A mountain pine beetle population can double every year if conditions are ideal. Assume the forest in Jasper National Park,AB has a population of 20 000 beetles.

The formula P=20 000(2)^n can model the population,P,after n years.

a) How many beetles were there in the forest four years ago? Eight years ago?

b) If the conditions remain ideal, how many beetles will there be two years from now?

Please help I don't understand how to do this

P(n) = 20000(2^n)

Let now be when n=0
four years ago, n = -4, so
P(-4) = 20000(2^-4) = 20000/16 = 1250

Now do the same for n = -8 and n=2

You don't really need to set n=0. Suppose we started k years ago. That means that now would be P(k).

So, 4 years ago would be P(k-4) = P(k)/2^4 = 20000/16

Ah, the pesky mountain pine beetles in Jasper National Park! Let's dig into your questions:

a) To find out how many beetles were in the forest four years ago, we need to find the value of P when n is -4 in the formula P = 20,000(2)^n. Substituting -4 into the equation, we get P = 20,000(2)^(-4). To calculate this, we raise 2 to the power of -4, giving us P = 20,000(1/16) = 1,250 beetles. Therefore, there were 1,250 beetles in the forest four years ago!

Similarly, to find out how many beetles were in the forest eight years ago, we need to find the value of P when n is -8 in the same formula. Substituting -8 into the equation, we get P = 20,000(2)^(-8). Calculating this, we raise 2 to the power of -8, resulting in P = 20,000(1/256) = 78.125 beetles. So, there were approximately 78 beetles eight years ago! That's quite the beetle boom, isn't it?

b) If we want to know how many beetles there will be two years from now, we substitute n = 2 into the formula P = 20,000(2)^n. Plugging in 2, we get P = 20,000(2)^2. Evaluating this, we need to square 2, which gives us P = 20,000(4) = 80,000 beetles. In two years, if conditions remain ideal, brace yourself for a whopping 80,000 beetles!

Remember, these calculations assume ideal conditions, so keep an eye out for any beetle-related surprises. Good luck with your beetle adventures in Jasper National Park!

To answer these questions, we will use the formula P = 20,000(2)^n, where P represents the population after n years.

a) To find the number of beetles in the forest four years ago, we need to plug in n = -4 into the formula:
P = 20,000(2)^(-4)
P = 20,000 * 1/16
P = 1,250 beetles (approximately)

Similarly, to find the number of beetles in the forest eight years ago, we need to plug in n = -8 into the formula:
P = 20,000(2)^(-8)
P = 20,000 * 1/256
P = 78.125 beetles (approximately)

b) To find the number of beetles in the forest two years from now, we need to plug in n = 2 into the formula:
P = 20,000(2)^2
P = 20,000 * 4
P = 80,000 beetles

Therefore, if the conditions remain ideal, there will be approximately 80,000 beetles in the forest two years from now.

To solve this problem, we need to substitute the values given into the formula P = 20,000(2)^n, where P represents the beetle population after n years.

a) To find the number of beetles four years ago, we need to substitute n = -4 into the equation:

P = 20,000(2)^(-4)
P = 20,000(1/16)
P = 1,250

Therefore, there were 1,250 beetles in the forest four years ago.

To find the number of beetles eight years ago, we substitute n = -8 into the equation:

P = 20,000(2)^(-8)
P = 20,000(1/256)
P = 78.125

Since we're dealing with a population of beetles, we need to round down to the nearest whole number. Therefore, there were 78 beetles in the forest eight years ago.

b) To determine the predicted population two years from now, we substitute n = 2 into the equation:

P = 20,000(2)^2
P = 20,000(4)
P = 80,000

Thus, if the conditions remain ideal, there will be an estimated 80,000 beetles in the forest two years from now.

In summary:

a) There were 1,250 beetles in the forest four years ago and 78 beetles eight years ago.
b) If the current conditions remain ideal, there will be an estimated 80,000 beetles in the forest two years from now.