The population of bacteria present after t hours is given by the function P(t) = 2530e^(0.24t)

.
(a) What is the initial population of the bacteria?
(b) How long will it take the population to reach 10,000?

e^0 = 1

so initial is
2530e^(0) = 2530

2530e^(0.24t) = 10,000

e^.24 t = 3.953
ln [e^.24 t ]= ln 3.953
.24 t = 1.374

(a) The initial population of the bacteria can be found by plugging in t = 0 into the function P(t).

P(0) = 2530e^(0.24 * 0) = 2530e^0 = 2530 * 1 = 2530

So, the initial population of the bacteria is 2,530.

(b) To find out how long it will take the population to reach 10,000, we need to solve the equation P(t) = 10,000.

10,000 = 2530e^(0.24t)

To isolate the variable t, divide both sides of the equation by 2530:

10,000/2530 = e^(0.24t)

Now take the natural logarithm (ln) on both sides of the equation to remove the exponential function:

ln(10,000/2530) = ln(e^(0.24t))

Using the property of logarithms, ln(e^(0.24t)) simplifies to 0.24t:

ln(10,000/2530) = 0.24t

Now divide both sides of the equation by 0.24:

ln(10,000/2530)/0.24 = t

Using a calculator to find ln(10,000/2530)/0.24, we get:

t ≈ 9.37

So, it will take approximately 9.37 hours for the population to reach 10,000 bacteria.

To determine the initial population of the bacteria, we can substitute in t = 0 into the given function P(t) = 2530e^(0.24t):

P(0) = 2530e^(0.24*0)
= 2530e^0
= 2530 * 1
= 2530

So, the initial population of the bacteria is 2530.

To find out how long it will take for the population to reach 10,000, we can set P(t) = 10,000 and solve for t in the equation P(t) = 2530e^(0.24t):

10,000 = 2530e^(0.24t)

Now, divide both sides of the equation by 2530:

10,000/2530 = e^(0.24t)

Simplify the left-hand side:

3.952 = e^(0.24t)

To solve for t, take the natural logarithm of both sides:

ln(3.952) = ln(e^(0.24t))

Using the logarithmic property ln(e^x) = x, we get:

ln(3.952) = 0.24t

Now, divide both sides by 0.24:

t = ln(3.952)/0.24

Using a calculator, we can approximate this value as:

t ≈ 9.054

Therefore, it will take approximately 9.054 hours for the population to reach 10,000.

To find the answers to these questions, we will use the given function P(t) = 2530e^(0.24t) for the population of bacteria.

(a) The initial population refers to the population at time t = 0. To find the initial population, we substitute t = 0 into the function P(t).

P(0) = 2530e^(0.24 * 0) = 2530e^0 = 2530 * 1 = 2530

Therefore, the initial population of the bacteria is 2530.

(b) To find how long it will take for the population to reach 10,000, we need to find the value of t when P(t) = 10,000.

10,000 = 2530e^(0.24t)

Divide both sides of the equation by 2530:

e^(0.24t) = 10,000 / 2530

To isolate the exponential term, take the natural logarithm (ln) of both sides:

ln(e^(0.24t)) = ln(10,000 / 2530)

Using the property of logarithms, ln(e^(0.24t)) simplifies to 0.24t:

0.24t = ln(10,000 / 2530)

Now, we divide both sides of the equation by 0.24:

t = ln(10,000 / 2530) / 0.24

Using a calculator, we can evaluate the right side of the equation:

t ≈ 12.65

Therefore, it will take approximately 12.65 hours for the population to reach 10,000.