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yroc
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I'm confused about what sign to use for an inflating or deflating sphere. The question describes a deflating spherical balloon and asks for the instantaneous rate of change of its volume when its radius is 5 cm.

Using the definition of instantaneous rate of change, $\lim_{h\to0}\frac{\frac{4}{3}\pi(5+h)^3-\frac{4}{3}\pi(5)^3}{h}$, which evaluates to $100\pi$

The fact that I'm getting a positive rate of change seems a bit counterintuitive since the balloon is deflating, but on the other hand it makes sense because even though volume is decreasing during deflation, radius is also decreasing, so the two negatives "cancel out".

There are other signs that the rate of change should be positive. Like the fact that the derivative function of $V(r)=\frac{4}{3}\pi r^3$ is $V'(r)=4\pi r^2$, which is always positive, and that the tangent to $V(r)=\frac{4}{3}\pi r^3$ is always positive.

However the textbook gives the answer $-100\pi$, so I'm not sure.

The only other thing I tried was to use $-h$ in the definition, so $\lim_{h\to0}\frac{\frac{4}{3}\pi(5-h)^3-\frac{4}{3}\pi(5)^3}{h}$. When I did this, I got $-100\pi$, but I always thought the variable $h$ could be positive or negative by itself (i.e., $h\in R$). What is the implication of using $+h$ vs. $-h$ in the definition?

Edit: Here's the exact wording of the question:

Determine the instantaneous rate of change in the volume of a spherical balloon, as it's deflated, at the point in time when the radius reaches 5 cm

I'm confused about what sign to use for an inflating or deflating sphere. The question describes a deflating spherical balloon and asks for the instantaneous rate of change of its volume when its radius is 5 cm.

Using the definition of instantaneous rate of change, $\lim_{h\to0}\frac{\frac{4}{3}\pi(5+h)^3-\frac{4}{3}\pi(5)^3}{h}$, which evaluates to $100\pi$

The fact that I'm getting a positive rate of change seems a bit counterintuitive since the balloon is deflating, but on the other hand it makes sense because even though volume is decreasing during deflation, radius is also decreasing, so the two negatives "cancel out".

There are other signs that the rate of change should be positive. Like the fact that the derivative function of $V(r)=\frac{4}{3}\pi r^3$ is $V'(r)=4\pi r^2$, which is always positive, and that the tangent to $V(r)=\frac{4}{3}\pi r^3$ is always positive.

However the textbook gives the answer $-100\pi$, so I'm not sure.

The only other thing I tried was to use $-h$ in the definition, so $\lim_{h\to0}\frac{\frac{4}{3}\pi(5-h)^3-\frac{4}{3}\pi(5)^3}{h}$. When I did this, I got $-100\pi$, but I always thought the variable $h$ could be positive or negative by itself (i.e., $h\in R$). What is the implication of using $+h$ vs. $-h$ in the definition?

I'm confused about what sign to use for an inflating or deflating sphere. The question describes a deflating spherical balloon and asks for the instantaneous rate of change of its volume when its radius is 5 cm.

Using the definition of instantaneous rate of change, $\lim_{h\to0}\frac{\frac{4}{3}\pi(5+h)^3-\frac{4}{3}\pi(5)^3}{h}$, which evaluates to $100\pi$

The fact that I'm getting a positive rate of change seems a bit counterintuitive since the balloon is deflating, but on the other hand it makes sense because even though volume is decreasing during deflation, radius is also decreasing, so the two negatives "cancel out".

There are other signs that the rate of change should be positive. Like the fact that the derivative function of $V(r)=\frac{4}{3}\pi r^3$ is $V'(r)=4\pi r^2$, which is always positive, and that the tangent to $V(r)=\frac{4}{3}\pi r^3$ is always positive.

However the textbook gives the answer $-100\pi$, so I'm not sure.

The only other thing I tried was to use $-h$ in the definition, so $\lim_{h\to0}\frac{\frac{4}{3}\pi(5-h)^3-\frac{4}{3}\pi(5)^3}{h}$. When I did this, I got $-100\pi$, but I always thought the variable $h$ could be positive or negative by itself (i.e., $h\in R$). What is the implication of using $+h$ vs. $-h$ in the definition?

Edit: Here's the exact wording of the question:

Determine the instantaneous rate of change in the volume of a spherical balloon, as it's deflated, at the point in time when the radius reaches 5 cm

Source Link
yroc
yroc
  • 1.1k
  • 1
  • 12
  • 24

Rate of change sign ($+$ or $-$) for inflating/deflating sphere?

I'm confused about what sign to use for an inflating or deflating sphere. The question describes a deflating spherical balloon and asks for the instantaneous rate of change of its volume when its radius is 5 cm.

Using the definition of instantaneous rate of change, $\lim_{h\to0}\frac{\frac{4}{3}\pi(5+h)^3-\frac{4}{3}\pi(5)^3}{h}$, which evaluates to $100\pi$

The fact that I'm getting a positive rate of change seems a bit counterintuitive since the balloon is deflating, but on the other hand it makes sense because even though volume is decreasing during deflation, radius is also decreasing, so the two negatives "cancel out".

There are other signs that the rate of change should be positive. Like the fact that the derivative function of $V(r)=\frac{4}{3}\pi r^3$ is $V'(r)=4\pi r^2$, which is always positive, and that the tangent to $V(r)=\frac{4}{3}\pi r^3$ is always positive.

However the textbook gives the answer $-100\pi$, so I'm not sure.

The only other thing I tried was to use $-h$ in the definition, so $\lim_{h\to0}\frac{\frac{4}{3}\pi(5-h)^3-\frac{4}{3}\pi(5)^3}{h}$. When I did this, I got $-100\pi$, but I always thought the variable $h$ could be positive or negative by itself (i.e., $h\in R$). What is the implication of using $+h$ vs. $-h$ in the definition?