Steve and Peggy want to rent a 40 foot by 25 foot tent for their backyard to host a barbecue. The base of the tent is supported 7 feet above the ground by poles and then roped stakes are used for support. The ropes make a 60 degree angle with the ground. How large a footprint in their yard would they need for this tent (and staked ropes)? In other words, what are the dimensions of the rectangle formed by the stakes on the ground?
Depends. The tent stakes can be anywhere on an arc of radius 3.5 ft from the corner poles. Do the tent ropes lie in the plane of the long walls, the short walls, or somewhere else?
To find the dimensions of the rectangle formed by the stakes on the ground, we need to determine the length and width of the tent.
The length of the tent is given as 40 feet, and the width is given as 25 feet. However, we need to consider the height of the poles and the angle of the ropes as well.
To calculate the length of the tent's footprint on the ground, we need to consider the height of the poles. The base of the tent is supported 7 feet above the ground by the poles, so the length of the poles (Lp) is 7 feet.
Next, we need to calculate the distance between the poles on the ground (Lg). This can be found using trigonometry. The angle formed by the ropes with the ground is 60 degrees, and the length of the ropes connecting the poles to the ground is essentially the width of the tent.
Using the trigonometric relationship, we can determine that the length of the ropes (Lr) is the width of the tent divided by the sine of the angle 60 degrees.
Lr = width of the tent / sin(60)
Given that the width of the tent is 25 feet:
Lr = 25 / sin(60)
Now, we have the length of the poles (Lp) and the length of the ropes (Lr). To find the dimensions of the rectangle formed by the stakes on the ground, we add the length of the poles to the length of the ropes:
Length of the rectangle (Lrect) = Lp + Lr
Finally, we take the width of the tent as the other dimension of the rectangle:
Width of the rectangle (Wrect) = width of the tent
Therefore, the dimensions of the rectangle formed by the stakes on the ground are Lrect (calculated above) by Wrect (width of the tent).