The value of an investment A
(in dollars) after t years is given by the function A(t) = A0ekt
. If it takes 10 years for an investment of $1,000 to triple, how many years will it take for the investment to be $9,000? Simplify your answer completely.
I am very confused. So please show all work thank you
3 = 1 e^(10k)
10k = ln 3
k = ln3/10
so amount = 1000 e^( (ln3/10)t)
9000 =1000 e^( (ln3/10)t)
9 = e^( (ln3/10)t)
ln 9 = (ln3/10)t
ln9/(ln3/10)= t
t = 10ln9/ln3 = 20 years
Well, let's start by figuring out the value of k. We know that after 10 years, the investment tripled, so A(10) = 3A0.
Substituting the values, we have:
3A0 = A0e^10k
Dividing both sides by A0, we get:
3 = e^10k
To solve for k, we can take the natural logarithm of both sides:
ln(3) = ln(e^10k)
Using the property of logarithms that ln(e^x) = x, we have:
ln(3) = 10k
Now we can solve for k:
k = ln(3) / 10
Ok so we have found the value of k, which is approximately 0.0462.
Now, we need to find out how many years it will take for the investment to reach $9000.
We know that A(t) = A0e^kt, and we want to find the value of t when A(t) = $9000.
So we have:
9000 = 1000e^(0.0462t)
Divide both sides by 1000:
9 = e^(0.0462t)
Again, we can take the natural logarithm of both sides:
ln(9) = ln(e^(0.0462t))
Using the same property as earlier, we have:
ln(9) = 0.0462t
Now we can solve for t:
t = ln(9) / 0.0462
Calculating that, we find t is approximately 24 years.
So, it will take approximately 24 years for the investment to be $9000.
And there you have it, the answer without any clowning around!
To solve this problem, we need to use the given formula A(t) = A0ekt, where:
A(t) represents the value of the investment after t years,
A0 is the initial value of the investment (in this case, $1,000),
k is the growth or decay constant,
t is the number of years.
We know that it takes 10 years for the investment to triple, meaning that after 10 years the value of the investment is 3 times the initial value. Therefore, we have:
A(10) = 3A0
Substituting A0 = $1,000 into the equation, we get:
A(10) = 3($1,000)
Now we can solve for the value of k. From the given formula, we have:
A(t) = A0ekt
Dividing both sides of the equation by A0, we get:
A(t)/A0 = ek
Since A(10)/A0 = 3, we can rewrite this as:
3 = ek
To isolate k, we need to take the natural logarithm (ln) of both sides:
ln(3) = ln(ek)
Using the property ln(ab) = b ln(a), we can rewrite the equation as:
ln(3) = k ln(e)
Since ln(e) = 1, the equation simplifies to:
ln(3) = k
Now we know the value of k. To find out how many years it will take for the investment to reach $9,000, we can substitute A0 = $1,000, k = ln(3), and solve for t:
A(t) = A0ekt
$9,000 = $1,000e(ln(3))t
Dividing both sides by $1,000, we get:
9 = e(ln(3))t
Taking the natural logarithm of both sides again:
ln(9) = ln(e(ln(3))t)
Using the property ln(ab) = b ln(a) and the fact that ln(e) = 1, the equation simplifies to:
ln(9) = ln(3)t
Now we can solve for t by dividing both sides by ln(3):
t = ln(9)/ln(3)
Calculating this using a calculator, we find:
t ≈ 2.7095 years
Therefore, it will take approximately 2.7095 years for the investment to reach $9,000.
To find out how many years it will take for the investment to be $9,000, we need to solve for t in the equation A(t) = 9,000.
Given:
A(t) = A0ekt
A0 = initial investment = $1,000
A(t) = final investment = $9,000
Substituting these values into the equation, we have:
9,000 = 1,000 * ekt
To simplify this equation, we can divide both sides by 1,000:
(9,000 / 1,000) = ekt
Now, we can take the natural logarithm of both sides to eliminate the exponential term:
ln(9) = k * t
Next, we divide both sides by k to isolate t:
ln(9) / k = t
To determine the value of k, we can use the given information that it takes 10 years for the investment to triple. Therefore, we can set up another equation using the same formula:
3,000 = 1,000 * e^(10k)
We can simplify this equation by dividing both sides by 1,000:
3 = e^(10k)
Now, we can take the natural logarithm of both sides:
ln(3) = 10k
Dividing both sides by 10 isolates k:
ln(3) / 10 = k
Now that we have the value of k, we can substitute it back into the equation we derived earlier:
ln(9) / (ln(3) / 10) = t
Simplifying this equation gives us the final answer:
t = 10 * ln(9) / ln(3)
Approximating the value, we get:
t ≈ 10 * 2.197 / 1.099
t ≈ 20 years
Therefore, it will take approximately 20 years for the investment to reach $9,000.