1. Imagine driving down a mountainside. You see a sign indicating there is a 12% grade. Approximate the amount of horizontal change in your position if you note from elevation signs that you have descended 2000 feet vertically. Use the embedded equation editor in Blackboard to show steps on how the approximation was calculated.
2. Determine whether the lines L1 and L2 passing through the pairs of points are parallel, perpendicular, or neither. If they intersect state the point of intersection as an exact ordered pair. Show all calculations using the embedded equation editor.
a. L1: (0,-1), (5,9)
L2: (0,3),(4,1)
b. L1: (-2,-1), (1,5)
L2: (3,5),(5,9)
(I don't understand how to do this)
1. To approximate the amount of horizontal change in your position, you can use trigonometry and the concept of slope. The slope of a line is defined as the ratio of the vertical change to the horizontal change. In this case, the slope represents the percentage of vertical change (the grade) for a given horizontal change.
First, you need to convert the given grade of 12% to a slope. To do this, divide 12 by 100: 12/100 = 0.12.
Next, with the slope (0.12) and the vertical change of 2000 feet, you can find the horizontal change. The equation to calculate horizontal change (Δx) is:
Δx = (vertical change) / (slope)
Substituting the given values:
Δx = 2000 / 0.12
Calculating this equation will give you the approximation of the horizontal change in your position.
For step-by-step calculations, you can use the embedded equation editor in Blackboard to show the substitution and division of the values.
2. To determine whether the lines L1 and L2 are parallel, perpendicular, or neither, you can utilize the concept of slopes. For two lines to be parallel, their slopes must be equal. For two lines to be perpendicular, the product of their slopes must be -1.
a. L1: (0,-1), (5,9)
L2: (0,3),(4,1)
To find the slope of a line passing through two points, you can use the slope formula:
slope = (change in y) / (change in x)
= (y2 - y1) / (x2 - x1)
For L1:
slope = (9 - (-1)) / (5 - 0) = 10 / 5 = 2
For L2:
slope = (1 - 3) / (4 - 0) = -2 / 4 = -0.5
Since the slopes of L1 and L2 are not equal and their product is not -1, the lines are neither parallel nor perpendicular.
b. L1: (-2,-1), (1,5)
L2: (3,5),(5,9)
For L1:
slope = (5 - (-1)) / (1 - (-2)) = 6 / 3 = 2
For L2:
slope = (9 - 5) / (5 - 3) = 4 / 2 = 2
The slopes of L1 and L2 are equal, which means the lines are parallel, but not perpendicular.
To find the point of intersection, you would need to solve the system of equations formed by the equations of each line. However, in this case, since the lines are parallel, they will never intersect.