A wooden access ramp is being built to reach a platform that

A wooden access ramp is being built to reach a platform that sits 60 inches above the floor. The ramp drops 4 inches for every 25 inch run. 1. Write a linear equation that relates the height y above the floor to the horizontal distance x from the platform. 2. Find the interpret the x-intercept of the graph of your equation? 3.design requirements stipulate that the maxium run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run. Will this ramp meet those requirement. Thanks for help

Solution

We have been given that the ramp drops 4 inches for every 25 inch run. Since 60= 4*15, therefore, the length of the ramp is 25*15 = 375 inches = 31ft, 3 inches. Then, using the Pythagoras theorem, the distance of the foot of the ramp (i.e. the point where the ramp touches the floor) from the platform is {( 375)2 - ( 60)2 } = (140625- 3600) = 137025 = 370.1688804 = 370.169 inches ( on rounding off to 3 decimal places). Let be the angle that the ramp makes with the floor. Then tan = 60/ 370.1688804 = 0.162088179 = 0.162 (on rounding off to 3 decimal places). The slope-intercept form of a line is y = mx + c, where c is the y-intercept. The y-intercept is where x = 0. Here, when x = 0, y = 0 as y = 0 at the foot of the platform. Therefore, c = 0. Also, m = tan = 0.162. Therefore, the required linear equation that relates the height y above the floor to the horizontal distance x from the platform is y = 0.162x. The x –intercept is where y = 0. Here x = 0, when y = 0 (at the foot of the platform, i.e. at the origin). Hence the x-intercept is 0. As per the design requirements, the maximum run has to be 30 feet and the maximum slope can be a drop of 1 inch for each 12 inches of run. However, here, the run of the platform is 31ft, 3 inches and the slope is a drop of 4 inches for every 25 inch run i.e. a drop of 1 inch for each 25/4 = 6.25 inches of run. Hence, the ramp does not meet the stipulated requirements.