What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is r=4.84±0.05m?
I keep getting 6% and it says i am wrong.
To calculate the percent uncertainty in the volume of a spherical beach ball, we need to find the percent uncertainty in the radius and use it to find the percent uncertainty in the volume.
The percent uncertainty is calculated using the formula:
Percent uncertainty = (Uncertainty / Measurement) * 100
Given that the radius of the beach ball is measured as r = 4.84 ± 0.05 m, we can calculate the percent uncertainty in the radius:
Percent uncertainty in radius = (0.05 / 4.84) * 100 ≈ 1.0331%
Now, to find the percent uncertainty in the volume, we use the formula for the volume of a sphere:
V = (4/3) * π * r^3
Taking the derivative with respect to r, we have:
dV/dr = 4 * π * r^2
We can rewrite this equation in terms of percent uncertainties:
Percent uncertainty in volume = 3 * (Percent uncertainty in radius)
So, the percent uncertainty in the volume of the beach ball is:
Percent uncertainty in volume ≈ 3 * 1.0331% ≈ 3.0994%
Therefore, the correct answer is approximately 3.1%, not 6%.
To calculate the percent uncertainty in the volume of a spherical beach ball, we first need to find the percent uncertainty in the radius.
The formula for percent uncertainty is given by:
Percent Uncertainty = (Error / Measurement) x 100
Here, the error is the uncertainty in the radius, which is ±0.05 m, and the measurement is the actual radius, which is 4.84 m.
Plugging the values into the formula:
Percent Uncertainty = (0.05 m / 4.84 m) x 100 = 1.03%
So, the percent uncertainty in the radius is approximately 1.03%.
Now, to find the percent uncertainty in the volume, we need to consider the formula for the volume of a sphere:
V = (4/3) x π x r^3
First, let's calculate the percent uncertainty in r^3:
Percent Uncertainty in r^3 = 3 x Percent Uncertainty in r = 3 x 1.03% = 3.09%
Next, let's calculate the volume using the maximum and minimum values of r to account for the uncertainty:
Maximum volume = (4/3) x π x (4.84 + 0.05)^3 = (4/3) x π x 4.89^3
Minimum volume = (4/3) x π x (4.84 - 0.05)^3 = (4/3) x π x 4.79^3
Now, let's find the difference between the maximum and minimum volume:
Volume Difference = Maximum volume - Minimum volume
= [(4/3) x π x 4.89^3] - [(4/3) x π x 4.79^3]
Finally, we can calculate the percent uncertainty in the volume using the following formula:
Percent Volume Uncertainty = (Volume Difference / Average Volume) x 100
Average Volume = [(Maximum volume + Minimum volume) / 2]
By plugging in the values, we can calculate the final answer for the percent uncertainty in the volume of the beach ball.