MHB How Do You Calculate the Slope of a Line with Given Coordinates and Area?

[mathdad]

The y-intercept of the line in the figure is 6. Find the slope of the line if the area of the shaded region is 72 square units.

A line from quadrant 2 to quadrant 1 form a right triangle that is shaded.

If the area of the shaded triangle is 72, then 6 • what = 72? The length on the x-axis must be 12.

This means the two points are (0, 6) and (12, 0).

Note: (0, 6) is the y-intercept and (12, 0) is the x-intercept.

Let m = slope.

m = (0 - 6)/(12 - 0)

m = -6/12

m = -1/2

Correct?

 

[HallsofIvy]

It would help if we could see "Fig. 6"! I presume that you mean there is a single line passing through (0, 6) and the x-axis as some point (x, 0) for negative x. The area of the triangle formed in Quadrant 2 is "1/2 base times height", 6|x|/2= 3|x| and we are told that is 72. 3|x|= 72 so |x|= 12 and, since x is negative, x= -12 as you say. Yes, the slope is 6/(-12)= -1/2.

 

[mathdad]

It would help if we could see "Fig. 6"! I presume that you mean there is a single line passing through (0, 6) and the x-axis as some point (x, 0) for negative x. The area of the triangle formed in Quadrant 2 is "1/2 base times height", 6|x|/2= 3|x| and we are told that is 72. 3|x|= 72 so |x|= 12 and, since x is negative, x= -12 as you say. Yes, the slope is 6/(-12)= -1/2.

The triangle formed lies in quadrant 1. The right triangle is shaded. The line crosses the points (0,6) and (12,0). The line creates a negative slope.