A bead is made by drilling a cylindrical hole of radius $3$ through a sphere of radius $5$. Find the volume of the bead.
*Give your answer as a fraction in terms of $\pi$

A vase maker decides to construct a vase whose contour is a cubic polynomial according to the following specifications:

• The vase is 4 feet tall.
• The vase is 4 feet in diameter at its widest (which occurs 1 foot from the base).
• The vase is 2 feet in diameter at its narrowest (which occurs 1 foot from the top).
• The vase has a flat circular bottom.
  *Give your answer as a fraction in terms of $\pi$. Don’t include units ($ft^2$)
  
  Check

A particle starts from rest and travels $A$ km with constant acceleration, then travels $2A$ km with constant velocity, then comes to rest after traveling $5A$ km with constant deceleration. Find the value of $\frac{\text{average velocity}}{\text{maximum velocity}}$.
*Give your answer as a fraction in simplest terms

Evaluate $\displaystyle \int_0^\infty{\left(1+2x\right)e^{-x}}\;dx$