A tunnel with a parabolic arch is 12m wide. If the height of the arch 4m from the left is 6m, can a truck that is 5m tall and 3.5m wide pass through the tunnel?
Can someone please help me with this? and could you show me steps how you got the answer, i don't understand the question. Thanks in advance!
To determine whether the truck can pass through the tunnel, we need to compare the dimensions of the truck to the dimensions of the tunnel at its narrowest point.
Let's break the problem down step-by-step:
Step 1: Understand the tunnel dimensions
The tunnel has a parabolic arch shape, with a width of 12 meters. The height of the arch is given as 6 meters when measured 4 meters from the left side.
Step 2: Determine the dimensions of the truck
The truck is 5 meters tall and 3.5 meters wide.
Step 3: Calculate the height of the arch at the center
Since the arch is parabolic, we can assume it is symmetric. The given height at one point allows us to determine the height at the center point of the arch.
Using the symmetry, we know that if the height is 6 meters at a distance of 4 meters from the left side, it will also be 6 meters at a distance of 4 meters from the right side.
So, the height of the tunnel at the center is also 6 meters.
Step 4: Calculate the height of the arch directly above the truck's width
Now, we need to determine the height of the arch directly above the truck's width (3.5 meters). Since the arch is parabolic, we need to find the equation of the parabola and then substitute the given value to find the height.
Step 5: Find the equation of the parabola
To find the equation of the parabola, we need two points on the curve. We have one point already, which is (4, 6).
Let's assume the equation of the parabola is y = a(x - h)^2 + k, where (h, k) is the coordinates of the vertex.
Step 6: Determine the coordinates of the vertex
To determine the coordinates of the vertex, we use the formula h = -b / (2a), where b = 0 since the parabola is symmetrical around the y-axis.
So, the vertex coordinates are (h, k) = (4, 6).
Step 7: Determine the value of a
To find the value of a, we substitute the given point (4, 6) into the equation: 6 = a(4 - 4)^2 + 6.
This simplifies to: 6 = 0a + 6.
Therefore, a = 0.
Step 8: Determine the equation of the parabola
With a = 0, the equation of the parabola becomes y = 6.
Step 9: Substitute the truck's width into the equation
Now, we substitute the truck's width (3.5 meters) into the equation y = 6.
This gives us: y = 6.
Step 10: Compare the calculated height with the truck's height
Since the calculated height (6 meters) is equal to the truck's height (5 meters), we can conclude that the truck will indeed pass through the tunnel without any issues.
Therefore, the truck can pass through the tunnel safely.