A spherical balloon is being inflated.Let r(t)=3t can represents its radius at time t seconds and let g(r)=4/3 pie r be the volume of the same balloon if its radius is r.Write (g.r) in terms of t and describe what it represents.
that's PI, not pie!!
and I have no idea what g.r means.
g(r) = 4/3 pi r^3 = 4/3 pi (3t)^3
If you want, you can expand that out. This is just Algebra I, right?
To find (g⦁r) in terms of t, we need to substitute the expression for r(t) into the equation for g(r).
Given:
r(t) = 3t (radius at time t seconds)
g(r) = (4/3)πr (volume of the balloon with radius r)
Substituting r(t) into the equation for g(r), we get:
g(r(t)) = (4/3)πr(t)
Replacing r(t) with 3t:
g(r(t)) = (4/3)π(3t)
Simplifying:
g(r(t)) = 4πt
Therefore, (g⦁r) in terms of t is 4πt.
This expression represents the volume of the spherical balloon at time t seconds, given that its radius is increasing at a rate of 3 units per second.
To find (g∘r)(t), which represents the volume of the balloon at time t, we need to substitute the expression for r(t) into the function g(r).
Given:
r(t) = 3t (radius of the balloon at time t)
g(r) = (4/3)πr (volume of the balloon with radius r)
Substituting r(t) into the function g(r):
(g∘r)(t) = g(r(t))
= g(3t)
= (4/3)π(3t) (substituting 3t for r)
= 4πt (simplifying)
So, (g∘r)(t) = 4πt represents the volume of the balloon at time t seconds.