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.

we are told that

dP/dt = -k * P^(1/2)
P^(-1/2) dP = -k dt
2P^(1/2) = -kt + c
P = (2c - 2kt)^2 = (a-bt)^2
if
a = 2c
b = 2k

Now, we are told that P(0) = 400

(a-0)^2 = 400
a = 20 or -20

P(30) = 100

If a=20
(20-30b)^2 = 100
so b = 1 or 1/3

If a = -20
(-20-30b)^2 = 100
so b = -1 or -1/3

So far, we have 4 combinations of values for a and b

I'll let you figure out P(50). Maybe you have more info that eliminates some of the choices.

To find the solution to the differential equation and solve parts (a), (b), and (c), we can follow these steps:

Step 1: Form the differential equation.
The differential equation representing the given scenario is:
dP/dt = -k * sqrt(P)

Step 2: Solve the differential equation.
Separate the variables:
1 / sqrt(P) * dP = -k * dt

Integrate both sides:
∫(1/sqrt(P)) dP = -∫k dt

Simplify the integration:
2 * sqrt(P) = -kt + C

Step 3: Find the constant term.
Substitute the initial condition when t = 0, P = 400, into the equation:
2 * sqrt(400) = -k(0) + C
2 * 20 = C
C = 40

Therefore, the equation becomes:
2 * sqrt(P) = -kt + 40

Step 4: Solve for P.
Square both sides of the equation:
4 * P = k^2 * t^2 - 80k * t + 1600

Rearrange the equation:
P = (k^2 * t^2 - 80k * t + 1600) / 4
P = 1/4 * (k^2 * t^2 - 80k * t + 1600)

Simplify the equation:
P = (1/4) * (k^2 * t^2 - 80k * t + 1600)
P = (1/4) * (k^2 * t^2 - 2k * 40t + 40^2)

Now, let a = 40 and b = k:
P = (a - bt)^2

(a) The differential equation and its solution are P = (a - bt)^2, where a and b are constants.

(b) Given t = 0 and P = 400:
400 = (a - b(0))^2
400 = a^2
√400 = a
a = ±20

(c) Given t = 30 and P = 100:
100 = (20 - b(30))^2
10 = (20 - 30b)^2
√10 = 20 - 30b
30b = 20 - √10
b = (20 - √10) / 30

Substitute the values of a and b into the equation:
P = (20 - (20 - √10) / 30 * t)^2

Finally, when t = 50:
P = (20 - (20 - √10) / 30 * 50)^2
Simplify the expression to find the value of P.