The population of rabbits in a national forest is modeled by the formula P=8,600+650 ln(t+1) where "t" is the number of years from the present.
a)How many rabbits are now in the forest
b)Estimate when it will take for the number of rabbits to reach 12,000?
c)Estimate how many rabbits there will be 50 years from now
d)Estimate how many rabbits there will be in 100 years from now.
Please show as much work as possible.
at "present time" t = 0
P = 8600 + 650(ln(1), but ln1 = 0
= 8600
b) 8600 + 650 ln(t+1) = 12000
ln(t+1) = 5.23077
t + 1 = e^5.23077 = 186.9
t = 185.9 years ???? (my math is correct)
c) plug in 50 for t, evaluate
(I got 11,156)
d) same as c, plug in t = 100
thank you very much, it all makes sense now except for a. could you explain how you get that number. I plugged it into a calculator and it is correct but i just don't understand how it works.
a )
Now t = 0
8,600 + 650 ln( t + 1 ) =
8,600 + 650 ln( 0 + 1 )=
8,600 + 650 ln ( 1 ) =
8,600 + 650 * 0 =
8,600 + 0 = 8,600
b )
8,600 + 650 ln( t + 1 ) = 12,000
650 ln( t + 1 ) = 12,000 - 8,600
650 ln( t + 1 ) = 3,400
ln( t + 1 )= 3,400 / 650
ln( t + 1 )= 5.23076923
t + 1 = e ^ 5.23076923
t + 1 = 186.93655
t = 186.93655 - 1 = 185.93655
c )
8,600 + 650 ln( 50 + 1 ) =
8,600 + 650 ln( 51 ) =
8,600 + 650 * 3.93182563 =
8,600 + 2,555.6866595 = 11,155.6866595
Rounded 11,156
d )
8,600 + 650 ln( 100 + 1 ) =
8,600 + 650 ln( 101 ) =
8,600 + 650 * 4.61512 =
8,600 + 2999.828 = 11,599.828
Rounded 11,600
Ohh my gosh, thank you. I was just confused about "a" but the extra information cant hurt. Thank you so much.
To find the answers to the given questions, we will use the formula P = 8,600 + 650 ln(t + 1), where P represents the population of rabbits and t represents the number of years from the present. Let's solve each part of the problem step by step:
a) How many rabbits are now in the forest?
To find the current population of rabbits, we need to substitute t = 0 into the formula:
P = 8,600 + 650 ln(0 + 1)
P = 8,600 + 650 ln(1)
Since ln(1) equals 0, the equation simplifies to:
P = 8,600 + 650(0)
P = 8,600 + 0
P = 8,600
Hence, there are currently 8,600 rabbits in the forest.
b) Estimate when it will take for the number of rabbits to reach 12,000?
To determine the estimated time when the rabbit population reaches 12,000, we need to solve the equation P = 12,000:
12,000 = 8,600 + 650 ln(t + 1)
Subtracting 8,600 from both sides, we get:
3,400 = 650 ln(t + 1)
Now, divide both sides of the equation by 650:
5.23 = ln(t + 1)
To remove the natural logarithm, we need to take the exponent of both sides:
e^5.23 = t + 1
Using a calculator, evaluate e^5.23 ≈ 187.43
t + 1 = 187.43
Subtracting 1 from both sides, we find:
t = 187.43 - 1 ≈ 186.43
Therefore, it will take approximately 186.43 years for the rabbit population to reach 12,000.
c) Estimate how many rabbits there will be 50 years from now?
To determine the estimated rabbit population 50 years from now, we substitute t = 50 into the formula:
P = 8,600 + 650 ln(50 + 1)
P = 8,600 + 650 ln(51)
Using a calculator, evaluate ln(51) ≈ 3.9318
P ≈ 8,600 + 650(3.9318)
P ≈ 8,600 + 2,550.97
P ≈ 11,150.97
Therefore, the estimated rabbit population 50 years from now is approximately 11,151.
d) Estimate how many rabbits there will be in 100 years from now?
To determine the estimated rabbit population 100 years from now, we substitute t = 100 into the formula:
P = 8,600 + 650 ln(100 + 1)
P = 8,600 + 650 ln(101)
Using a calculator, evaluate ln(101) ≈ 4.61512
P ≈ 8,600 + 650(4.61512)
P ≈ 8,600 + 3,000.78
P ≈ 11,600.78
Therefore, the estimated rabbit population in 100 years from now is approximately 11,601.
In conclusion, using the given formula, we can estimate the rabbit population for different time intervals in the future and determine the corresponding number of rabbits.