What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is r = 0.84 ± 0.04m ?
v = (4/3) pi r^3
dv/dr = 4 pi r^2
(which is the surface area of course :)
dv = 4 pi r^2 dr
dv/v = 4 pi r^2 dr/(4/3)pir^3
dv/v = 3 dr/r
here dr = .04
and r = .84
so 100 * 3 dr/r = 12/.84 = 14.3 %
how do u get dv/v = 3 dr/r ?
To calculate the percent uncertainty in the volume of a spherical beach ball, we can use the formula for the volume of a sphere, V = 4/3πr³.
Given that the radius is r = 0.84 ± 0.04 m, the percent uncertainty in the volume can be calculated using the following steps:
1. Calculate the volume with the maximum radius: V_max = (4/3)π(r + Δr)³
where Δr is the uncertainty in the radius, given as ±0.04 m.
V_max = (4/3)π(0.84 + 0.04)³
2. Calculate the volume with the minimum radius: V_min = (4/3)π(r - Δr)³
V_min = (4/3)π(0.84 - 0.04)³
3. Calculate the absolute uncertainties in the volume:
ΔV = V_max - V_min
4. Calculate the average volume:
V_avg = (V_max + V_min) / 2
5. Calculate the percent uncertainty:
Percent uncertainty = (ΔV / V_avg) * 100
By plugging in the values, we get:
V_max = (4/3)π(0.88)³
V_min = (4/3)π(0.8)³
ΔV = V_max - V_min
V_avg = (V_max + V_min) / 2
Percent uncertainty = (ΔV / V_avg) * 100
Please note, you can substitute the values and calculate the final answer based on the given radius.
To determine the percent uncertainty in the volume of a spherical beach ball, we need to calculate the relative uncertainty in its radius.
The relative uncertainty is given by the formula:
Relative uncertainty = (Uncertainty / Measurement) * 100%
In this case, the uncertainty in the radius is given as ± 0.04 m and the measurement is 0.84 m.
Substituting these values into the formula:
Relative uncertainty = (0.04 / 0.84) * 100%
Calculating this expression:
Relative uncertainty = 0.0476 * 100%
The relative uncertainty is equal to 4.76%.
Therefore, the percent uncertainty in the volume of the spherical beach ball is approximately 4.76%.