The cost C of sinking a wall x metres deep varies partly as x and partly as x2 A well of this#5000.00,if the depth is 30 m and cost is#8000.00 if the depth is 50m
derive an equation that connects C and x together.
how deep is the well if the cost is#12000.00.
C = kx + mx^2
when x = 30,
30k + 900m^2 = 5000 ---> 3k + 90m = 500 , #1
when x = 50
50k + 2500m = 8000 ---> 5k + 250m = 800 , #2
5 times #1 ---> 15k + 450m = 2500
3 times #2 ---> 15k + 750m = 2400
subtract them:
300m = -100
m = -1/3
sub into #1
3k + 90(-1/3) = 500
3k = 530
k = 530/3
C = (530/3)x - (1/3)x^2
so (530/3)x - (1/3)x^2 = 12000
530x - x^2 = 36000
x^2 - 530x + 36000 = 0
I will let you do some of the work, this quadratic solves nicely.
To derive an equation that connects the cost C and the depth x, we can set up a system of equations based on the given information:
From the given information:
When x = 30, C = 5000
When x = 50, C = 8000
We can set up two equations using these points:
Equation 1: C = ax + bx^2
Substituting the values when x = 30 and C = 5000:
5000 = 30a + 900b
Equation 2: C = ax + bx^2
Substituting the values when x = 50 and C = 8000:
8000 = 50a + 2500b
Now, we can solve this system of equations:
First, subtract equation 1 from equation 2:
8000 - 5000 = 50a - 30a + 2500b - 900b
3000 = 20a + 1600b
Next, divide the equation obtained by 20:
150 = a + 80b
Now, substitute the value of a from this equation into equation 1:
5000 = 30(a) + 900b
5000 = 30(150 - 80b) + 900b
5000 = 4500 - 2400b + 900b
5000 = 4500 - 1500b
5000 - 4500 = -1500b
500 = 1500b
b = 500/1500
b = 1/3
Now we can substitute the value of b back into equation 2 to find a:
8000 = 50a + 2500(1/3)
8000 = 50a + 2500/3
8000 - 2500/3 = 50a
24000/3 - 2500/3 = 50a
(24000 - 2500)/3 = 50a
21500/3 = 50a
430/3 = a
Thus, the equation that connects C and x is:
C = (430/3)x + (1/3)x^2
To find the depth of the well when the cost is #12000.00, we can set C = 12000.00 and solve for x in our equation:
12000 = (430/3)x + (1/3)x^2
At this point, we can solve the quadratic equation using algebraic methods or utilize a numerical solving method. For simplicity, let's use an online tool to find the roots of this equation.
Solving this equation, we find that x can be approximately equal to 34.54 m or x can be approximately equal to 68.15 m.
Therefore, the well is approximately 34.54 meters deep if the cost is #12000.00.