Can anyone tell me if this answer is correct?
Solve the problem.
Let's imagine that in 2001, a surgeon performs the first brain transplant, the only transplant performed that year. If in 2010, there are 983 brain transplants, find an exponential growth function that fits the data. (Round decimals to three places.)
Answer= N(t) = e^0.766t, where t is the number of years after 2001.
To determine if the answer is correct, we can try to find the exponential growth function by analyzing the given data and verifying if it matches the provided function.
To begin, we'll consider the form of an exponential growth function: N(t) = N0 * e^(kt), where N(t) represents the current population, N0 represents the initial population, t represents the time elapsed, and k is the growth rate.
Now let's apply this information to our problem:
Given that in 2001, there was only 1 brain transplant, we can assume N0 = 1.
In 2010, the number of brain transplants was 983.
First, we need to find the elapsed time from 2001 to 2010. By subtracting 2001 from 2010, we find that 9 years have passed.
Now, let's substitute the values into the exponential growth function and see if it aligns with the given answer:
N(t) = N0 * e^(kt)
N(9) = 1 * e^(9k)
983 = e^(9k)
To solve for k, we need to take the natural logarithm (ln) of both sides of the equation:
ln(983) = ln(e^(9k))
Simplifying further:
ln(983) = 9k * ln(e)
Now, ln(e) equals 1, so the equation becomes:
ln(983) = 9k
Dividing both sides by 9:
k ≈ ln(983) / 9
Evaluating this expression on a calculator (rounding to three decimal places) gives us k ≈ 0.109.
Now let's compare this value to the given answer of 0.766. Since they are not equal, it suggests that the provided answer is incorrect.
Therefore, the correct exponential growth function for this problem would be:
N(t) ≈ e^(0.109t)