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 can use the formula for uncertainty propagation. The formula is given as:
uncertainty in volume = average volume × (√(uncertainty in diameter/diameter )² + (√(uncertainty in thickness/thickness)²))
Given data:
Average volume = 4.12 cm^3
Diameter = 8.92 × 10^(-2) cm
Uncertainty in diameter = 1 × 10^(-3) cm
Thickness = 6.6 × 10^(-2) cm
Uncertainty in thickness = 1 × 10^(-3) cm
A) Find the uncertainty in the volume of a cookie:
uncertainty in volume = 4.12 cm^3 × (√((1 × 10^(-3) cm)/(8.92 × 10^(-2) cm))^2 + (√((1 × 10^(-3) cm)/(6.6 × 10^(-2) cm) )^2))
uncertainty in volume = 4.12 cm^3 × (√(1.12 × 10^(-3))^2 + (√(1.52 × 10^(-3))^2))
uncertainty in volume = 4.12 cm^3 × (√(1.2544 × 10^(-6) + 2.3104 × 10^(-6)))
uncertainty in volume = 4.12 cm^3 × (√3.5648 × 10^(-6)))
uncertainty in volume = 4.12 cm^3 × (5.97 × 10^(-4)))
uncertainty in volume ≈ 2.46084 × 10^(-3) cm^3
Therefore, the uncertainty in the volume of a cookie is approximately 2.46084 × 10^(-3) cm^3.
B) Find the uncertainty in the ratio of the diameter to the thickness:
uncertainty in ratio = ratio × (√(uncertainty in diameter/diameter )² + (√(uncertainty in thickness/thickness)²))
Given:
Ratio of diameter to thickness = 135
uncertainty in ratio = 135 × (√((1 × 10^(-3) cm)/(8.92 × 10^(-2) cm))^2 + (√((1 × 10^(-3) cm)/(6.6 × 10^(-2) cm) )^2))
uncertainty in ratio = 135 × (√(1.12 × 10^(-3))^2 + (√(1.52 × 10^(-3))^2))
uncertainty in ratio = 135 × (√(1.2544 × 10^(-6) + 2.3104 × 10^(-6)))
uncertainty in ratio = 135 × (√3.5648 × 10^(-6)))
uncertainty in ratio = 135 × (5.97 × 10^(-4)))
uncertainty in ratio ≈ 8.055 × 10^(-2)
Therefore, the uncertainty in the ratio of the diameter to the thickness is approximately 8.055 × 10^(-2).
To find the uncertainty in the volume of a cookie, we need to calculate the range of possible values based on the given uncertainties in the diameter and thickness. The formula for the volume of a cookie is V = πr^2h, where r is the radius and h is the height.
A) Let's first calculate the uncertainty in the volume of a cookie.
1. Calculate the uncertainty in the radius:
The ratio of the diameter to the thickness is given as 135. Since the diameter is 8.92 × 10^(-2) cm, divide it by the ratio (135) to find the radius:
Radius (r) = diameter / 2 = (8.92 × 10^(-2) cm) / 2 = 4.46 × 10^(-2) cm.
To find the uncertainty in the radius, we use the formula:
Uncertainty in radius = (uncertainty in diameter) / 2 = (1 × 10^(-3) cm) / 2 = 5 × 10^(-4) cm.
2. Calculate the uncertainty in the height:
The height (h) is given as 6.6 × 10^(-2) cm.
The uncertainty in the height is given as 1 × 10^(-3) cm.
3. Calculate the uncertainty in the volume:
Using the formula for the uncertainty in the volume of a cylinder, which is given by:
Uncertainty in volume = volume * √((uncertainty in radius / radius)^2 + (uncertainty in height / height)^2)
Volume = πr^2h = π(4.46 × 10^(-2) cm)^2 * (6.6 × 10^(-2) cm)
Uncertainty in volume = 4.12 cm^3 * √((5 × 10^(-4) cm / 4.46 × 10^(-2) cm)^2 + (1 × 10^(-3) cm / 6.6 × 10^(-2) cm)^2)
By substituting the values and computing, you can find the uncertainty in the volume of a cookie.
B) To find the uncertainty in the ratio of the diameter to the thickness, we need to determine the relative uncertainties of each value and then calculate the combined relative uncertainty.
1. Calculate the relative uncertainty in the diameter:
Relative uncertainty in diameter = (uncertainty in diameter) / diameter = (1 × 10^(-3) cm) / (8.92 × 10^(-2) cm)
2. Calculate the relative uncertainty in the thickness:
Relative uncertainty in thickness = (uncertainty in thickness) / thickness = (1 × 10^(-3) cm) / (6.6 × 10^(-2) cm)
3. Calculate the combined relative uncertainty:
Combined relative uncertainty = √( (relative uncertainty in diameter)^2 + (relative uncertainty in thickness)^2 )
By substituting the values and computing, you can find the uncertainty in the ratio of the diameter to the thickness.