For P = x²y¹⁄³z/w⁴, percentage measurement errors in x, y, z and w are 2%, 3%, 1% and 2%, respectively. If the maximum percentage error in P is 2n%, find n.
Physics · Mixed Chapter
JEE Mains

Question

For P = x²y¹⁄³z/w⁴, percentage measurement errors in x, y, z and w are 2%, 3%, 1% and 2%, respectively. If the maximum percentage error in P is 2n%, find n.

G1 Physics · Accepted Answer

Step 1 — Source expression: P = x²y¹⁄³z/w⁴. All symbols represent measured quantities. Step 2 — Take logarithms: lnP = 2lnx + ⅓lny + lnz − 4lnw. Step 3 — For maximum uncertainty, add the magnitudes of each contribution: (δP/P)ₘₐₓ = 2(δx/x) + ⅓(δy/y) + (δz/z) + 4(δw/w). Step 4 — Substitute percentage uncertainties: δP% = 2(2%) + ⅓(3%) + 1% + 4(2%) = 4% + 1% + 1% + 8% = 14%. Step 5 — Because 14% = 2n%, we have 2n = 14, so n = 7. Answer: 7.