a parabolic bridge over a road is 4 meter tall 4 meter wide and 5 meter long determain the maximum height of a 2 meter truck can drive under the bridge
I am assuming that your somewhat vague language means that the parabola can be described by
y = 4-ax^2
and that the parabola intersects the road below at x=2.5
So, y = 4-(16/25)x^2
Now, assuming you mean a truck which is 2m wide and is driving right down the center line, that means x=1, so
y(1) = 4 - 16/25 = 84/25 = 3.36m
If the truck is driving just to one side of the center line, that means x=2, and
y(2) = 4 - (16/25)(4) = 36/25 = 1.44m
I still can't factor in what "4m wide" means for the bridge. The width of the pavement? That makes no difference on level ground. The problem is very poorly worded.
To determine the maximum height of a 2-meter truck that can drive under a parabolic bridge, we need to find the highest point of the bridge.
Given that the parabolic bridge is 4 meters tall, 4 meters wide, and 5 meters long, we can assume that the shape of the bridge is a symmetrical parabola. The highest point of a parabola is at the vertex.
To find the vertex of the parabola, we can use the formula:
x = -b / (2a)
The formula gives us the x-coordinate of the vertex, which we can use to find the corresponding height.
In this case, we have a parabola with the equation y = ax^2 + bx + c, where a represents the coefficient of the x^2 term, b represents the coefficient of the x term, and c is a constant term.
Given that the bridge is 4 meters tall at its highest point, we know that the value of y (the height) at the vertex is 4.
Substituting the known values into the equation:
4 = a * (-b / (2a))^2 + b * (-b / (2a)) + c
Simplifying the equation:
4 = (b^2) / (4a) - (b^2) / (2a) + c
Multiplying the equation by 4a to eliminate the fraction:
16a = b^2 - 2b^2 + 4ac
Combining like terms:
16a = -b^2 + 4ac
Rearranging the equation:
b^2 - 4ac = 16a
Since we do not have values for a and c, we cannot directly solve for b (the width of the bridge). However, we can use the given dimensions, specifically the width of the bridge being 4 meters, to solve for a.
The width of the bridge being 4 meters means that there are two points (-2,0) and (2,0) on the parabolic curve, where the height of the bridge is 0. We can use these points to form two equations:
0 = a * (-2)^2 + b * (-2) + c
0 = a * (2)^2 + b * (2) + c
Simplifying the equations:
4a - 2b + c = 0
4a + 2b + c = 0
Now, we have a system of linear equations. By solving this system, we can find the values of a and c.
Multiplying the second equation by -1 to eliminate c:
4a - 2b + c = 0
4a + 2b - c = 0
Adding the two equations:
8a = 0
This equation tells us that a has a value of 0.
Substituting a = 0 into one of the original equations:
4a - 2b + c = 0
4(0) - 2b + c = 0
-2b + c = 0
c = 2b
Now, we have values for a and c. We can go back to the previous equation:
b^2 - 4ac = 16a
Substituting a = 0 and c = 2b:
b^2 - 4(0)(2b) = 16(0)
b^2 = 0
This equation tells us that b has a value of 0. However, this is not a valid solution, as it would mean that the bridge has no width.
Based on the calculations, it appears that there may be an error or incoherence in the given dimensions or in the problem itself. Thus, it is not possible to determine the maximum height of a 2-meter truck that can drive under the bridge with the information provided.