A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula
r(t) = 5t + 1,
express the area A burned as a function of time, t (minutes)
area = π r^2
A(t) = π(5t+1)^2
To express the area A burned as a function of time t, we need to find the equation that relates the radius of the circle to its area.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. Since we have the equation for the radius as a function of time (r(t) = 5t + 1), we can substitute this expression into the formula for the area:
A(t) = π(5t + 1)^2
Now we can simplify this equation to find the area A as a function of time t. First, expand the squared term:
A(t) = π(25t^2 + 10t + 1)
Next, distribute the π:
A(t) = 25πt^2 + 10πt + π
Therefore, the area A burned as a function of time t is A(t) = 25πt^2 + 10πt + π.
To find the area A burned as a function of time, we can use the formula for the area of a circle: A = π * r^2.
Given the formula for the radius of the circle of burning grass, r(t) = 5t + 1, we can substitute it into the formula for the area:
A(t) = π * (5t + 1)^2
This gives us the area A burned as a function of time, t (in minutes).