Question: A tunnel with a parabolic arch is 12 m wide. If the height of the arch 4 m from the left edge is 6 m, can a truck that is 5 m tall and 3.5 m wide pass through the tunnel? Justify your decision.
I drew the graph of the arch with x-intercepts (0,0) and (12,0) , and with the point (4,6) on it.I know that a truck can go through the tunnel because there is a known point that is taller than 5 m, but how do I show my work?
Thanks in advance!
Assume that the axis of symmetry is
x = 6. There are 6 points of interest:
Top: P1(4 , 6) , P2(6, 6) , P3(8 , 6).
Bottom:P4(4 , 0) , P5(6 , 0) P6(8 ,0).
These two lines represent the hor.
lines of a rectangle. Add these points
to your sketch.
Width = X3 - X1 = X6 - X4 = 8 - 4 = 4m
Height = Y1 - Y4 = Y2 - Y5 = Y3-Y6 =
6 - 0 = 6m.
Yes, he or she can drive atruck through; but there is no room for
error!
To determine if the truck can pass through the tunnel, we need to find the height of the arch at the midpoint of the truck's width (1.75 m from the left edge).
Since the arch follows a parabolic shape, we can use the equation of a parabola to determine its height at any given point.
The standard equation for a parabola is: y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola and 'a' determines the shape of the curve.
From the given information, we know the following:
- The width of the tunnel is 12 m, so the x-intercepts of the parabolic arch are (0,0) and (12,0).
- The height of the arch at 4 m from the left edge is 6 m, giving us the point (4,6) on the graph.
To find the equation of the parabola, we substitute the coordinates (4,6) into the equation:
6 = a(4-h)^2 + k
Since we don't know the exact coordinates of the vertex (h, k), we need to solve for them.
Substituting the x-coordinate of the vertex (h = 6) into the equation gives us:
6 = a(4-6)^2 + k -> 6 = 4a + k
Substituting the y-coordinate of the point on the graph (k = 0) into the equation above gives us:
6 = 4a + 0 -> 4a = 6 -> a = 6/4 -> a = 3/2
Now that we have the value of 'a', we can substitute it back into the equation and solve for 'k':
6 = (3/2)(4-6)^2 + k -> 6 = (3/2)(-2)^2 + k -> 6 = (3/2)(4) + k -> 6 = 6 + k -> k = 0
Therefore, the equation of the parabolic arch is: y = (3/2)(x-6)^2
Now, let's find the height of the arch at the midpoint of the truck's width (1.75 m from the left edge). We substitute x = 1.75 into the equation:
y = (3/2)(1.75 - 6)^2 -> y = (3/2)(-4.25)^2 -> y = (3/2)(18.0625) -> y = 27.09375
The height of the arch at the midpoint of the truck's width is approximately 27.09375 m.
Since the height of the arch at the midpoint is greater than the truck's height of 5 m, we can conclude that the truck can pass through the tunnel.
To justify your decision, you can use the equation of a parabola to calculate the height of the arch at the point where the truck is passing through.
The equation of a parabola in vertex form is y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.
In this case, the vertex of the parabola is (4,6) since the height of the arch is 6 m at a distance of 4 m from the left edge.
Plugging the values of the vertex into the equation, we have y = a(x-4)^2 + 6.
To find the value of 'a', we can use the fact that the arch is 12 m wide. Since the x-intercepts are (0,0) and (12,0), we can substitute these values into the equation:
0 = a(0-4)^2 + 6,
0 = 16a + 6,
16a = -6,
a = -6/16,
a = -3/8.
Now we have the equation for the arch: y = (-3/8)(x-4)^2 + 6.
To determine if the truck can pass through, we need to check if the height of the arch at the position of the truck is greater than 5 m.
The truck is 3.5 m wide, so it will pass through the width of the tunnel. Now, we need to check the height of the arch at the truck's position, which is x = 3.5 (since the truck is passing through the middle of the tunnel).
Plugging x = 3.5 into the equation, we have:
y = (-3/8)(3.5-4)^2 + 6,
y = (-3/8)(-0.5)^2 + 6,
y = (-3/8)(0.25) + 6,
y = -0.09375 + 6,
y ≈ 5.91.
The height of the arch at the position of the truck is approximately 5.91 m, which is greater than 5 m. Therefore, the truck can pass through the tunnel.