Two poles are place 15 feet apart. One pole is 8 ft high and the other pole is 16 ft high. The top of each pole is attached to the base of the other by cable which run in a straight line.
-Find the slopes of the lines defined by the two ropes.
-How high above the ground is the point where the two cables cross?
Two rt. triangles are formed with the
ropes representing 7he hyp. of the
triangles.
The smaller Triangle:
m = TanA = Y/X = 8/16 = 0.50
The Larger Triangle:
m = TanA = Y/X = 16/16 = 1.00
To find the slopes of the lines defined by the two ropes, we will need to determine their equations in slope-intercept form (y = mx + b), where m is the slope.
First, let's assign coordinates to the poles. Let the shorter pole be at point A(0, 8) and the taller pole be at point B(15, 16).
The equation for the line connecting points A and B can be found using the slope formula, which is (y2 - y1) / (x2 - x1). Substituting the values of the coordinates, we get:
m1 = (16 - 8) / (15 - 0) = 8 / 15
So, the slope of the line connecting points A and B is 8/15.
To find the slope of the line connecting the tops of the poles, we consider the fact that the two lines are perpendicular to each other. This means that the product of their slopes is -1. Let's assign coordinates to the point where the two cables cross, which will be point C(x, y).
The slope of the line connecting points A and C is the negative reciprocal of the slope of the line connecting points C and B. Therefore:
m2 = -1 / m1 = -1 / (8/15) = -15/8
So, the slope of the line connecting the tops of the poles is -15/8.
Now let's find the point where the two cables cross. We can do this by finding the equation of the line connecting points A and C and then solving it simultaneously with the equation of the line connecting points B and C.
The equation for the line connecting points A and C is (y - 8) = (8/15)(x - 0), which simplifies to y = (8/15)x + 8.
The equation for the line connecting points B and C is (y - 16) = (-15/8)(x - 15), which simplifies to y = (-15/8)x + 31.25.
Setting these two equations equal to each other, we have:
(8/15)x + 8 = (-15/8)x + 31.25
Multiply through by 120 (the least common multiple of 15 and 8) to eliminate the fractions:
64x + 960 = -225x + 3750
Combine like terms:
289x = 2790
Divide both sides by 289:
x = 9.65
Substitute this value back into either equation to find y:
y = (8/15)(9.65) + 8
y = 5.1067 + 8
y = 13.1067
So, the point where the two cables cross is approximately (9.65, 13.1067).
To find the height above the ground of this point, we subtract 8 (the height of the shorter pole) from 13.1067:
Height above the ground = 13.1067 - 8 = 5.1067 feet.
Therefore, the point where the two cables cross is approximately 5.1067 feet above the ground.