Find the slope of the curve at the indicated point.
-A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant $80.0 \mathrm {~mm}$. Find the rate of change of surface area with respect to radius when the radius is $0.090 \mathrm {~mm}$. (Answer can be left in terms of $\pi$ ).
A) $160.36 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$
B) $160.0 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$
C) $80.36 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$
D) $80.18 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$

Question

Quizplus · Accepted Answer

The surface area of a cylinder is given by the formula $A=2\pi rh+2\pi r^2$. Since the length remains constant at $80.0$ mm, we can rewrite this as $A=160\pi r+2\pi r^2$.
Taking the derivative of this with respect to $r$ gives $\frac{dA}{dr}=160\pi+4\pi r$.
When the radius is $0.090$ mm, this becomes $\frac{dA}{dr}=160\pi+4\pi (0.090)= 160.36\pi$.
Therefore, the slope (or rate of change) of the curve at this point is $160.36 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$, which is option A.

Find the Slope of the Curve at the Indicated Point 80.0 mm80.0 \mathrm {~mm}80.0 mm

Find the Slope of the Curve at the Indicated Point $80.0 \mathrm {~mm}$