#1**+3 **

The volume of a cylinder: \(\pi r^2 \cdot h\). h = height

The radius is 6 cm, and the height is 4*12 = 48 cm. The volume of the cylinder is \(36\pi \cdot 48 = 1728\pi\).

The volume of a sphere: \(\dfrac{4}{3} \pi r^3\).

The radius is 6 cm, and \(6^3 \) is 216. \(\dfrac{4}{3} \cdot 216 = 288\). The volume of one sphere is \(288\pi\). If we multiply by 4, the volume of the four spheres is \(1152\pi\).

We can calculate the empty cylinder space by subtracting 4 VolumeSphere from VolumeCylinder. We have \(1728\pi - 1152\pi\), which simplifies into \(\boxed{576\pi}\).

PartialMathematician
Nov 27, 2018

#2**+1 **

For the second one we have

96 = (1/3) (base area) (height)

96 = (1/3) [ x (3x + 1) ] (12)

96 = 4 [x (3x + 1) ] divide both sides by 4

24 = x (3x + 1)

24 = 3x^2 + x rearrange as

3x^2 + x - 24 = 0 factor

(3x - 8) ( x + 3) = 0

Setting each factor to 0 and solve for x and we have that

x = 8/3 cm or x = - 3cm

Reject the second answer

So...the base area is (x) (3x + 1) = (8/3) (3(8/3) + 1 ) = (8/3)(8 + 1) = (8/3)(9) =

24 cm^2

CPhill
Nov 27, 2018