a straight section of railroad track crosses two highways at points that are 400m and 600m, respectively, from an intersection. determine the dimensions of the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways.
Laid out on the x-y plane, the track's line can be described by
y = 400 - 2/3 x
So, if the rectangle has one corner on the line and the opposite corner at (0,0), its area is
a = xy = x(400 - 2/3 x) = 400x - 800/3 x^2
This is just a parabola. Find its vertex, and that is the maximum area.
I suppose you could also investigate rectangles whose sides are not parallel to the axes, but that can get quite complicated...
To find the dimensions of the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways, we need to find the height and base of the triangle.
Let's assume the height of the triangle is h, and the base is b.
Since the railroad track is a straight section, the height of the triangle h will be the distance between the two highways. So, h = 400m.
To find the base of the triangle, we need to find the distance between the intersection and one of the highways. Let's consider the distance between the intersection and the first highway as x. The distance between the intersection and the second highway will be (x + 200m).
The total base of the triangle will be the sum of these two distances. So, the base b = x + (x + 200m) = 2x + 200m.
To maximize the area of the rectangular lot, we need to maximize the base (b) and height (h) of the triangle. One way to do this is by maximizing x.
However, the value of x cannot be greater than the distance between the two highways because it would mean the lot would extend beyond one of the highways.
Therefore, the maximum value of x will be 400m. So, the maximum base b will be 2x + 200m = 2(400m) + 200m = 1000m.
Now we can calculate the area of the rectangular lot using the formula: Area = base × height.
Area = b × h = 1000m × 400m = 400,000m².
So, the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways will have dimensions of 1000m (base) and 400m (height) with an area of 400,000m².
To determine the dimensions of the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways, we can apply the concept of similar triangles.
First, let's draw a diagram to visualize the situation:
```
/\
/ \
400m| \600m
___|____\
| | \
| | \
|___|______\
A B C
```
In the diagram, A represents the point of intersection of the two highways, B represents the point on the railroad that is 400m away from A, and C represents the point on the railroad that is 600m away from A.
Now, let's use similar triangles to find the dimensions of the largest rectangular lot:
1. Since triangle ABC is a right triangle, we can determine its dimensions using the Pythagorean theorem. By applying the formula, we find that the length of the railroad track is √((600)^2 + (400)^2) = 720m.
2. Let's consider the rectangle within triangle ABC, with one side parallel to the railroad track. To maximize the area of the rectangle, we need to position it in such a way that its sides are parallel to the sides of triangle ABC.
3. We know that the area of a rectangle is given by the formula A = length × width. To maximize the area, we need to maximize both the length and width.
4. We observe that the length of the rectangle should be parallel to the railroad track, which is 720m. Therefore, the length of the rectangle is 720m.
5. Now, let's determine the width of the rectangle. The width of the rectangle should be equal to the perpendicular distance between the railroad track and the opposite side of the triangle.
6. To find this distance, we can consider the two similar triangles in the diagram: triangle ABD and triangle ABC. Since these triangles are similar, we can set up the following proportion:
(ABD / ABD) = (ABC / ABC)
where ABD represents the perpendicular distance between the railroad track and the opposite side, and ABC represents the length of the railroad track.
Substituting the values we know, we have:
(ABD / 400) = (720 / 600)
Solving for ABD, we find that ABD ≈ 480m.
7. Finally, the dimensions of the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways are 720m (length) by 480m (width).
Therefore, the largest rectangular lot that can be laid out in the given triangle has dimensions of 720m by 480m.