The measurement of the radius of a circle is 16 inches, with a possible error of .25 inches. (Use differentials to approximate the possible propagated error and the percent error.

Propagated error= 1.6%

Percent error= 4.6%

the propagated error (dC) is 2π*dr = 1.57 inches, not 1.57%

the % error (dC/C) = .25/16 = 1.57%

Since circumference is a linear function of radius, its % error cannot be greater than the % error in the radius.

To find the propagated error and percent error, we can use differentials.

1. Propagated error:
The formula for the propagated error is given by: propagated error = (differential of the dependent variable) * (differential of the independent variable)

Here, the dependent variable is the circumference of the circle, and the independent variable is the radius of the circle. Recall that the circumference of a circle is given by the formula: C = 2πr.

Taking the differential of both sides with respect to r, we get: dC = 2πdr

Now, let's substitute the given values:
r = 16 inches
dr = 0.25 inches

Substituting these values, we get: dC = 2π(0.25) ≈ 1.57 inches

Therefore, the propagated error is approximately 1.57 inches.

2. Percent error:
The formula for percent error is given by: percent error = (propagated error / measured value) * 100%

Here, the measured value is the radius of the circle, which is 16 inches.

Substituting the values, we get: percent error = (1.57 / 16) * 100% ≈ 9.81%

Therefore, the percent error is approximately 9.81%.

To approximate the propagated error and percent error, we can use differentials. Let's break down the problem step by step:

1. Propagated Error:
The formula for the propagated error is given by:
Δy ≈ |dy/dx| * Δx
where Δy is the propagated error, dy/dx is the derivative of the function with respect to x, and Δx is the error in x.

In this case, we need to find the propagated error for the radius of the circle, so y represents the radius, and x represents the measured value.

Given:
Radius (y) = 16 inches
Error in radius (Δx) = 0.25 inches

To find the propagated error, we need to find |dy/dx|, which represents the absolute value of the derivative of the function with respect to x. The derivative of the radius (y) with respect to the measured value (x) is simply 1 (since the radius is directly proportional to the measured value).

So, |dy/dx| = 1

Now we can plug in the values into the formula:
Δy ≈ |dy/dx| * Δx
Δy ≈ 1 * 0.25 = 0.25 inches

Therefore, the propagated error is 0.25 inches. To calculate the percentage, we divide the propagated error (0.25 inches) by the measured value (16 inches) and multiply by 100:
Percentage propagated error = (0.25 / 16) * 100 ≈ 1.6%

2. Percent Error:
The formula for percent error is:
Percent Error = (Δy / y) * 100
where Δy is the error, and y is the measured value.

In this case, we need to find the percent error for the radius of the circle, so y represents the radius.

Given:
Radius (y) = 16 inches
Error in radius (Δy) = 0.25 inches

Plugging the values into the formula:
Percent Error = (Δy / y) * 100
Percent Error = (0.25 / 16) * 100 ≈ 1.56%

Therefore, the percent error is approximately 1.56%.

To summarize:
- The propagated error is approximately 0.25 inches, which corresponds to a percentage propagated error of approximately 1.6%.
- The percent error is approximately 1.56%.