# Math Help - Please Check: Rate of Change problem

1. ## Please Check: Rate of Change problem

For Part B of the attached question;

Terms: E = Change d = Partial derivative D = Derivative

a = sqrt (h^2 - b^2) = (h^2 - b^2) ^1/2

Ea = da/dh . Eh + da/db . Eb

Da/Dt = da/dh . Da/Dt + da/db . Da/Dt

Therefore;

da/dh = 1/2 * (h^2 - b^2) ^ -1/2 . (2h)

da/db = 1/2 * (h^2 - b^2) ^ -1/2 . (-2b)

i.e:

Da/Dt = h / sqrt (h^2 - b^2) * Dh/Dt - b / sqrt (h^2 - b^2) * Db/Dt

Values are:

h = 5cm Dh/Dt = (+3 cm/s)
b = 3cm Db/Dt = (-2 cm/s)

Da/Dt = 5 / sqrt (5^2 - 3^2) * 3 - 3 / sqrt (5^2 - 3^2) * (-2)

Da/Dt = 3.75 -(-1.5) = 5.25 cm/s

2. Your answer is correct, but it seems you took a very convoluted path to get to the answer. Here is a much simpler method:

You are given:

$\frac{dh}{dt}=3$

$\frac{db}{dt}=-2$

$h=5$

$b=3$

$a=\sqrt{h^2-b^2} = 4$

So, differentiate with respect to t:

$h^2=a^2+b^2$

$\implies 2h \frac{dh}{dt} = 2a \frac{da}{dt} + 2b \frac{db}{dt}$

Then substitute in all known values and you are left with your solution.

3. thats a great help. many thanks