A balloon in the shape of a sphere is deflating. Given that t represents the time, in minutes, since it began losing air, the radius of the ballon (in cem) is r(t)=16-t. Let the equation V(r)= 4/3 pi r^3 represent the volume of a sphere of radius r. Find (V (open dot) r)(t)
V[open dot]r(t) (read V of r) is a shorthand for V(r(t)). The symbol [open dot] is a composition operator.
So
V[open dot]r(t)
=V(r(t))
=V(16-t)
=4/3 pi (16-t)^3
Simplify to complete the problem.
288
To find (V⋅r)(t), we need to differentiate the function V(r) with respect to t.
First, let's express V(r) in terms of t by substituting the expression for r(t) into the volume formula:
V(r(t)) = V(16-t) = (4/3)π(16-t)^3
Next, we differentiate V(r(t)) with respect to t. For this, we'll need to use the chain rule. The chain rule states that if we have a composition of functions, we need to take the derivative of the outer function multiplied by the derivative of the inner function.
d/dt[(V⋅r)(t)] = d/dr[V(r(t))] * d/dt[r(t)]
First, let's find d/dr[V(r(t))]:
The volume function is V(r) = (4/3)πr^3.
Taking the derivative of V(r) with respect to r gives: d/dr[V(r)] = d/dr[(4/3)πr^3] = 4πr^2.
Now, let's find d/dt[r(t)]:
The radius function is r(t) = 16 - t.
Taking the derivative of r(t) with respect to t gives: d/dt[r(t)] = d/dt[16 - t] = -1.
Finally, we can combine the two derivatives together using the chain rule:
d/dt[(V⋅r)(t)] = (d/dr[V(r(t))]) * (d/dt[r(t)]) = (4π(16-t)^2) * (-1)
= -4π(16-t)^2
Therefore, (V⋅r)(t) = -4π(16-t)^2.