Question

sphere and a cylinder have the same radius and height. The volume of the cylinder is 54 meters cubed. Amie found the volume of the sphere. A sphere with height h and radius r. A cylinder with height h and radius r. Her work is shown below. V = two-thirds + 54. V = two-thirds + StartFraction 162 Over 3 EndFraction. V = StartFraction 164 Over 3 EndFraction meter cubed. What is Amie’s error? Amie should have multiplied 54 by Two-thirds. Amie should have multiplied 54 by Four-thirds pi. Amie should have multiplied 54 by Four-thirds. Amie should have multiplied 54 by Two-thirds pi.

Answer 1

the answer is Amie should have multiplied 54 by two thirds i.e Amie should have multiplied 54 by 2/3.

Explanation:

We find the formula for the Volume of the Sphere and Cylinder.

a. Volume the Sphere = 4/3 π r³........Equation 1

b. Volume the Cylinder = π r² h........Equation 2

We would find the ratio of the Volume of the Sphere to the Volume of the Cylinder .

= Volume of the Sphere : Volume of the Cylinder

= 4/3 π r³ : π r² h.........Equation 3

Divide both sides by π r²

= 4/3 r : h.............Equation 4

It is important to note that the height of the sphere = the diameter of the sphere .

The diameter of a Sphere (D) = 2r

In the question, remember we were told that the height of the Sphere is the same as the height of the Cylinder.

i.e Height of the Sphere = Height of the Cylinder

Hence, the height of the Cylinder is also = 2r

Step 3

We would be substituting 2r for the Height of the Cylinder represented as h in Equation 4

Therefore, the ratio will be given as :

= 4/3 r : 2r ..............Equation 5

We would divide both sides by 2r

= 2/3 : 1 ..............Equation 6

From Equation 6 we can see that the volume of the sphere = 2/3 the volume of the cylinder

In the question above, we were told the volume of the cylinder = 54 m³

Hence, the volume of the sphere = 2/3 × 54 = 36 m³

Therefore, the answer is Amie should have multiplied 54 by two thirds i.e Amie should have multiplied 54 by 2/3.

Answer 2

Amie should have multiplied 54 by 2/3

Explanation:

The formula of the volume of a sphere is given by

`(4)/(3)\pi r^3`, where r is the radius of the sphere.

to find the volume of the sphere, we must find
`r^3`

We know that the formula of a cylinder of height h and radius r is given by

`\pi r^2h`

So
`54 = \pi r^2 h`

We know that the sphere and the cylinder have the same height. Recall that the height of the sphere of radius r, is the diameter of a circle that passes through the heighest and lowest points of the sphere. Then, h=2r. This means that

`54 = 2\pi r^3`

From here, we know that
`\pi r^3 = (54)/(2)`.

So the volume of the sphere is
`(4)/(3)(54)/(2) = (2)/(3)\cdot54`.

So Amie should have multiplied 54 by 2/3.