Amount after time t A (t) = yobt

Where:
yo = starting amount
b = growth constant (percent remaining)
t = number of time units

In the 1978 the population of blue whale in the southern hemisphere was thought to number 5000. Since whaling has been outlawed, and an abundant food supply is available the population N(t) is expected to grow exponentially according to the formula N(t) =5000e0.0036t where t is in years and t=0 corresponds to 1978

a) predict the population in the
year 2010
b) how may years will it take the
whole population to exceed
6000?

your equation should be

N(t) =5000e^(0.0036t)

a) 2010 - 1978 = 32
N(32) = 5000 e^(32(.0036)) = 5610

b)
6000 = 5000 e^(.0036t)
1.2 = e^(.0036t)
ln 1.2 = .0036t
t = ln 1.2/.0036 = 50.6

so it will take 50.6 years from 1978 to reach 6000
so to go over 6000 would take 51 years or in the years 2029

To answer these questions, we will use the given formula: N(t) = 5000e^(0.0036t).

a) To predict the population in the year 2010, we need to find the value of N(t) when t = 2010 - 1978 = 32 (since t=0 corresponds to 1978).

We plug the value of t into the equation:
N(t) = 5000e^(0.0036 * 32)

To calculate this, we need to compute e^(0.0036 * 32) and then multiply it by 5000.

Now, let's explain how to calculate e^(0.0036 * 32):
1. Raise the mathematical constant e to the power of 0.0036 * 32.
e^(0.0036 * 32)

2. Calculate 0.0036 * 32.
0.0036 * 32 = 0.1152

3. Raise e to the power of 0.1152.
e^(0.1152)

To calculate e^(0.1152), you can use a scientific calculator or a mathematical software. If you have access to a scientific calculator, you can directly input 0.1152 and calculate the exponential value. If you are using a mathematical software (e.g., Python, MATLAB), you can use the exponential function to calculate it.

Once you have the value of e^(0.1152), multiply it by 5000 to predict the population in the year 2010.

b) To determine how many years it will take the whole population to exceed 6000, we need to find the value of t when N(t) = 6000.

We set up the equation:
6000 = 5000e^(0.0036t)

To solve this equation and find the value of t, we'll have to use logarithms.

1. Divide both sides of the equation by 5000:
6000 / 5000 = e^(0.0036t)

2. Take the natural logarithm of both sides of the equation:
ln(6000 / 5000) = ln(e^(0.0036t))

3. Use the property of logarithms to bring down the exponent:
ln(6000 / 5000) = 0.0036t * ln(e)

4. Since ln(e) is equal to 1, the equation simplifies to:
ln(6000 / 5000) = 0.0036t

Now, divide both sides by 0.0036 to solve for t:
t = ln(6000 / 5000) / 0.0036

Again, you can use a scientific calculator or mathematical software to calculate this value.

Once you have the value of t, you will know how many years it will take for the whole population to exceed 6000.