Josh finds that his new ball fits tightly inside his cylindrical can of diameter 6 inches and height 5 inches. What is the exact amount of empty space is there inside the can?

The exact amount of empty space is  

Step-by-step explanation:

we know that

To find out  the exact amount of empty space, subtract the volume of the ball from the volume of the cylindrical can

step 1

Find the volume of teh ball

The volume of the sphere (ball) is given by

we have

---> the radius is half the diameter

substitute

step 2

Find the volume of the cylinder

The volume of the cylinder is given by

we have

---> the radius is half the diameter

substitute

step 3

Subtract the volumes

let's call the solutions x₁ = (5 - 2√7)/3 and x₂ = (5 + 2√7)/3

then the equation is (x-x₁)(x-x₂) = 0

if you fill that in you get a lot of numbers which you can recognize as ax²+bx+c=0.

step-by-step explanation:

you can see that a=1 (there is just an x²)

b = (-5 -2√7 + 2√7 - 5)/3 = -10/3

c = -2√7)/3 ) * (-5+2√7)/3) = -1/3

so your answer is 3x^2 – 10x – 1 = 0