As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of = 8.92 1×10^−2 and a thickness of = 6.6×10^−2 1×10^−3 .
the average volume of a cookie; 4.12 cm^3
ratio of the diameter to the thickness: 135
A)Find the uncertainty in the volume of a cookie.
B) find the uncertainty in this ratio.
To find the uncertainty in the volume of a cookie, we need to use the formula for the uncertainty propagation. The formula is as follows:
δV = V * √((δd/d)^2 + (δh/h)^2)
where:
- δV is the uncertainty in the volume of the cookie,
- V is the average volume of the cookie (4.12 cm^3),
- δd is the uncertainty in the diameter of the cookie (given as 8.92 × 10^(-2)),
- d is the average diameter of the cookie,
- δh is the uncertainty in the thickness of the cookie (given as 6.6 × 10^(-3)),
- h is the average thickness of the cookie.
Now, let's solve for the uncertainty in the volume of a cookie (δV):
Step 1: Calculate the average diameter (d) and average thickness (h) of the cookie:
The ratio of diameter to thickness is given as 135.
d = 135 * h
Step 2: Calculate the uncertainty in the diameter (δd) and thickness (δh) of the cookie:
δd = d * (uncertainty in the ratio)
= d * (uncertainty in the ratio) * h/h
= 135 * h * (uncertainty in the ratio)
δh = (uncertainty in the thickness)
Step 3: Substitute the values into the formula:
δV = V * √((δd/d)^2 + (δh/h)^2)
= 4.12 * √(((135 * h * (uncertainty in the ratio))/d)^2 + ((uncertainty in the thickness)/h)^2)
Now, substitute the given values into the formula to get the uncertainty in the volume of a cookie (δV).