Question

A physical quantity 'y' is represented by the formula $y = m^x r^{-4} g^p l^{3/2}$. If the percentage errors found in y, m, r, l and g are 18, 1, 0.5, 4 and p respectively, then find the value of x and p.

• A. 4 and $\pm 3$
• B. 5 and $\pm 2$
• C. 8 and $\pm 2$
• D. $\frac{16}{3}$ and $\pm \frac{3}{2}$

Hint:

When dealing with error propagation problems, be vigilant for potential typos. If a variable is used for both an exponent and an error percentage, it's likely a mistake. Assume a plausible value for the error (like 1%) and see if it leads to one of the given options.

Answer 1

The Correct Option is C

The formula for propagation of relative error is given by:
$\frac{\Delta y}{y} = |x|\frac{\Delta m}{m} + |-4|\frac{\Delta r}{r} + |p|\frac{\Delta g}{g} + |\frac{3}{2}|\frac{\Delta l}{l}$.
Multiplying by 100 gives the formula for percentage error:
$%\text{error in y} = |x|(%\text{error in m}) + 4(%\text{error in r}) + |p|(%\text{error in g}) + \frac{3}{2}(%\text{error in l})$.
The problem states "the percentage errors found in y, m, r, l and g are 18, 1, 0.5, 4 and p respectively". There is a likely typo here, as the percentage error in 'g' is given as the variable 'p', which is also an exponent. This is highly unusual. A common mistake in question papers is to intend a numerical value for the error. Let's assume the percentage error in g was intended to be 1%.
With the assumption that the percentage error in g is 1%:
$18 = |x|(1) + 4(0.5) + |p|(1) + \frac{3}{2}(4)$.
$18 = |x| + 2 + |p| + 6$.
$18 = |x| + |p| + 8$.
$|x| + |p| = 10$.
Now we check the given options to see which pair of (x, p) satisfies this equation.
(A) $|4| + |\pm 3| = 4 + 3 = 7 \neq 10$.
(B) $|5| + |\pm 2| = 5 + 2 = 7 \neq 10$.
(C) $|8| + |\pm 2| = 8 + 2 = 10$. This pair satisfies the condition.
(D) $|\frac{16}{3}| + |\pm \frac{3}{2}| = \frac{16}{3} + \frac{3}{2} = \frac{32+9}{6} = \frac{41}{6} \neq 10$.
Based on this logical deduction assuming a typo, the values for x and p are 8 and $\pm 2$.