When placed inside a cube, a sphere is tangent to all the faces of the cube. If the surface area of the cube is \(150\) cm\(^2,\) what is the radius of the sphere in centimeters?

(A) \(\ \ 2.5\)

(B) \(\ \ 5\)

(C) \(\ \ 10\)

(D) \(\ \ 25\)

(E) \(\ \ 75\)

A plastic tube is \(10\) cm long. If the inside of the tube is empty, its inner radius is \(r=7\) cm and its outer radius is \(R=9\) cm, what is the volume of plastic used to create the tube, in cubic centimeters?

(A) \(\ \ 32\pi\)

(B) \(\ \ 320\pi\)

(C) \(\ \ 490 \pi\)

(D) \(\ \ 810\pi\)

(E) \(\ \ 1300\pi\)

A regular hexagonal pyramid has height \(h=6,\) and base edges \(b = 10.\) What is the volume of the pyramid? (The volume of a pyramid can be found using the formula \(V = \frac{1}{3}Bh,\) where \(B\) is the area of the base.)

(A) \(\ \ 50\sqrt{3}\)

(B) \(\ \ 150\sqrt{3}\)

(C) \(\ \ 300\)

(D) \(\ \ 300\sqrt{2}\)

(E) \(\ \ 300\sqrt{3}\)

Shown above is a model of a staircase that is to be made of concrete. If each step has the shape of a rectangular solid, and \(s=2\) in, how many cubic inches of concrete will be needed to create the stairs?

(A) \(\ \ 8\)

(B) \(\ \ 24\)

(C) \(\ \ 40\)

(D) \(\ \ 80\)

(E) \(\ \ 120 \)

A cylinder and a sphere have the same radius, \(r=27.\) If the volume of the sphere is twice bigger than that of the cylinder, what is the height of the cylinder? (The volume of a sphere can be found using the formula \(V = \frac{4}{3}\pi r^3.\))

(A) \(\ \ 13.5\)

(B) \(\ \ 18\)

(C) \(\ \ 36\)

(D) \(\ \ 72\)

(E) \(\ \ 486\)

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