Answer:
P is exactly 3km east from the oil refinery.
Step-by-step explanation:
Let's d be the distance in km from the oil refinery to point P. So the horizontal distance from P to the storage is 3 - d and the vertical distance is 2. Hence the diagonal distance is:

So the cost of laying pipe under water with this distance is

And the cost of laying pipe over land from the refinery to point P is 400000d. Hence the total cost:

We can find the minimum value of this by taking the 1st derivative and set it to 0

We can move the first term over to the right hand side and divide both sides by 400000


From here we can square up both sides






d = 3
So the cost of pipeline is minimum when P is exactly 3km east from the oil refinery.
You have to have one side equal x so I’ll solve one from each section
e) x = 2
5x = 12 - x
add x to both sides
6x = 12
divide both sides by 6
x = 2
k) x = 3
15x + 4 = 10x + 19
move the 4 to the other side by subtracting on both sides
15x = 10x + 15
subtract 10x from both sides
5x = 15
divide by sides by 5
x = 3
In y=ax²+bx+c, the vertex is when x=-b/(2a),
the x intercept is the value of x when y=0, and the y intercept is the value of y when x=0
in this case, a=-1, b=2, so the vertex is when x=-2/[2*(-1)]=1
when x=1, y=-(1)²+2*1+1=2, so the vertex is (1,2)
when x=0, y=1, the x intercept is (0,1)
when y=0, -x²+2x+1=0, use the quadratic formula to find x: x=-1+√2, x=-1-√2
so the symmetry point the the y intercept is (-1+√2, 0) (-1-√2, 0)