In the following exercise, you are asked about the growth factor of an exponential function
N = N(t).
One entry in a data table for N gives
t = 3
and
N = 9.
Another entry gives
t = 5
and
N = 27.
What value of N should be given for
t = 6?
(Round your answer to one decimal place.)
Given a growth factor of r, we have
N(5)/N(3) = r^2
so r=√3
and N(6) = N(5)*√3 = 27√3
To find the value of N for t = 6, we need to determine the growth factor of the exponential function.
The growth factor can be calculated using the formula:
Growth Factor = (N2 / N1) ^ (1 / (t2 - t1))
Let's use the given data to find the growth factor:
N1 = 9 (from t = 3)
t1 = 3
N2 = 27 (from t = 5)
t2 = 5
Substituting these values into the formula, we get:
Growth Factor = (27 / 9) ^ (1 / (5 - 3))
Simplifying the expression inside the parentheses:
Growth Factor = 3 ^ (1 / 2)
Taking the square root of 3:
Growth Factor = √3
Now, we can find N for t = 6 using the growth factor:
N = N1 * (Growth Factor) ^ (t - t1)
Substituting the known values:
N = 9 * (√3) ^ (6 - 3)
Simplifying the exponent:
N = 9 * (√3) ^ 3
Calculating (√3) ^ 3:
N = 9 * √(3 * 3 * 3)
N = 9 * √27
Calculating √27:
N ≈ 9 * 5.2
Rounding the answer to one decimal place:
N ≈ 46.8
Therefore, the value of N for t = 6 is approximately 46.8.
To find the value of N for t = 6 in an exponential function N = N(t), we need to determine the growth factor of the function. The growth factor is the constant base raised to the power of the exponent.
Let's start by finding the growth factor based on the given data. We are given two data points:
1. When t = 3, N = 9.
2. When t = 5, N = 27.
We can use these data points to find the growth factor. The formula for an exponential function is N = N₀ * (growth factor)^t, where N₀ is the initial value of N.
Using the first data point (t = 3, N = 9), we can plug in these values into the equation:
9 = N₀ * (growth factor)^3
Similarly, using the second data point (t = 5, N = 27), we get:
27 = N₀ * (growth factor)^5
Now, we can divide the second equation by the first equation to eliminate N₀:
27/9 = (growth factor)^5 / (growth factor)^3
3 = (growth factor)^(5-3)
3 = (growth factor)^2
Taking the square root of both sides, we find:
√3 = growth factor
Now that we have the growth factor, we can use it to find N for t = 6. Plugging in the values into the equation N = N₀ * (growth factor)^t, we get:
N = N₀ * (√3)^6
Since we don't have the initial value N₀, we can't determine N precisely. However, we can still calculate an approximate value by using the given data. Let's calculate it:
N ≈ 9 * (√3)^6
Using a calculator or a computer, we can calculate:
N ≈ 9 * 10.39 ≈ 93.51
Therefore, the value of N (rounded to one decimal place) for t = 6 is approximately 93.5.