Start by substituting 2r for the height into the formula for the volume of a cone. Then evaluate the expression.
False
Practice makes perfect
We are asked to verify the following statement.
The volume of a sphere is two-thirds the volume of a cylinder with the same radius r and height of 2r.
Let's start with the volume of a sphere. According to the formula we learned, the volume V_s of a sphere is four-thirds the product of π and the cube of the radius r.
V_s = 4/3 π r^3Now, the volume of a cone V_c is one-third the product of π, a square of the radius of the base r, and the height of the cone h.
V_c = 1/3 π r^2h
We are told that our cone has a height of 2r, where r is its radius. Let's then substitute 2r for h in the formula and simplify the expression.
We found the volume of the cone V_c. Now it is time for us to check if the volume of the sphere V_s is two-thirds of the volume of the cylinder V_c.
V_s ? = 2/3V_c
Let's then calculate two-thirds of the volume of the cylinder.
We are ready to compare the expressions.
V_s &= 43 π r^3
23V_c &= 49 π r^3
We see that the expressions are not equal. This is why the statement is false.
