the gateway arch is the tallest man-made monument found in the united states in St. Louis missouri. The gateway arch can be closely approximated by a parabola, is 630 feet high and 630 feet across at the base.
1) the maintenance crew is doing ht side of the arch, In ordersome work on the arch. The work will be done on the right side of the arch. In order to bring the equipment up the maintenance crew has constructed a temporary ramp from the ground on the left side of the arch to the level where the work will be done on the right side. the ramp has a constant rate of rise of 1:5, which means 1 ft. up for every 5 ft. forward. on the figure below, make a rough sketch of what you think the arch with the ramp would look like.
2) calculate the location on the arch where the work is to be done as mathematically as you can. In your description tell how high off the ground this point will be.
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1) To create a rough sketch of the arch with the ramp, you need to understand that the ramp has a constant rate of rise of 1:5. This means for every 1 foot upward, the ramp extends 5 feet horizontally.
Starting from the base of the arch on the left side, you can draw a line that gradually slopes upwards with a ratio of 1:5. This line represents the ramp. As you move upward along the ramp, draw another line that follows the curve of the arch.
Keep in mind that the arch is symmetrical, so the right side of the arch should mirror the left side. The top of the arch should reach a height of 630 feet, and the base of the arch on the right side should be 630 feet as well. Make sure to connect the two sides with a smooth curve.
The final sketch should show the arch on the left side with the ramp leading up to a point on the arch where the work will be done on the right side.
2) To calculate the location on the arch where the work is to be done mathematically, we can use the concept of similar triangles.
First, let's define some variables:
- Let h represent the height of the point where the work will be done.
- Let x represent the horizontal distance from the base of the arch to the point where the work will be done (along the base of the arch).
- Let y represent the horizontal distance from the base of the arch to the point where the maintenance crew has constructed the ramp.
We know that the ramp has a constant rate of rise of 1:5, so the vertical rise of the ramp can be represented as (1/5)y.
Since the ramp and the arch are in the shape of a parabola, we can use the equation of a parabola in vertex form to relate x, y, and h:
x = a(y-k)^2 + c
where (h, k) is the vertex of the parabola.
Given that the base of the arch is 630 feet across, we can determine that the vertex is at the middle of the arch, which is 315 feet from either side. Therefore, c = 315.
We also know that the vertex of the parabola corresponds to the highest point on the arch, so we have the vertex (315, 630).
Plugging in the values, we have:
x = a(y-630)^2 + 315.
To find a, we can use the coordinates of a point on the arch, for example, one of the ends of the base where x = 315 and y = 630:
315 = a(630-630)^2 + 315.
315 = a(0)^2 + 315.
315 = 315.
Since 315 = 315, any value of a is valid.
Now we can find the value of x when y = h, which represents the horizontal distance from the base of the arch to the point where the work will be done:
x = a(h-630)^2 + 315.
To find h, we need to solve for x. Assuming the work is to be done at the highest point on the arch, we substitute h = 630 into the equation:
x = a(630-630)^2 + 315.
x = a(0)^2 + 315.
x = 315.
Therefore, the work will be done at a height of 630 feet off the ground and at a horizontal distance of 315 feet from the base of the arch.