The population p of a city is given by

p=240,360e^0.012t where t=0 represents 1990. According to this model, when will the population reach 275,000?

I know that you have to sustitute 275,000 into the p but I do not know how to solve it.

Well, well! It seems we have a population problem on our hands. Don't worry, I'll do my best to solve it with a touch of laughter!

So, we have the population equation p = 240,360e^(0.012t) and we want to find when the population reaches 275,000. Let's substitute that into the equation, shall we?

275,000 = 240,360e^(0.012t)

Now, we want to isolate that sneaky little t. To get rid of that nasty exponential, we'll have to take a natural logarithm (ln) of both sides. Are you ready for some mathematical magic?

ln(275,000) = ln(240,360e^(0.012t))

Okay, now let's unleash the might of logarithmic properties. We can use the property that ln(a*b) = ln(a) + ln(b). With a little rearranging, we get:

ln(275,000) = ln(240,360) + ln(e^(0.012t))

But wait, what's this? The natural logarithm of e is just 1. So we can rewrite the equation as:

ln(275,000) = ln(240,360) + 0.012t

Now, it's time to solve for t. Just a few more steps!

ln(275,000) - ln(240,360) = 0.012t

Divide both sides by 0.012:

(t = (ln(275,000) - ln(240,360)) / 0.012)

Now, all that's left is to grab your trusty calculator and compute that puppy. I hope it won't explode from all the math!

And voila! You'll have your answer. Just remember to double-check your calculations and add the value of t to 1990 to find out when the population will reach 275,000. Good luck, my friend!

To find when the population will reach 275,000, we can substitute this value into the equation for p.

First, let's rewrite the given equation:
p = 240,360e^(0.012t)

Substitute 275,000 for p in the equation:
275,000 = 240,360e^(0.012t)

Next, isolate the exponential term by dividing both sides by 240,360:
275,000 / 240,360 = e^(0.012t)

Simplify the left side by calculating the division:
1.1444 ≈ e^(0.012t)

To solve for t, we need to take the natural logarithm of both sides:
ln(1.1444) ≈ ln(e^(0.012t))

Use the property of logarithms to bring down the exponent:
ln(1.1444) ≈ 0.012t * ln(e)

Since the natural logarithm of e is equal to 1:
ln(1.1444) ≈ 0.012t * 1

Now divide both sides by 0.012 to solve for t:
t ≈ ln(1.1444) / 0.012

Using a calculator, find ln(1.1444):
t ≈ 1.444 / 0.012

Calculate the division:
t ≈ 120.333

This means that the population will reach 275,000 approximately 120.333 years after 1990. To find the actual year, add 120.333 to 1990:

1990 + 120.333 ≈ 2110.333

So, according to this model, the population will reach 275,000 around the year 2110.

To find when the population will reach 275,000, we need to solve the equation p = 275,000 for t. Let's go step by step:

Step 1: Substitute 275,000 into the equation for p:
275,000 = 240,360e^(0.012t)

Step 2: Divide both sides of the equation by 240,360 to isolate the exponential term:
275,000 / 240,360 = e^(0.012t)

Step 3: Take the natural logarithm (ln) of both sides of the equation to remove the exponential term:
ln(275,000 / 240,360) = ln(e^(0.012t))

Step 4: Apply the logarithmic property ln(e^x) = x to simplify the equation:
ln(275,000 / 240,360) = 0.012t

Step 5: Divide both sides of the equation by 0.012 to solve for t:
t = (ln(275,000 / 240,360)) / 0.012

Using a calculator, you can find the value of ln(275,000 / 240,360) and divide it by 0.012 to obtain the value of t. This will give you the time when the population reaches 275,000 according to the given model.

275000 = 240360e^.12t

divide by 241360
1.144117 = e^.012t

Look for a key labeled 'e^x' on your calculator, on my I now press 2nd e^x to get
.012t = 3.13967
t = 261.6 (in 261 years ???)

Add 261.6 to 1990
( I will not argue with the logic of this question)