To dig a well of x dcm deep , one part of the total expenses varies directly with x and other part varies directly with `x^2`. If the expenses of digging wells of 100 dcm and 200 dcm depths are rs5000 and rs12000 respectively , calculate the expenses of digging a well of 250 dcm depth.

Explanation

Let total expenses = rs E, two parts of which are p and q, where p `prop ` x and q `prop x^2` .
`therefore p =k_1 x, k_1` =non-zero variation constant and `q=k_2x^2, k_2` = non -zero varaition constant.
`therefore E =p+q=k_1x+k_2x^2( where k_1,k_2 ne0=`variation constant )............(1)
As per question , expenses of 100 dcm is rs 5000 .
`therefore` form (1) we get , 5000 =`k_1xx100+k_2xx(100)^2`
or , 50=`k_1+100k_2`
or ,` k_1 +100k_2 =50 ...........(2)`
Again expenses of 200 dcm is rs 12000.
form (1) we get , 12000 =`k_1xx200+k_2xx(200)^2`
or , `12000=k_1+200k_1+40000k_2`
or ,`60= k_1 +200k_2`
`thereforek_1+200k_2=60 ...........(3)`
Now , subtracting (2) from (3) we get , 100 `k_2 =10 or , k_2 =(10)/(100)=(1)/(10) `
Then from (2) we get , `k_1+100 xx1/(10)=50 or , k_1 +10 =50 or , k_1=40 `.
`therefore` (1) becomes , E =40 x+`1/(10)x^2 ..........(4)`
Now , putting x =250 in (4) we get , E `40 xx250 +1/(10)xx(250)^2 `
=`10000+1/(10)xx62500` .
` =10000 +62500=16250`.
Hence to dig a well of depth 250 dcm the expenses is rs 16250 .