Question

We made the following measurements for three parameters a,b, and c. a=6±1,b=20±2,c=15±1 Assuming that these measurements are independent, estimate the following quantities (I, J, K,L) and their standard deviations. Note that a star (∗) indicates a multiplication, and loge​ is the natural logarithm.

I=a+b∗c

Answer 1

The quantity I=a+b*c for the given values a=6±1, b=20±2, and c=15±1 results in an estimated value of I=306. The standard deviation for I, calculated considering the uncertainties in a, b, and c, is approximately 36.06.

Step-by-step explanation:

To determine the value of I = a + b*c with their uncertainties, we can use the formula for the propagation of errors for independent measurements. Since a, b, and c are independent, the standard deviation of I, which we can denote as σ(I), will be the square root of the sum of the squares of the individual standard deviations after they have been multiplied by the square of the coefficient in front of each variable (if one exists). There is no coefficient for a, but for b*c we take into consideration both variables b and c.

First, let's calculate the estimate for I:

1. I = a + b*c = 6 + (20 * 15) = 6 + 300 = 306

Now, let's calculate the standard deviation for I (σ(I)):

1. σ(I) = √(σ(a)² + (b² * σ(c)²) + (c² * σ(b)²))
2. σ(I) = √((1)² + (20² * 1²) + (15² * 2²))
3. σ(I) = √(1 + 400 + 900) = √1301 ≈ 36.06

The estimate for I = 306 with a standard deviation of ≈ 36.06.