FILL IN THE BLANK WITH AN INTEGER
A spherical balloon is being inflated with helium
at a steady rate of .
The rate at which the surface area is changing is

Ask by Montgomery Evans. in Guyana

Mar 27,2025

Question

FILL IN THE BLANK WITH AN INTEGER
A spherical balloon is being inflated with helium
at a steady rate of .
The rate at which the surface area is changing is

Ask by Montgomery Evans. in Guyana

Mar 27,2025

UpStudy AI · Accepted Answer

We begin by recalling that the volume $$V$$ and the surface area $$A$$ of a sphere are given by

$$
V=\frac{4}{3}\pi r^3 \quad \text{and} \quad A=4\pi r^2.
$$

Since the balloon is inflated at a rate

$$
\frac{dV}{dt}=150\,\text{cm}^3/\text{s},
$$

differentiate $$V$$ with respect to time $$t$$:

$$
\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}.
$$

Thus,

$$
\frac{dr}{dt}=\frac{150}{4\pi r^2}.
$$

Next, differentiate the surface area $$A$$ with respect to time:

$$
\frac{dA}{dt}=\frac{d}{dt}(4\pi r^2)=8\pi r\frac{dr}{dt}.
$$

Substitute $$\frac{dr}{dt}$$ into the above:

$$
\frac{dA}{dt}=8\pi r\left(\frac{150}{4\pi r^2}\right).
$$

Simplify the expression:

$$
\frac{dA}{dt}=\frac{8\cdot150}{4}\cdot\frac{1}{r}=\frac{300}{r}.
$$

Notice that the given expression in the problem was

$$
\pi r\left(\frac{1}{4\pi r^{2}}\right)(150),
$$

which is missing a factor compared to the correct derivative. In the correct expression, the factor $$8$$ appears from differentiating $$4\pi r^2$$ to get $$8\pi r$$. Thus, the blank should be filled with the integer

$$
\boxed{8}.
$$
The blank should be filled with the integer $$8$$.

FILL IN THE BLANK WITH AN INTEGER
A spherical balloon is being inflated with helium
at a steady rate of .
The rate at which the surface area is changing is

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FILL IN THE BLANK WITH AN INTEGER A spherical balloon is being inflated with helium at a steady rate of . The rate at which the surface area is changing is

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Solución

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FILL IN THE BLANK WITH AN INTEGER
A spherical balloon is being inflated with helium
at a steady rate of .
The rate at which the surface area is changing is