(a) To find the decrease in volume, we first need to find the volume of the cylinder before and after the piston moves. The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height (or in this case, the distance the piston moves).

The diameter of the piston is given as 7.500 cm, so the radius r is half of this, or 3.750 cm. The distance the piston moves is given as 3.250 cm. So the decrease in volume is:

V = π(3.750 cm)²(3.250 cm) = 144.845 cm³

(b) To find the uncertainty in this volume, we need to use the formula for the propagation of uncertainty in a product or quotient, which is √[(ΔA/A)² + (ΔB/B)² + ...], where A, B, etc. are the quantities being multiplied or divided, and ΔA, ΔB, etc. are their uncertainties.

The uncertainties in the radius and height are half the uncertainties in the diameter and distance, respectively, or 0.001 cm and 0.0005 cm. So the relative uncertainties in the radius and height are (0.001 cm)/(3.750 cm) and (0.0005 cm)/(3.250 cm), respectively.

The relative uncertainty in the volume is then √[(0.001 cm/3.750 cm)² + (0.0005 cm/3.250 cm)²] = 0.000307.

So the uncertainty in the volume is (0.000307)(144.845 cm³) = 0.044 cm³.

Question

(a) To find the decrease in volume, we first need to find the volume of the cylinder before and after the piston moves. The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height (or in this case, the distance the piston moves).

The diameter of the piston is given as 7.500 cm, so the radius r is half of this, or 3.750 cm. The distance the piston moves is given as 3.250 cm. So the decrease in volume is:

V = π(3.750 cm)²(3.250 cm) = 144.845 cm³

(b) To find the uncertainty in this volume, we need to use the formula for the propagation of uncertainty in a product or quotient, which is √[(ΔA/A)² + (ΔB/B)² + ...], where A, B, etc. are the quantities being multiplied or divided, and ΔA, ΔB, etc. are their uncertainties.

The uncertainties in the radius and height are half the uncertainties in the diameter and distance, respectively, or 0.001 cm and 0.0005 cm. So the relative uncertainties in the radius and height are (0.001 cm)/(3.750 cm) and (0.0005 cm)/(3.250 cm), respectively.

The relative uncertainty in the volume is then √[(0.001 cm/3.750 cm)² + (0.0005 cm/3.250 cm)²] = 0.000307.

So the uncertainty in the volume is (0.000307)(144.845 cm³) = 0.044 cm³.

Knowee AI · Accepted Answer

(a) To find the decrease in volume, we first need to find the volume of the cylinder before and after the piston moves. The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height (or in this case, the distance the piston moves).

The diameter of the piston is given as 7.500 cm, so the radius r is half of this, or 3.750 cm. The distance the piston moves is given as 3.250 cm. So the decrease in volume is:

V = π(3.750 cm)²(3.250 cm) = 144.845 cm³

(b) To find the uncertainty in this volume, we need to use the formula for the propagation of uncertainty in a product or quotient, which is √[(ΔA/A)² + (ΔB/B)² + ...], where A, B, etc. are the quantities being multiplied or divided, and ΔA, ΔB, etc. are their uncertainties.

The uncertainties in the radius and height are half the uncertainties in the diameter and distance, respectively, or 0.001 cm and 0.0005 cm. So the relative uncertainties in the radius and height are (0.001 cm)/(3.750 cm) and (0.0005 cm)/(3.250 cm), respectively.

The relative uncertainty in the volume is then √[(0.001 cm/3.750 cm)² + (0.0005 cm/3.250 cm)²] = 0.000307.

So the uncertainty in the volume is (0.000307)(144.845 cm³) = 0.044 cm³.

A car engine moves a piston with a circular cross section of 7.500±0.002cm7.500±0.002cm diameter a distance of 3.250±0.001cm3.250±0.001cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

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