# Lindsay's shape

Calculus Level 3

After studying various 3D shapes and finding formulas for their volumes, I challenged my students to invent a new shape. Lindsay created a shape $$($$with height $$2 \text{ cm})$$ that is circular at the top $$($$with radius $$1\text{ cm})$$ but square at the bottom $$($$with side length $$2\text{ cm}).$$ Lindsay created this shape from a paraboloid: sliced four times parallel to the paraboloid's axis and two times perpendicular to the paraboloid's axis. Lindsay's shape is pictured below.

Find the volume of Lindsay's shape in $$\text{cm}^{3},$$ which can be written as $$\frac{A}{B}+C\pi$$ with $$A,B,C$$ integers, $$A$$ and $$B$$ coprime, and $$B$$ positive.

Give the value of $$A+B+C.$$

×