write the exponential function y=71(1.19)^t in the form y=ae^kt.
find k accuarate to 4 decimal places. find the exact annual growth rate and the continuous % growth rate accurate to 2 decimal places.
1.19 = e^.1739
y = 71 e^.1739t
To write the exponential function y=71(1.19)^t in the form y=ae^kt, we need to find the constant values a and k.
In the given function, a is equal to 71, and we need to find the value of k.
To find k, we can start by taking the natural logarithm (ln) of both sides of the equation:
ln(y) = ln(71(1.19)^t)
Using the properties of logarithms, we can simplify this equation as:
ln(y) = ln(71) + ln(1.19^t)
Now, using the properties of logarithms again, we can bring the exponent t down in front:
ln(y) = ln(71) + t * ln(1.19)
Rearranging the equation, we get:
ln(y) = t * ln(1.19) + ln(71)
This equation now fits the form ln(y) = kt + ln(a), where k = ln(1.19) and a = 71.
So, the exponential function y=71(1.19)^t can be written as y = 71e^(ln(1.19)t), where k is approximately equal to ln(1.19).
Using a calculator, we find k ≈ 0.1753 (accurate to 4 decimal places).
To find the exact annual growth rate, we can use the formula (e^k - 1) * 100%.
Substituting the value of k, we have:
Growth rate = (e^0.1753 - 1) * 100%
Calculating this using a calculator, we get:
Growth rate ≈ 19.03% (accurate to 2 decimal places).
To find the continuous % growth rate, we simply use the value of k multiplied by 100:
Continuous % growth rate = k * 100
Substituting the value of k, we have:
Continuous % growth rate ≈ 17.53% (accurate to 2 decimal places).
To rewrite the exponential function y = 71(1.19)^t in the form y = ae^kt, we need to convert the base from 1.19 to e.
First, let's rewrite the function using the identity "e^(ln(x)) = x":
y = 71(1.19)^t
= 71(e^(ln(1.19)))^t
Now, since we want to rewrite it in the form y = ae^kt, we can set a = 71 and k = ln(1.19).
Therefore, the exponential function y = 71(1.19)^t can be written as y = 71e^(ln(1.19)t).
To find the exact value of k, we can calculate ln(1.19) using a calculator:
k ≈ ln(1.19)
≈ 0.1758
To obtain the annual growth rate, we can subtract 1 from the base (1.19) and then multiply by 100:
Annual growth rate ≈ (1.19 - 1) × 100
≈ 0.19 × 100
≈ 19
Therefore, the exact annual growth rate is 19%.
The continuous percent growth rate can be found by multiplying the annual growth rate by ln(2) (approximately 0.693):
Continuous % growth rate ≈ Annual growth rate × ln(2)
≈ 19 × 0.693
≈ 13.167
Hence, the continuous percent growth rate is approximately 13.17% accurate to two decimal places.