A quantity has the value P at time t seconds and is decreasing at a rate proportional to sqrt(P).
a) By forming and solving a suitable differential equation, show that P= (a - bt)^2 , where a and b are constants.
Given that when t= 0, P = 400,
b) find the value of a.
Given also that when t= 30, P = 100,
c) find the value of P when t = 50.
decreasing rate dP/dt= k sqrt (P)
dP/(sqrtP)= k dt
integrate both sides.
-1/2 sqrtP=kT+ C
square both sides
1/4 P= (C+kT)^2
P= 4 (C+kT)^2 and by choosing the constansts C, k
P= (a-bt)^2
400=(a-b*o)^2
a= 20
100=(20-b30)^2
10=20-30b
b=1/3
P=(20-1/3*50)^2=(20-17.7)^2=...
To answer this question, we need to start by forming a differential equation based on the given information.
Let's denote the quantity as P(t) and its rate of decrease as dP/dt. We are told that the rate of decrease is proportional to the square root of P, so we can write this relationship as:
dP/dt = -k * sqrt(P)
Where k is a constant of proportionality.
To solve this differential equation, we can use the method of separation of variables. We'll start by separating the variables and integrating both sides of the equation.
1/sqrt(P) dP = -k dt
Integrating both sides:
∫ 1/sqrt(P) dP = ∫ -k dt
To integrate the left side, we can use a substitution u = sqrt(P), du = 1/(2 sqrt(P)) dP:
∫ du = ∫ -k dt
u = -kt + C
Now substituting back u = sqrt(P):
sqrt(P) = -kt + C
Solving for P:
P = (-kt + C)^2
Since we are given that when t = 0, P = 400, we can substitute these values into the equation to find the value of constant C:
400 = (0 + C)^2
400 = C^2
Taking the square root of both sides, we get C = ±20.
Now we can rewrite the equation in terms of a and b, by substituting C = 20 and rearranging the equation:
P = (-kt + C)^2
P = (-kt + 20)^2
P = (20 - kt)^2
So, for part a), the general solution to the differential equation is P = (a - bt)^2, where a = 20 and b = k.
Moving on to part b), we are given that when t = 30, P = 100. Substituting these values into our equation:
100 = (20 - 30b)^2
Taking the square root of both sides, we have:
10 = 20 - 30b
30b = 20 - 10
30b = 10
b = 1/3
So, for part b), the value of b is 1/3.
Finally, for part c), we need to find the value of P when t = 50. Using the equation P = (a - bt)^2, with a = 20 and b = 1/3, we have:
P = (20 - (1/3) * 50)^2
P = (20 - (50/3))^2
P = (20 - 16.67)^2
P ≈ 3.33^2
P ≈ 11.11
Therefore, when t = 50, P ≈ 11.11.