Question

The happiness(H) of a person depends upon the money he earned(m) and the time spend by him with his family(h) and is given by equation H=f(m,h)=400mh^2 whereas the money earned by him is also depends upon the time spend by him with his family and is given by m(h)=√(1-h^2). Find the time spend by him with his family so that the happiness of a person is maximum.

(a) √(^1⁄3)

(b) √(^2⁄3)

(c) √(^4⁄3)

(d) 0

This question was posed to me by my school teacher while I was bunking the class.

This intriguing question comes from Partial Differentiation topic in section Partial Differentiation of Engineering Mathematics

Answer 1

The correct choice is (b) √(^2⁄3)

Explanation: Given,H=400mh^2 and m=√(1-h^2)

Let, ϑ=m^2 + h^2 + 1 = 0, hence

By lagrange’s method,

^∂H⁄∂m + α^∂v⁄∂m = 0

400h^2 + 2m(α) = 0 …….(1)

Similarly,

^∂H⁄∂h + α ^∂v⁄∂h=0

800mh + 2h(α) = 0 ………(2)

Multiply by m and h in eq’1’ and eq’2’ respectively and adding them

α=-600mh^2

Now from eq. 1 and 2 we get putting value of α,

m = ^1⁄√3 and h = √(^2⁄3).

Hence, total time spend by a person with his family is √(^2⁄3).