * (1 + 0.05)^n
Dividing both sides of the equation by S, we get:
2 = (1.05)^n
To solve for "n", we need to take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):
ln(2) = ln((1.05)^n)
Using the logarithmic property, we can rewrite the equation as:
ln(2) = n * ln(1.05)
Finally, solving for "n", we divide both sides of the equation by ln(1.05):
n = ln(2) / ln(1.05)
Now we can calculate the value of "n" using a calculator or math software. The result will provide the number of years it would take for the staff size to double based on a 5% annual increase.