a diameter of a sphere can be measured with a maximum error of 0.2cm . Find the corresponding error volume of a sphere whose diameter is 90 cm.
Why did the sphere go to therapy? Because it had a lot of unresolved circular issues! Let's calculate the corresponding error volume, shall we?
To find the corresponding error volume, we need to calculate the error in the radius first. Since the maximum error in diameter is 0.2 cm, the maximum error in radius will be half of that, which is 0.1 cm.
Now, let's calculate the volume of the sphere. The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius. So, for the original sphere with a radius of 45 cm:
V = (4/3)π(45)³
≈ (4/3)π(91,125)
≈ (4/3)π(91125)
≈ 121,500π cm³
Now, let's calculate the maximum error volume. To do that, we need to use the formula for the volume of a sphere, but this time with the radius plus the maximum error:
V with error = (4/3)π(45 + 0.1)³
≈ (4/3)π(45.1)³
≈ 121,520.166π cm³
Finally, let's calculate the corresponding error volume:
Corresponding error volume = V with error - V
≈ 121,520.166π - 121,500π
≈ 20.166π cm³
So, the corresponding error volume of a sphere with a diameter of 90 cm is approximately 20.166π cm³.
To find the corresponding error volume of a sphere, we first need to calculate the error in the radius of the sphere.
Since the maximum error in the diameter is 0.2 cm, the maximum error in the radius would be half of that.
Maximum error in radius = 0.2 cm / 2 = 0.1 cm
Next, we calculate the volume of the sphere using the given diameter.
The formula for the volume of a sphere is:
Volume = (4/3) * π * radius^3
Now, let's substitute the values:
Diameter = 90 cm
Radius = Diameter / 2 = 90 cm / 2 = 45 cm
Using the formula, we can calculate the volume of the sphere:
Volume = (4/3) * π * 45^3
Now, let's calculate the error in the volume.
To find the error in volume, we need to differentiate the volume formula with respect to the radius and multiply it by the maximum error in the radius.
dVolume/dRadius = (4/3) * π * 3 * radius^2 (applying differentiation)
dVolume/dRadius = 4π * radius^2
Now, we multiply this by the error in radius:
Error in Volume = dVolume/dRadius * Error in radius
Error in Volume = 4π * radius^2 * 0.1 cm
Now, let's substitute the value of radius and calculate the error in volume:
Error in Volume = 4π * (45 cm)^2 * 0.1 cm
Finally, we can calculate the error in volume using a calculator.
To find the corresponding error in the volume of a sphere, we need to consider the formula for the volume of a sphere:
V = (4/3) * π * (r^3)
where V is the volume and r is the radius of the sphere.
Since the given error is for the diameter, we first need to find the radius of the sphere. The radius is half of the diameter, so in this case, the radius (r) would be 90 cm / 2 = 45 cm.
Next, we need to consider the error in the diameter (0.2 cm) and how it affects the radius. The error in the radius will be half of the error in the diameter. Therefore, the error in the radius (Δr) is 0.2 cm / 2 = 0.1 cm.
Now, we can find the error in the volume of the sphere. The formula for the error in the volume can be written as:
ΔV = (4/3) * π * [(r + Δr)^3 - r^3]
Substituting the values we have:
ΔV = (4/3) * π * [(45 cm + 0.1 cm)^3 - 45 cm^3]
Calculating the expression inside the brackets:
[(45.1 cm)^3 - 45 cm^3] = 91,366.23 cm^3
Now, finding the error in the volume:
ΔV = (4/3) * π * 91,366.23 cm^3 ≈ 383,845.97 cm^3
Therefore, the corresponding error in the volume of the sphere with a diameter of 90 cm is approximately 383,845.97 cm^3.