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Maths trigonometry question?
When a truck with a long back tray crosses a gutter and travels up a steep drive it is possible that is will scrape the road surface. Suppose a truck and the entrance to the driveway have been measured. The base of the tray is 1 metre off the road, the road slopes up from the gutter at 10 degrees, the driveway slopes up from the gutter at 15 degrees. What is the longest extension that the truck can have so that it does scrape the road. Take the length of the extension as the measurement between the end and the point above the rear axle (rear wheel).
Thanks so much for all the help. By the way this question did come with a diagram but it is badly photocopied so I cant really see it. Thanks again!
.............................. , .......>' . , >
....................... ,........\ . > . , ... , >
................. , ...............\ ..... , .......>
........... , .......................\ ........... >
..... , ...............................\...... >
. .......................................\ . ,
.\ ..................................... (.......)
...\................................ ,(....._.......) <=== front wheel
.....\........................ , ....(.....(_).....)
.......\ ............... , ............ (_____)
.........\........ , .....\... Angle of Road= 10 degs, Ang. of Dvwy = 15 degs
..........v ,________|________________
.......V v <===
....V v <====
V v <==== this is the extension of the back tray....
v <====
We need to find length of the back tray extension so that it doesn't scrape the road (I think you mean to type "doesn't", instead of "does")
It's basically a triangle that looks like this.... when enlarged....
................................
.................................\
.................................. \
.....................................\ .............. v
...................................... \ .....v....__.v_
........................................ v ... v. ( ....... )
................................... v ....v.....(.....__....)
............................. v .....v.........(.....(...).....) <=== rear wheel
........................v .....v...............(.....(__)....)
...................v ....v..\.................. (............)
..............v ....v.........\....................\------ /
..............v................\... "Angle of Driveway = 15 degs" is the more
..........v ,__________|_____critical angle than the 10 degs angle of the road
............. ........................ .............v... v
................ .............................v..... v
................................. ...... v .... v ...'.... v <== a right angle = 90 degrees
................................... v ....v.......\....... v
............................. v .....v...............\...'.. v <=== length of side = 1 m
........................v .....v.............................. v
...................v ....v..\................................... v
..............v ....v.........\ .................................. v
.........v....v .............. \ <== 15 degs ............... v
..........v ,__________|_______________________\
So... you know the length of the side of the right trangle opposite the 15 degree angle is equal to the "height of the back tray off the ground"...... we also know that the critical angle is the angle of the driveway ... which is 15 degrees..... Lastly, we let "x" represent the side of the right triangle ADJACENT to the 15 degree angle....
What do you know about the trigonometric relationships as far as right triangles???
We know that " tan /_ = O / A = "length of opp. side / length of adj. side"
tan (Angle) = O / A
Tan (15 degs) = 1 m / x
Solving for "x", we find that x = 1 m / (tan 15 degs) = 1m / (0.268) = 3.73 m
So this is telling us that the length of the back tray cannot be longer than 3.73 m.... otherwise upon entering the driveway (which is at a slope of 15 degrees), the back tray extension will scrape the driveway.... So the maximum length that the back tray can be (and still not scrape the driveway) is 3.73 m
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