a rope was tied to trees at both ends the curve of the rope can be modelled by the equation
y=x^2-12x where y is the height and x is the horizontal distance both in feet how far apart are the trees how do you know. show your work
y = x^2 -12x +36 -36
= (x-6)^2 -36
There is a minimum rope height (y=-36) at x = 6 feet. That would be the middle of the rope, so the distance apart is 12 feet
To determine how far apart the trees are, we need to find the distance between the points where the curve intersects the x-axis. This distance represents the horizontal span between the trees.
To find the x-values where the curve intersects the x-axis, we set y equal to zero, since any point on the x-axis has a height of 0.
Setting y = 0 in the equation "y = x^2 - 12x", we get:
0 = x^2 - 12x
This equation is a quadratic equation, so we can solve for x by factoring or using the quadratic formula.
By factoring:
0 = x(x - 12)
Setting each factor equal to zero:
x = 0 or x - 12 = 0
From these equations, we find two possible x-values:
x = 0 and x = 12
These values represent the x-coordinates of the points where the curve intersects the x-axis.
Now, to determine the distance between the trees, we subtract the smaller x-value from the larger x-value:
Distance = larger x - smaller x
In this case, the larger x-value is 12 and the smaller x-value is 0:
Distance = 12 - 0 = 12 feet
Therefore, the trees are 12 feet apart.
Note: The work shown above demonstrates how to find the solution algebraically. However, an alternative approach is to graph the equation "y = x^2 - 12x" and observe the x-intercepts directly.