A fire has started in a dry open field and is spreading in the form of a circle. If the radius of this circle increases at the rate of 6ft/min, express the toatal fire area A as a function of time t (in minutes
"the radius of this circle increases at the rate of 6ft/min" tells us
dr/dt = 6
then r = 6t + c
but when t=0, r was probably zero, so c=0
r = 6t
but Area of circle = pir^2
= pi(6t)^2
= 36pi(t^2)
thank you! but what do
"dr" and "dt" mean?
and "c"?
Are you not in Calculus?
What grade level is this from?
dr/dt is the rate of change of r, the radius in feet, with respect to t, the time in minutes
my little equation of r = 6t should be easy to understand without knowing Calculus
e.g. in 2 minutes, the radius would be 12 feet. (r = 6(2) )
in 4 minutes it would be r = 6(4) or 24 feet.
To express the total fire area A as a function of time t, we need to understand how the area of a circle changes with respect to its radius.
The area of a circle is given by the formula A = πr^2, where A represents the area and r represents the radius.
We know that the radius of the fire circle is increasing at a rate of 6ft/min. This means that for each minute that passes, the radius increases by 6ft.
Let's denote the initial radius as r0 and the time as t:
r0 = initial radius of the fire circle
t = time in minutes
Since the radius increases at a constant rate, we can express the radius at any given time t as:
r = r0 + 6t
Substituting this expression for r into the formula for the area of a circle, we get:
A = π(r0 + 6t)^2
Simplifying further, we have:
A = π(r0^2 + 12r0t + 36t^2)
Therefore, the total fire area A can be expressed as a function of time t as:
A(t) = π(r0^2 + 12r0t + 36t^2)