Write the exponential function y=6(1.21)t in the form y=aekt.
(a) Once you have rewritten the formula, give k accurate to at least four decimal places.
k= help (numbers)
If t is measured in years, indicate whether the exponential function is growing or decaying and find the annual and continuous growth/decay rates. The rates you determine should be positive in the case of growth or decay (by choosing decay the negative rate is implied).
(b) The annual rate is % per year (round to the nearest 0.01%).
(c) The continuous rate is % per year (round to the nearest 0.01%).
To rewrite the exponential function y=6(1.21)^t in the form y=aekt, we need to find the value of k.
Let's start by rewriting the given function:
y = 6(1.21)^t
We know that a = 6 and e is the base of the natural logarithm, approximately equal to 2.71828.
Now, we can rewrite this equation as:
y = 6e^(kt)
Comparing this with the given function, we can see that:
6e^(kt) = 6(1.21)^t
To find k, we can equate the exponents on both sides:
e^(kt) = 1.21^t
Taking the natural logarithm (ln) of both sides:
ln(e^(kt)) = ln(1.21^t)
Using the property of logarithms that ln(a^b) = b ln(a):
kt ln(e) = t ln(1.21)
Since ln(e) = 1, we can simplify this equation to:
kt = t ln(1.21)
Dividing both sides by t:
k = ln(1.21)
Using a calculator, we can evaluate ln(1.21) = 0.19021 (accurate to at least four decimal places).
Therefore, k = 0.19021.
(a) k = 0.19021
Since the value of k is positive, the exponential function y = 6(1.21)^t is growing.
(b) To find the annual growth rate as a percentage, we can use the formula:
Annual growth rate = (e^k - 1) * 100%
Substituting the value of k:
Annual growth rate = (e^0.19021 - 1) * 100%
Using a calculator, we can compute e^0.19021 ≈ 1.20996.
Annual growth rate = (1.20996 - 1) * 100% ≈ 20.996%.
Rounded to the nearest 0.01%, the annual growth rate is approximately 21.00%.
(c) To find the continuous growth rate as a percentage:
Continuous growth rate = k * 100%
Substituting the value of k:
Continuous growth rate = 0.19021 * 100% ≈ 19.021%.
Rounded to the nearest 0.01%, the continuous growth rate is approximately 19.02%.
To write the exponential function y=6(1.21)^t in the form y=aekt, we need to find the value of k.
Step 1: Rewrite the expression as y = 6e^(kt), where e is the base of the natural logarithm (approximately 2.71828).
Step 2: Since 1.21 = e^k, we need to solve for k. Take the natural logarithm of both sides of the equation:
ln(1.21) = ln(e^k)
Step 3: Use the property of logarithms that states ln(e^x) = x to simplify the equation:
k = ln(1.21)
To find k accurate to at least four decimal places, let's calculate it.
k = ln(1.21) ≈ 0.1917
So k is approximately 0.1917 (rounded to four decimal places).
Now let's move to part (b) and (c) of the question.
Since the exponential function y = 6(1.21)^t represents population growth, it is a growing function.
Now, to find the annual growth rate, we need to calculate the percentage growth rate per year. This can be done using the formula:
Percentage growth rate per year = (e^k - 1) * 100
(b) The annual growth rate is approximately (e^0.1917 - 1) * 100.
Calculating it:
Annual growth rate = (e^0.1917 - 1) * 100 ≈ (1.2112 - 1) * 100 ≈ 21.12%
So the annual growth rate, rounded to the nearest 0.01%, is approximately 21.12%.
To find the continuous growth rate, we use the formula:
Continuous growth rate per year = k * 100
(c) The continuous growth rate is approximately k * 100 = 0.1917 * 100 = 19.17%
Therefore, the continuous growth rate, rounded to the nearest 0.01%, is approximately 19.17%.
In summary:
(a) k is approximately 0.1917 (rounded to four decimal places).
(b) The annual growth rate is approximately 21.12%.
(c) The continuous growth rate is approximately 19.17%.
y = 6(1.21)^t
1.21 = e^ln(1.21) = e^0.19
y = 6*(e^0.19)^t = 6*e^(0.19t)
Now you can work the rest