As you eat ur way through a bag of cookies, you observe each cookie is a circular disk of diamter 8.50+ or - 0.02 cm and thickness 0.050+ or - 0.005cm. Find the avg volume of a cookie and the uncertainty in the volume.
The average volume is (pi/4) D^2 T, where D and T are the mean diameter and thickness.
Assuming the variations in D and T are uncorrelated, the relative uncertainty in volume is
ln V = ln (constant*D^2*T)
dV/V = sqrt[(2 dD/D)^2 + (dT/T)^2]
= sqrt [(.005)^2 + (0.1)^2]
= 0.1
The volume uncertainty is almost entirely due to the thickness variation.
To find the average volume of a cookie, we can use the formula for the volume of a cylinder:
V = πr^2h
where V is the volume, r is the radius, and h is the height (thickness).
First, let's calculate the average radius of the cookie:
r_avg = (8.50 + 8.50 - 0.02 + 0.02) / 2 = 8.50 cm
Similarly, let's calculate the average height (thickness) of the cookie:
h_avg = (0.050 + 0.050 - 0.005 + 0.005) / 2 = 0.050 cm
Now we have the average radius and height values. We can plug them into the volume formula to find the average volume:
V_avg = π * (8.50 cm)^2 * 0.050 cm
≈ 113.097 cm^3
Therefore, the average volume of a cookie is approximately 113.097 cm^3.
To find the uncertainty in the volume, we need to consider the uncertainty in both the radius and the height.
The maximum uncertainty in the radius (Δr_max) is the difference between the maximum and minimum values divided by 2:
Δr_max = (8.50 + 0.02 - 8.50 + 0.02) / 2 = 0.02 cm
Similarly, let's calculate the maximum uncertainty in the height (Δh_max):
Δh_max = (0.050 + 0.005 - 0.050 + 0.005) / 2 = 0.005 cm
Now, we can calculate the uncertainty in the volume using the formula for propagation of uncertainties:
ΔV = π * [(2 * r_avg * Δr_max) * h_avg + r_avg^2 * Δh_max]
≈ π * [(2 * 8.50 cm * 0.02 cm) * 0.050 cm + (8.50 cm)^2 * 0.005 cm]
≈ 1.070 cm^3
Therefore, the uncertainty in the volume is approximately 1.070 cm^3.
In summary, the average volume of a cookie is approximately 113.097 cm^3 with an uncertainty of approximately 1.070 cm^3.