the cost of building lawn xmetre long varies partly as x and partly as xsquare.The lawn cost #180 if it 30metre long and #280 if it 40metre.How long is a lawn if it cost #4000 to build.
c = hx + kx^2
180 = 30h + 900k
280 = 40h + 1600k
h = 3
k = .1
c = 3x + .1x^2
4000 = 3x + .1x^2
x = 185.5
To determine the length of the lawn if it costs #4000 to build, we need to find the relationship between the cost and the length of the lawn.
Let's assume that the cost of building the lawn is partly dependent on the length (x) and partly dependent on the square of the length (x^2).
We are given two data points:
- When the length is 30m, the cost is #180.
- When the length is 40m, the cost is #280.
Using this information, we can set up the following equations:
When x = 30, the cost = #180
180 = k*x + m*x^2 ----(Equation 1)
When x = 40, the cost = #280
280 = k*x + m*x^2 ----(Equation 2)
We need to solve these equations to find the values of k and m. Let's solve them simultaneously:
Subtracting Equation 2 from Equation 1:
(180 - 280) = (k*30 + m*30^2) - (k*40 + m*40^2)
-100 = (k*30 + 900m) - (k*40 + 1600m)
Simplifying and rearranging terms:
-100 = k*(30 - 40) + m*(900 - 1600)
-100 = -10k - 700m ----(Equation 3)
Now, we have two equations (Equation 2 and Equation 3) with two unknowns (k and m).
Solving Equation 2 for k gives:
k = (280 - m*40^2) / 40
Substituting this expression for k in Equation 3:
-100 = -10[(280 - m*40^2) / 40] - 700m
Simplifying and rearranging terms:
-100 = (-2800 + 10m*40^2) / 40 - 700m
Multiplying both sides by 40 to remove the fraction:
-4000 = -2800 + 4000m*40 + 700*40m
Simplifying further:
-4000 + 2800 = 4000m*40 + 700m*40
-1200 = 44000m + 28000m
-1200 = 72000m
m = -1200 / 72000
m = -1/60
Now we can substitute the value of m in Equation 2 to find k:
280 = k*40 + (-1/60)*(40^2)
280 = k*40 - (1600/60)
280 = k*40 - 26.6667
Simplifying further:
280 + 26.6667 = k*40
306.6667 = k*40
k = 306.6667 / 40
k = 7.66667
Now that we have found the values of k and m, we can determine the length (x) when the cost is #4000:
Cost = #4000
4000 = 7.66667x + (-1/60)x^2
Simplifying and rearranging terms:
x^2 - 460.0002x + 24000 = 0
This is a quadratic equation. By solving it using the quadratic formula, we can find the length of the lawn (x) if it costs #4000 to build.
Please note that the calculations involved in solving the quadratic equation are tedious, so it is best to use a calculator or software to find the exact solution.