Please help me asap! Could you please explain how you got the answer
1 answer:
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A) 30 minutes after the air conditioner started to cool the house, the temperature inside the house has dropped to 76°F
B)
Step-by-step explanation:
A)
In this problem, f(x) represents temperature, in degrees Fahrenheit, of Sam’s house x minutes after the air conditioner started to cool the house.
So we have:
x = minutes after the air conditioner started to cool the house
f(x) = temperature
Here we want to find what is the meaning of
Which implies that:
, which means that 30 minutes have passed after the conditioner has started to cool the house
, which means that the temperature of the room at that time is 76°F
So, the interpretation of f(30) =76 is that:
"30 minutes after the air conditioner started to cool the house, the temperature inside the house has dropped to 76°F"
B)
Here we want to write explicitely a function that tells us how the temperature inside the house varies as a function of x, the number of minutes passed after the air conditioner has turned on.
We know the following information:
- At time x = 0, the temperature inside the house is 88°F
- At time x = 30 min, the temperature inside the hosue is 76°F
We can assume that the temperature inside the house changes linearly with time; so, we can describe the temperature function with the equation of a line:
where
is the value of the temperature at x = 0, so q = 88
m is the rate of change of the temperature
The rate of change can be found by calculating the rate of change of the temperature with respect to the change in time; so we find:
which means that the temperature decreases by 0.4°F every minute. So, the function can be written as
A logistic differential equation can be written as follows:
![\frac{dP}{dt} = rP[1- \frac{P}{K}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdP%7D%7Bdt%7D%20%3D%20rP%5B1-%20%5Cfrac%7BP%7D%7BK%7D%5D%20)
where r = growth parameter and K = carrying parameter.
In order to write you equation in this form, you have to regroup 2:
![\frac{dP}{dt} = 2P[1- \frac{P}{10000}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdP%7D%7Bdt%7D%20%3D%202P%5B1-%20%5Cfrac%7BP%7D%7B10000%7D%5D%20)
Therefore, in you case r = 2 and K = 10000
To solve the logistic differential equation you need to solve:
![\int { \frac{1}{[P(1- \frac{P}{K})] } } \, dP = \int {r} \, dt](https://tex.z-dn.net/?f=%20%5Cint%20%7B%20%5Cfrac%7B1%7D%7B%5BP%281-%20%5Cfrac%7BP%7D%7BK%7D%29%5D%20%7D%20%7D%20%5C%2C%20dP%20%3D%20%20%5Cint%20%7Br%7D%20%5C%2C%20dt%20%20)
The soution will be:
P(t) =
where P(0) is the initial population.
In your case, you'll have:
P(t) = <span>
Now you have to calculate the limit of P(t).
We know that
</span>
hence,
<span>
</span><span>
</span>
where r = growth parameter and K = carrying parameter.
In order to write you equation in this form, you have to regroup 2:
Therefore, in you case r = 2 and K = 10000
To solve the logistic differential equation you need to solve:
The soution will be:
P(t) =
where P(0) is the initial population.
In your case, you'll have:
P(t) = <span>
Now you have to calculate the limit of P(t).
We know that
</span>
hence,
</span><span>
</span>