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 .
Since#A,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#
