A power dissipated in a resistor is given by p=E^2/R. Using calculus, find the approximate percentage error in P when E is increased by 3% and R is decreased by 2%?

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Answer

Calculating Percentage Error in Power Dissipation

Given:
- Power dissipated in a resistor, p = E^2/R
- Increase in E by 3%
- Decrease in R by 2%

To find:
- Approximate percentage error in p

Solution:

To find the approximate percentage error in p, we need to use the formula for percentage error, which is given by:

Percentage Error = (Approximate Change / Actual Value) × 100

We can break down the given changes in E and R into their actual values as follows:

Actual Change in E = 3% of E
Actual Change in R = -2% of R

Substituting these values in the original equation for p, we get:

p_new = (E + 0.03E)^2 / (R - 0.02R)

Simplifying this equation, we get:

p_new = (1.0609E^2) / (0.98R)

Now we can find the approximate percentage error in p by using the formula for percentage error:

Percentage Error = ((p_new - p) / p) × 100

Substituting the new value for p and simplifying, we get:

Percentage Error = ((1.0609(E^2/R) - E^2/R) / (E^2/R)) × 100

Percentage Error = (0.0609 / 1) × 100

Percentage Error = 6.09%

Therefore, the approximate percentage error in p is 6.09%.

Conclusion:
- The approximate percentage error in power dissipation when E is increased by 3% and R is decreased by 2% is 6.09%.

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