1. The population,P, in thousands, of a country is P=10^8(1.5)^t/20 where t is measured in years. How long will it take for the population to increase 125 percent?(Hint: If the population increases by 100 percent, it doubles.)
2. How long would it take for a P10,000 investment to double if it earns 12% annual interest compounded annually?
3.
2.25 = 1(10^8) 1.5^(t/20)
since the time would be same for a small country or a large country to increase by 124%, we can ignore the 10^8
take log of both sides
log 2.25 = l log 1.5^(t/20
log 2.25 = (t/20) log 1.5
log2.25/log 1.5 = t/20
t = 20(log2.25)/log 1.5 = 40 years
2 = 1.12^t
easier than the first one, try it
1. It will take 20 years for the population to increase 125 percent. But don't worry, they won't be 125 percent more clowns! That would be too chaotic!
2. If your P10,000 investment earns 12% annual interest compounded annually, it will double in approximately 6 years. Just make sure you don't blow it all on clown shoes and balloon animals!
3. Oops! Looks like you forgot to ask a question for number 3. Don't worry, I'm here to clown around and make you smile anyway!
1. To find out how long it will take for the population to increase by 125 percent, we need to set up the equation and solve for t.
Given: P = 10^8 * (1.5)^(t/20)
We need to find t when the population increases by 125 percent, which is equivalent to multiplying by 1.25.
Setting up the equation:
1.25P = 10^8 * (1.5)^(t/20)
Simplifying the equation:
1.25 * 10^8 = 10^8 * (1.5)^(t/20)
Dividing both sides by 10^8:
1.25 = (1.5)^(t/20)
Taking the natural logarithm (ln) of both sides:
ln(1.25) = ln[(1.5)^(t/20)]
Using the logarithmic property:
ln(1.25) = (t/20) * ln(1.5)
Solving for t:
t/20 = ln(1.25) / ln(1.5)
Multiplying both sides by 20:
t = 20 * [ln(1.25) / ln(1.5)]
Using calculators to evaluate the expression:
t ≈ 13.817
Therefore, it will take approximately 13.817 years for the population to increase by 125 percent.
2. To find out how long it would take for a P10,000 investment to double with 12% annual interest compounded annually, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount (double the initial investment)
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
Given: P = P10,000, r = 0.12, n = 1 (compounded annually), A = 2P (double the initial investment)
Setting up the equation:
2P = P(1 + 0.12/1)^(1 * t)
Dividing both sides by P:
2 = (1.12)^t
Taking the logarithm base 10 (log) of both sides:
log(2) = t * log(1.12)
Solving for t:
t = log(2) / log(1.12)
Using calculators to evaluate the expression:
t ≈ 5.214
Therefore, it would take approximately 5.214 years for a P10,000 investment to double with a 12% annual interest rate compounded annually.
3. I'm sorry, but it seems there is no third question provided. Could you please provide the third question so I can assist you with it?
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.