A circular oil slick is expanding with radius, r in yards, at time t in hours given by r=2.5t−0.3t2, for 0≤t≤10.
Find a formula for the area in square yards, A=f(t), as a function of time.
f(t)=
HELP PLEASE!
Oh, don't worry, I'm here to clown around and help you out! So, we need to find the area of the circular oil slick as a function of time, right?
The formula for the area of a circle is A = πr^2, where r is the radius. In this case, we have a formula for the radius as a function of time, so we can substitute it into the formula for the area.
Using the given radius formula r = 2.5t - 0.3t^2, we just need to square it and multiply by π. Let's do some math!
A = π(2.5t - 0.3t^2)^2
Now, we can expand and simplify this expression further, but I'll leave that part for you to do. Remember to apply the power of algebra, and you'll figure it out!
Now go on, my friend, and solve that equation. You got this!
To find the formula for the area, A, as a function of time, we need to find the equation for the area of a circle in terms of the radius.
The formula for the area of a circle is given by A = π * r^2, where r is the radius.
Substituting the given equation for r=2.5t−0.3t^2 into the formula for the area, we get:
A = π * (2.5t−0.3t^2)^2
Simplifying further:
A = π * (2.5t−0.3t^2) * (2.5t−0.3t^2)
A = π * (2.5t)^2 - 0.3t^2 * (2.5t) - 0.3t^2 * (2.5t) + 0.3t^2 * (0.3t^2)
A = π * (6.25t^2) - 1.5t^3 + 0.09t^4
Therefore, the formula for the area in square yards, A = f(t), as a function of time is:
f(t) = π * (6.25t^2) - 1.5t^3 + 0.09t^4
To find a formula for the area, A, as a function of time, we need to use the formula for the area of a circle, which is A = πr^2.
Given that the radius, r, is given by the equation r = 2.5t - 0.3t^2, we can substitute this expression for r into the formula for the area:
A = π(2.5t - 0.3t^2)^2
Expanding and simplifying this expression, we get:
A = π(6.25t^2 - 1.5t^3 + 0.09t^4)
So, the formula for the area, A, as a function of time, t, is:
f(t) = π(6.25t^2 - 1.5t^3 + 0.09t^4)
I hope this helps!
area=PI r^2=PI (2.5t-.3t^2)^2
= pi t^2(6.25-1.5t+.09t^2)