(a) What is the continuous percent growth rate for P=130e^0.07t , with time, t , in years?
(b) Write this function in the form P=P0a^t. What is the annual percent growth rate?
Round your answer to two decimal places.
I tried it but now I'm just more confused. How do I do this???
a) 7%
b) something is missing
To calculate the continuous percent growth rate, given the equation P = 130e^0.07t, you need to follow these steps:
(a) Continuous Percent Growth Rate:
Step 1: Start with the equation P = 130e^0.07t.
Step 2: Take the natural logarithm (ln) on both sides to eliminate the exponential term. This gives us ln(P) = ln(130e^0.07t).
Step 3: Apply the logarithm properties: ln(ab) = ln(a) + ln(b). So, ln(130e^0.07t) = ln(130) + ln(e^0.07t).
Step 4: Simplify the equation further using the properties of logarithms. ln(e^0.07t) simplifies to 0.07t, and ln(e) is equal to 1, so it becomes ln(130) + 0.07t.
Step 5: Rearrange the equation to isolate t and the constant to find the continuous percent growth rate. t can be found by dividing both sides by 0.07. This will give us the continuous percent growth rate, which can then be converted to an annual percent growth rate.
(b) Annual Percent Growth Rate:
To write the function P = 130e^0.07t in the form P = P0a^t, we can use the formula P = P0e^rt, where r is the annual percent growth rate and t is time in years.
Step 1: Substitute the given values in the equation: P = 130e^0.07t.
Step 2: Compare this equation with P = P0e^rt. From the comparison, we can see that P0 = 130, and r = 0.07 (the continuous percent growth rate).
Step 3: Rearrange the equation to solve for a. P = P0a^t becomes 130 = 130a^t.
Step 4: Divide both sides by P0: 130/130 = 130a^t/130. This simplifies to 1 = a^t.
Step 5: To find a, we take the t-th root on both sides. This gives us a = 1^(1/t) = 1.
Step 6: The annual percent growth rate is given by r = ln(a) = ln(1) = 0 (no growth).
So, in summary:
(a) The continuous percent growth rate is 0.07.
(b) The annual percent growth rate is 0 (no growth).
Remember, when calculating the values, use the appropriate steps and mathematical properties to arrive at the correct solution.