What is the following sum? 2 (RootIndex 3 StartRoot 16 x cubed y EndRoot) + 4 (RootIndex 3 StartRoot 54 x Superscript 6 Baseline
y Superscript 5 Baseline) 4 x (RootIndex 3 StartRoot 2 y EndRoot) + 12 x squared y (RootIndex 3 StartRoot 2 y squared EndRoot) 8 x (RootIndex 3 StartRoot x y EndRoot) + 12 x cubed y squared (RootIndex 3 StartRoot 6 y EndRoot) 16 x cubed y (RootIndex 3 StartRoot 2 y squared EndRoot) 48 x cubed y (RootIndex 3 StartRoot 2 y EndRoot)
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Answer:
Step-by-step explanation:
<u>Rates of Change as Derivatives</u>
If some variable V is a function of another variable r, we can compute the rate of change of one with respect to the other as the first derivative of V, or
The volume of a sphere of radius r is
The volume of the balloon is growing at a rate of . This can be written as
We need to compute the rate of change of the radius. Note that both the volume and the radius are functions of time, so we need to use the chain rule. Differentiating the volume with respect to t, we get
solving for
We need to find the value of r, which can be obtained by using the condition that in that exact time
Simplifying and isolating r
Replacing in the rate of change