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Question

# Find the inner surface area A of a cylindrical 250.0 cubic cm cup as a function of the radius r of the base. Then if A = 175.0 square cm, solve for r. Write one or two paragraphs explaining your method for setting up the function, and how you used a graphing calculator to solve for r with the given value of A.

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It is given that the volume of the cylindrical cup is 250 cm$^3$. The volume of a cylinder is $V=\pi r^2h$ so $\pi r^2h=250$. Solving this for $h$ gives $h=\dfrac{250}{\pi r^2}$. The surface area of a cylinder is $A=2\pi r^2+2\pi rh$. A cup only has one base though since the top is open so the surface area of the cup is $A=\pi r^2+2\pi rh$. Substituting in $h=\dfrac{250}{\pi r^2}$ then gives $A=\pi r^2+2\pi r\left(\dfrac{250}{\pi r^2}\right)=\pi r^2+\dfrac{500}{r}$.

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