A 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.
2 answers:
The question above was not properly written in an explanatory form.
Find attached to this answer a diagram where the question is properly written for a better understanding.
Answer:
The Correct answer is the First option(Option A) which says:
Amie should have multiplied 54 by Two-thirds. This means Amie should have multiplied 54 by 2/3
Step-by-step explanation:
In the question we were given two solid shapes, a Sphere and a Cylinder. Both shapes had the same height and radius and the Volume of the Cylinder was given as 54 in³.
Step 1
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
Step 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.
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