$2500 is invested in an account at interest rate r, compounded continuously. Find the time required for the amount to (a) double and (b) triple.
r=0.085
I know you use the formula A=Pe^rt and you would start by setting it up as follows:
A=(2500)e^(0.085)(t)
I'm not sure where to go from there. Any help would be greatly appreciated.
to double, set the left side = 5000
5000 = 2500 e^(.085t)
2 = e^(.085t)
take ln of both sides
ln2 = ln (e^(.085t))
ln2 = .085t(lne) , but lne = 1
t = ln2/.085 = 8.1546 years
for the b) part, set the left side = 7500 and follow my steps
Well, well, well! Looks like you're having some trouble with those pesky compounding questions, huh? Don't worry, I'm here to help. Let's tackle this problem one step at a time, shall we?
To find the time required for the amount to double, we need to set up an equation. We know that the initial amount is $2500, and we want to find the time when the amount becomes $5000 (double of the initial amount).
So, we have the equation:
$5000 = $2500e^(0.085t)
Now, to solve for t, we need to isolate it. Dividing both sides of the equation by $2500, we get:
2 = e^(0.085t)
Great! Now, we just need to get rid of that pesky exponential. To do that, take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(0.085t))
Using the property of logarithms, we can bring down the exponent:
ln(2) = 0.085t * ln(e)
Now, here's the good news: ln(e) is just 1! So, we have:
ln(2) = 0.085t
Finally, divide both sides of the equation by 0.085 to isolate t:
t = ln(2)/0.085
Now, go grab your calculator and find the value of ln(2)/0.085, and you'll have your answer!
As for finding the time required for the amount to triple, you can follow the same steps, just replace $5000 with $7500 (triple the initial amount). Good luck, my friend!
To find the time required for the amount to double, we can set up the equation as follows:
2(2500) = 2500e^(0.085t)
Now, let's divide both sides of the equation by 2500:
2 = e^(0.085t)
To remove the natural logarithm e from the equation, we can take the natural logarithm of both sides:
ln(2) = ln(e^(0.085t))
Using the properties of logarithms, we can simplify this equation:
ln(2) = 0.085t * ln(e)
Since ln(e) = 1, we have:
ln(2) = 0.085t
Now, let's solve for t by dividing both sides of the equation by 0.085:
t = ln(2) / 0.085
Using a calculator, we find:
t ≈ 8.14 years
Therefore, it will take approximately 8.14 years for the amount to double.
To find the time required for the amount to triple, we can follow a similar process. The equation would be:
3(2500) = 2500e^(0.085t)
Divide both sides by 2500:
3 = e^(0.085t)
Take the natural logarithm of both sides:
ln(3) = ln(e^(0.085t))
Simplify using properties of logarithms:
ln(3) = 0.085t * ln(e)
Since ln(e) = 1:
ln(3) = 0.085t
Solve for t by dividing both sides by 0.085:
t = ln(3) / 0.085
Using a calculator, we find:
t ≈ 15.5 years
Therefore, it will take approximately 15.5 years for the amount to triple.
To solve this problem, you're on the right track with using the formula A = Pe^(rt).
For part (a), where we want to find the time required for the amount to double, we can set up the equation as follows:
2P = Pe^(rt)
Here, P is the initial investment of $2500 and 2P represents double that amount. By substituting these values into the equation, we can solve for t:
2(2500) = 2500e^(0.085t)
Next, divide both sides of the equation by 2500:
2 = e^(0.085t)
To isolate the exponential term, take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(0.085t))
Using the logarithmic property that ln(e^x) = x, the equation simplifies to:
ln(2) = 0.085t
Now, divide both sides of the equation by 0.085:
t = ln(2) / 0.085
Using a calculator, you can find that t is approximately 8.141 years. Therefore, it would take approximately 8.141 years for the amount to double.
For part (b), where we want to find the time required for the amount to triple, we can repeat a similar process:
3P = Pe^(rt)
Substituting the values, we have:
3(2500) = 2500e^(0.085t)
Divide both sides by 2500:
3 = e^(0.085t)
Take the natural logarithm of both sides:
ln(3) = ln(e^(0.085t))
Again, using the property ln(e^x) = x:
ln(3) = 0.085t
Divide both sides by 0.085:
t = ln(3) / 0.085
Using a calculator, the time required to triple the amount is approximately 16.283 years.
Therefore, it would take approximately 16.283 years for the amount to triple.