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Question

The value of z is calculated from the following formula:
z=(12a+5b)/(11a+19b)

a is increased by 2.2%
b is increased by 1.2% of a
Using partial differentiation, find an expression for the percentage change in z in terms of a and b.
b is measured as 11.
What non-zero value of a would result in a zero percentage change in z?

Answer 1

z = (12a + 5b) / (11a + 19b)
∂z/∂a = (132a + 228b - (132a + 55b) ) / (11a + 19b)^2
∂z/∂a = 173b / (11a + 19b)^2

∂z/∂b = (5(11a + 19b) - 19(12a + 5b)) / (11a + 19b)^2
∂z/∂b = (55a + 95b - (228a + 95b)) / (11a + 19b)^2
∂z/∂b = - 173a / (11a + 19b)^2

dz = (∂z/∂a)da + (∂z/∂b)db
dz = (173/(11a + 19b)^2)(bda - adb)
dz/z = (173(11a + 19b)/(12a + 5b)(11a + 19b)^2)(bda - adb)
dz/z = (173/(12a + 5b)(11a + 19b))(bda - adb)
dz/z = (173/(12a + 5b)(11a + 19b))(2.2ab - 1.2a^2)
dz/z = (173/(132a^2 + 283ab + 19b^2))(2.2ab - 1.2a^2)
b is measured as 11.
dz/z = (173/(12a + 55)(11a + 209))(24.2a - 1.2a^2)
What non-zero value of a would result in a zero percentage change in z?
12a = 242 = 121/6
a ≈ 20.16667