a species of orchid is being studied. The population 'p' at time 't' years after the study started is assumed to be p = 2800ae^0.2t/1 + ae^0.2t where 'a' is a constant. Given tnat there were 300 orchids when the study started.
a. show that a =0.12
b. use the equation with 'a' = 0.12 to predict the number of years before the population of orchids reaches 1850.
(a). when t = 0
300 = 2800a/1+a
300(1+a) = 2800a
300 + 300a = 2800a
a = 0.12
no idea anyone know :(
a) To show that a = 0.12, we can use the given information that there were 300 orchids when the study started (t = 0).
Given:
p = 2800ae^0.2t / (1 + ae^0.2t)
For t = 0:
p = 300
300 = 2800a(1) / (1 + a)
Now we can solve for a:
300(1 + a) = 2800a
300 + 300a = 2800a
300 = 2800a - 300a
300 = 2500a
a = 300 / 2500
a = 0.12
Therefore, a = 0.12.
b) To predict the number of years before the population of orchids reaches 1850, we can use the equation with a = 0.12.
Given:
p = 2800ae^0.2t / (1 + ae^0.2t)
We need to find t when p = 1850:
1850 = 2800(0.12)e^0.2t / (1 + (0.12)e^0.2t)
Now we can solve for t:
1850(1 + 0.12e^0.2t) = 2800(0.12)e^0.2t
1850 + 222e^0.2t = 336e^0.2t
336e^0.2t - 222e^0.2t = 1850
114e^0.2t = 1850
e^0.2t = 1850 / 114
Take the natural logarithm (ln) of both sides:
0.2t = ln(1850 / 114)
Now solve for t:
t = ln(1850 / 114) / 0.2
Using a calculator, we find that t ≈ 5.39 years.
Therefore, it will take approximately 5.39 years for the population of orchids to reach 1850.
To solve the problem, we need to use the given information and equation for the population of orchids.
a. Show that a = 0.12:
We are given that the population 'p' at time 't' is given by the equation:
p = 2800ae^(0.2t) / (1 + ae^(0.2t))
We also know that when the study started, there were 300 orchids. This implies that when t = 0, p = 300:
300 = 2800a * e^(0.2*0) / (1 + a * e^(0.2*0))
Simplifying the equation:
300 = 2800a / (1 + a)
Cross-multiplying:
300 + 300a = 2800a
300 = 2500a
Dividing both sides by 2500:
a = 0.12
Therefore, we have shown that a = 0.12.
b. Use the equation with 'a' = 0.12 to predict the number of years before the population of orchids reaches 1850:
We are given the equation for the population of orchids as:
p = 2800ae^(0.2t) / (1 + ae^(0.2t))
Substituting 'a' with 0.12:
p = 2800 * 0.12 * e^(0.2t) / (1 + 0.12 * e^(0.2t))
We want to find the number of years 't' when the population reaches 1850, so we can set up the equation:
1850 = 2800 * 0.12 * e^(0.2t) / (1 + 0.12 * e^(0.2t))
To solve for 't', we can cross-multiply and rearrange the equation:
1850 * (1 + 0.12 * e^(0.2t)) = 2800 * 0.12 * e^(0.2t)
1850 + 222 * e^(0.2t) = 336
222 * e^(0.2t) = 336 - 1850
222 * e^(0.2t) = -1514
Next, we can isolate e^(0.2t) by dividing both sides of the equation by 222:
e^(0.2t) = -1514 / 222
Now, to find 't', we can take the natural logarithm (ln) of both sides:
ln(e^(0.2t)) = ln(-1514 / 222)
Simplifying:
0.2t = ln(-1514 / 222)
Finally, we can solve for 't' by dividing both sides by 0.2:
t = ln(-1514 / 222) / 0.2
Please note that the result you obtain from this calculation will be approximately equal to the number of years before the population of orchids reaches 1850, using the given equation with 'a' = 0.12.