Correct volume: 20.57m^3 Mistake: Our friend substituted the diameter, not of the radius, into the formula for the volume of a sphere.
Practice makes perfect
Our friend says that the volume of a sphere with a diameter of 3.4 meters is 164.55 cubic meters. We know that this answer is incorrect. We want to find and correct the mistake that our friend made.
To do so, we will begin by calculating the correct volume. Recall that the volume of a sphere is four thirds the product of π and the cube of the radius.
V=4/3Ï€ r^3
From the diagram we can see that the diameter of the sphere is 3.4 meters. This means that the radius is 3.4÷ 2=1.7 meters. We can substitute this value for r in the formula and simplify the right-hand side.
The volume of the sphere is about 20.57 cubic meters. Let's compare this to the result that our friend got.
V &= 20.57 m^3 ✓
V &= 164.55 m^3 *
We see that the volume that our friend got is larger. Perhaps it was because they substituted the diameter of the sphere, not the radius. Let's check if our suspicions are correct.
This is the same result that our friend got. Our suspicions were correct.
Alternative Solution
Comparing the Volumes
We can find our friend's error in another way as well, by directly comparing the two volumes. Here are our volume and our friend's volume again.
V &= 20.57 m^3 ✓
V &= 164.55 m^3 *
We see that the correct volume is less than the volume that our friend found. Let's then find the quotient of the two volumes. This way we will know how many times bigger our friend's result is than ours.
164.55/20.57 ≈ 8
Their result is about 8 times more than the correct volume. Let's now take another look at the formula for the volume of a sphere.
V=4/3Ï€ r^3
In the formula the radius is raised to the 3^(rd) power.
V=4/3Ï€ r^3This means that if we substitute the diameter, which is twice as long as the radius, the volume will be 2^3=8 times more. This is another way of proving that our friend used the diameter instead of the radius.
