a spherical balloon's radius was measured to be 30 cm with a maximum possible error of 0.2 cm.
a) use differentials to estimate the maximum error in the calculated volume.
b) what is the relative error?
dv = 4pi r^2 dr
rel err = dv/v
To estimate the maximum error in the calculated volume of the spherical balloon, we can use differentials. The formula for the volume of a sphere is given by:
V = (4/3)πr^3
where V is the volume and r is the radius of the sphere.
a) Estimating the maximum error in the calculated volume:
First, we need to find the differential of the volume equation. The differential of a function is the linear approximation of a small change in the function's value.
dV = d((4/3)πr^3)
To find dV, we differentiate the volume equation with respect to r:
dV = (4/3)(3πr^2)dr
= 4πr^2dr
Now, let's substitute the radius and maximum possible error into the equation:
r = 30 cm (radius)
dr = 0.2 cm (maximum possible error)
dV = 4π(30 cm)^2(0.2 cm)
= 144π cm^3
Therefore, the maximum error in the calculated volume is estimated to be 144π cm^3.
b) To find the relative error, we can divide the maximum error in volume by the actual volume of the balloon.
The actual volume of the balloon can be calculated by substituting the radius into the volume equation:
V = (4/3)π(30 cm)^3
= 36,000π cm^3
Relative error = (Maximum error in volume) / (Actual volume)
= (144π cm^3) / (36,000π cm^3)
= 0.004 or 0.4%
Therefore, the relative error in the calculated volume of the spherical balloon is 0.4%.