To solve each of these questions, we need to use the respective formulas provided and solve for the variable (time) that satisfies the given condition.
1. In the first question, we are given the population formula: P = 10^8(1.5)^(t/20), where P represents the population in thousands, t is the time in years, and 1.5 is the growth factor. We need to find the time it takes for the population to increase by 125 percent.
To solve this, we need to set up the equation:
P + 125% of P = 10^8(1.5)^(t/20)
Let's simplify the equation:
P + 1.25P = 10^8(1.5)^(t/20)
Combine the like terms:
2.25P = 10^8(1.5)^(t/20)
Now, to solve for t, we need to isolate t on one side of the equation:
1.5^(t/20) = (2.25P) / 10^8
Take the logarithm (base 1.5) of both sides:
log base 1.5 (1.5^(t/20)) = log base 1.5 ((2.25P) / 10^8)
Simplify:
t/20 = log base 1.5 ((2.25P) / 10^8)
Finally, multiply both sides by 20 to isolate t:
t = 20 * log base 1.5 ((2.25P) / 10^8)
Now, substitute the given value for P and calculate t.
2. In the second question, we are given the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate as a decimal, n is the number of times interest is compounded per year, and t is the time in years. Here, we need to find the time it takes for a P10,000 investment to double with a 12% annual interest compounded annually.
Let's set up the equation:
A = 2P = P(1 + r/n)^(nt)
Now substitute the given values:
2P = P(1 + 0.12/1)^(1*t)
Simplify:
2 = (1.12)^t
Take the logarithm (base 1.12) of both sides:
log base 1.12 (2) = log base 1.12 ((1.12)^t)
Simplify:
t = log base 1.12 (2)
Now, calculate the logarithm, and you will get the time it takes for the investment to double.
3. Unfortunately, it seems like you haven't provided the third question. Please provide the details, and I will be happy to help you solve it.