Question

The tension T of a cable at its lowest point is given by: T =WL²/ 8а Where W is the weight of the cable, L is the length and a is how much it sags. Given that the error in W is 1%, in L is 2.1% and in a is 2%, find the maximum percentage error in the measurement of T.

Answer 1

To find the maximum percentage error in the measurement of T, calculate the individual percentage errors of each variable (W, L, a) and use the absolute values of their partial derivatives to determine the maximum percentage error in T.

Step-by-step explanation:

To find the maximum percentage error in the measurement of T, we need to determine how the error in each variable affects T. Let's start by calculating the individual percentage errors:

1. For W: The error in W is 1% of the weight. So the percentage error in W is 1%.
2. For L: The error in L is 2.1% of the length. So the percentage error in L is 2.1%.
3. For a: The error in a is 2% of the sag. So the percentage error in a is 2%.

Now let's use these percentage errors to find the maximum percentage error in T:

1. Calculate the partial derivative of T with respect to each variable:
2. dT/dW = L²/8a
3. dT/dL = WL/4a
4. dT/da = -WL²/8a²
5. Calculate the absolute values of these partial derivatives:
6. |dT/dW| = |L²/8a|
7. |dT/dL| = |WL/4a|
8. |dT/da| = |-WL²/8a²|
9. Calculate the maximum percentage error using these absolute values:
10. Maximum percentage error = (|dT/dW| * percentage error in W + |dT/dL| * percentage error in L + |dT/da| * percentage error in a) * 100%

Substituting the given values of the percentage errors, we can now calculate the maximum percentage error in the measurement of T.