We can try to solve the problem using Stefan-Boltzman law of radiation.According to this law we have the follwing equation.
rho=Asigmae(T^4-T_c^4)
"Where"
rho->"Net radiated power"
A->"Radiaditing area"
sigma->"Stefan's constant"
e->"Emissivity"
T->"Temperature of radiating surface"
T_c->"Temperature of surrounding "
So rho will be proportional to the rate of eletrical energy spent or wattage E of the lamp .
SinceA,sigma,e=90% (given) are remaining same in two cases.
So rhopropE=>(T^4-T_c^4)propE
In our problem
For first lamp
E_1->"Electrical power"=60W
T_1->"Temperature of lamp"=65^@C=(65+273)K=338K
T_c->"Temperature of surrounding"=18^@C=(18+273)K=291K
For 2nd lamp
E_2->"Electrical power"=150W
T_2->"Temperature of lamp"=?
T_c->"Temperature of surrounding"=18^@C=291K
So we can write
(T_2^4-T_c^4)/(T_1^4-T_c^4)=E_2/E_1=150/60=2.5
=>(T_2^4-291^4)/(338^4-291^4)=2.5
=>T_2^4=(338^4-291^4)xx2.5+291^4
T_2=384.6K=(384.6-273)^@C=111.6^@C
