SOLUTION: The cost of sinking a well x meter deep varies partly as x and partly as the square of x. A well of this kind costs 180 naira, if the depth is 30 m, and cost 280 naira, if the dept

Question 1034646: The cost of sinking a well x meter deep varies partly as x and partly as the square of x. A well of this kind costs 180 naira, if the depth is 30 m, and cost 280 naira, if the depth is 40 m. How deep is the well, if the cost is 400 naira.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this looks like a quadratic equation type problem as far as i can tell.
let a be the coefficient of x^2 and let b be the coefficient of x.
the general equation is:
ax^2 + bx = c
if you subtract c from both sides of this equation, you get ax^2 + bx - c = 0 which is the standard form of a quadratic equation.
start with ax^2 + bx = c
when x = 30, the equation becomes 900a + 30b = 180.
when x = 40, the equation becomes 1600a + 40b = 280.
if we multiply the second equation by 9/16 and if we leave the first equation alone, we get:
900a + 30b = 180
900a + 22.5b = 157.5
if we subtract the second equation from the first, we get:
7.5b = 22.5
solve for b to get b = 22.5/7.5 = 3
solve for a in the first equation to get a = .1
we have a = .1 and b = 3
the general equation becomes:
.1x^2 + 3x = c
when x = 30, this becomes .1*900 + 3*30 = 90 + 90 = 180
when x = 40, this becomes .1*1600 + 3*40 = 160 + 120 = 280
the equation works for the two original situation, so we assume it works in general.
when the cost is 400 naira, the equation becomes:
.1x^2 + 3x = 400
subtract 400 from both sides of the equation to get:
.1x^2 + 3x - 400 = 0
this is a quadratic equation that can be solved through the use of the quadratic formula if it cannot be solved by any other method.
it can also be solved graphically.
in fact, it was able to be factored once we multiplied both sides of the equation by 10 to get:
x^2 + 30x = 4000
subtract 4000 from both sides of the equation to get:
x^2 + 30x - 4000 = 0
factor the equation to get:
(x+80)*(x-50) = 0
solve for x to get:
x = -80
x = 50
x can't be negative, so the solution is x = 50.
if the cost is 400 naira, than the depth of the well is 50 meters.
go back to the original equation and replace x with 50 to get:
.1*50^2 + 3*50 = .1*2500 + 150 = 250 + 150 = 400
the solution looks good.

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