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
R = 6t, if R is in feet and t is in minutes.
A = pi R^2 = pi * 36 t^2
To express the total fire area A as a function of time t, we need to determine the relationship between the radius and the area of a circle. The formula for the area of a circle is given by:
A = π * r^2
where A is the area and r is the radius.
Given that the radius is increasing at a rate of 6ft/min, we can determine the current radius at any given time by multiplying the rate of increase (6ft/min) by the time (t).
So, the radius (r) in terms of time (t) can be expressed as:
r = 6t
Substituting this expression for radius into the formula for area, we have:
A = π * (6t)^2
Simplifying this equation, we have:
A = π * 36t^2
Therefore, the total fire area A as a function of time t is given by the equation:
A(t) = 36πt^2