(K^0.75/L^0.75)/2 = (3L^0.25/K^0.25)/10
MPL/W = MPK/ R
What ratio of capital to labor minimizes jakes cost?
$$\\(K^{0.75}/L^{0.75})/2 = (3L^{0.25}/K^{0.25})/10\\\\
5(K^{0.75}/L^{0.75}) = 3L^{0.25}/K^{0.25}\\\\
5(K/L)^{0.75} = 3(L/K)^{0.25}\\\\
5(K/L)^{0.75}= 3(K/L)^{-0.25}\\\\
$Let $ R=K/L\\\\
5(R^{0.75}) = 3(R)^{-0.25}\\\\
(R^{0.75})/(R)^{-0.25} = 3/5\\\\
R^1=3/5\\
r=3/5$$
I don't know about the minimize bit.
I just get the ratio of K to L is 3:5
$$\\(K^{0.75}/L^{0.75})/2 = (3L^{0.25}/K^{0.25})/10\\\\
5(K^{0.75}/L^{0.75}) = 3L^{0.25}/K^{0.25}\\\\
5(K/L)^{0.75} = 3(L/K)^{0.25}\\\\
5(K/L)^{0.75}= 3(K/L)^{-0.25}\\\\
$Let $ R=K/L\\\\
5(R^{0.75}) = 3(R)^{-0.25}\\\\
(R^{0.75})/(R)^{-0.25} = 3/5\\\\
R^1=3/5\\
r=3/5$$
I don't know about the minimize bit.
I just get the ratio of K to L is 3:5