A forest fire leaves behind an area of grass burned in an expanding circular pattern. The radius of the circle of burning grass is increasing with time according to the formula r(t)= 1t+5. What function expresses the area burned as a function of time t in minutes?
A(t)=?
A(t) = πr^2 = π(t+5)^2
odd that you wrote "1t" -- is that what you meant?
Well, since we're dealing with a circular pattern, we know that the area of a circle is given by the formula A = πr². So, we just need to substitute the given formula for r(t) into the area formula.
A(t) = π(1t + 5)²
But, since we're looking for the area burned as a function of time, we can simplify it a bit further:
A(t) = π(t² + 10t + 25)
So, the function that expresses the area burned as a function of time t in minutes would be:
A(t) = π(t² + 10t + 25)
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
We are given that the radius is increasing with time according to the formula r(t) = t + 5.
To find the expression for the area burned as a function of time, A(t), we can substitute the expression for r(t) into the formula for the area of a circle:
A(t) = π(r(t))^2.
Substituting r(t) = t + 5, we have:
A(t) = π(t + 5)^2.
Simplifying further, we expand the square:
A(t) = π(t^2 + 10t + 25).
Therefore, the function that expresses the area burned as a function of time t in minutes is A(t) = π(t^2 + 10t + 25).
To find the function that expresses the area burned as a function of time t, we need to use the formula for the area of a circle.
The formula for the area of a circle is A = πr^2, where A represents the area and r is the radius.
Given that the radius of the circle of burning grass is given as r(t) = t + 5, we can substitute this expression into the formula for the area of a circle to find the function A(t).
A(t) = π * (r(t))^2
Substituting the expression for r(t):
A(t) = π * (t + 5)^2
Expanding and simplifying the expression, we have:
A(t) = π * (t^2 + 10t + 25)
Therefore, the function that expresses the area burned as a function of time t in minutes is A(t) = π * (t^2 + 10t + 25).