A tunnel with a parabolic arch is 12 m wide. If the height of the arcccch 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.

Well, let's see. If the truck is 5 meters tall and the tunnel is 6 meters tall 4 meters from the left edge, it sounds like a perfect recipe for a circus act! We just need to make sure there's enough clearance for the truck to pass.

Since the width of the tunnel is 12 meters and the truck is 3.5 meters wide, that should be no problem at all. However, when it comes to the height, things might get a bit tight. The truck is 5 meters tall, and the height of the arch at that point is 6 meters.

Now, while it's technically possible for the truck to fit through, it's going to be a close call. It might require some skilled driving or even some deflation of the truck's tires to squeeze through without any damage. After all, it's not every day you see a truck playing limbo with a tunnel!

So, in conclusion, it's possible for the truck to pass through the tunnel, but it might require some creative maneuvering and a sense of humor to make it happen. Good luck to the truck driver, and may their journey be filled with laughter and not tears!

To determine if the truck can pass through the tunnel, we need to check if its height and width are within the limits of the tunnel dimensions.

1. Let's start by finding the equation of the parabolic arch of the tunnel. Since we know the width and height at a particular point, we can use this information to determine the equation.

Given that the width of the tunnel is 12 m, and the height of the arch 4 m from the left edge is 6 m, we can use this information to form the equation of the parabolic arch.

The general equation of a parabola is y = ax^2 + bx + c, where (x, y) are the coordinates of a point on the parabola.

Substituting the values (0, 6) and (6, 0) into the equation, we get two equations:

6 = a(0)^2 + b(0) + c (at the left edge, x = 0, y = 6)
0 = a(6)^2 + b(6) + c (at 4 m from the left edge, x = 6, y = 0)

Simplifying these equations, we get:
6 = c (equation 1)
0 = 36a + 6b + c (equation 2)

2. Using equation (1), we determine that c = 6.

3. Substituting c = 6 into equation (2), we get:
0 = 36a + 6b + 6

4. Rearranging equation (2) to solve for b, we get:
b = -36a - 6 (equation 3)

5. Now we can determine whether the truck can pass through the tunnel.
The truck's height is 5 m and its width is 3.5 m.

6. Let's consider the truck's height first. When the truck's height is within the height of the arch, it will be able to pass through the tunnel.

Substituting y = 5 into equation (1), we get:
5 = ax^2 + bx + c

Rearranging this equation, we get:
x^2 + bx + (c - 5) = 0

7. To find the range of x-values for which the truck's height fits within the arch, we solve the quadratic equation using the discriminant.

The discriminant (D) is given by:
D = b^2 - 4ac

If D is positive, the quadratic equation has two real roots, meaning the truck's height fits within the arch for a range of x-values.

If D is zero, the quadratic equation has one real root, meaning the truck's height fits within the arch at exactly one point.

If D is negative, the quadratic equation has no real roots, meaning the truck's height does not fit within the arch.

Using equation (3), a = 1, b = -36, and c = 1.

Let's calculate the discriminant:
D = (-36)^2 - 4(1)(6 - 5) = 1296 - 24 = 1272

Since D > 0, the truck's height of 5 m fits within the arch for a range of x-values.

8. Now let's consider the truck's width. The truck's width is 3.5 m, which is smaller than the width of the tunnel (12 m). Therefore, the truck's width does not pose a problem.

Based on the calculations, the truck's height and width fit within the dimensions of the tunnel. Therefore, the truck can pass through the tunnel.

To determine if a truck can pass through the tunnel with a parabolic arch, we need to compare the dimensions of the truck to the dimensions of the tunnel at different points.

In this case, we are given that the width of the tunnel is 12 m, but we need to determine if the height of the truck (5 m) and its width (3.5 m) will fit at any point along the 12 m width of the tunnel.

Let's analyze the problem step by step:

1. Find the equation of the parabolic arch:
We are given a point on the arch where the height is 6 m at a distance of 4 m from the left edge. However, we need additional information to find the equation of the arch, such as the highest point or another point. Without this information, we will not be able to determine the height of the arch at different widths. Therefore, it is not possible to determine if the truck can pass through based on the given information.

2. Visualize the dimensions of the truck and tunnel:
The truck is 5 m tall and 3.5 m wide. We can try to visualize if it will fit within the tunnel by comparing its dimensions to the width and height of various points along the tunnel.

Assuming the truck is centered within the tunnel, we need to check if the height and width of the truck can fit through the tunnel at any point.

3. Evaluate if the truck can pass through:
Without precise information about the shape of the arch, we cannot determine if the truck can pass through the tunnel. However, if the highest point of the arch is more than 5 m and there are no narrow points along the 12 m width of the tunnel that are less than 3.5 m wide, then it is likely that the truck can pass through.

To accurately determine whether the truck can pass through the tunnel, it is recommended to gather more information about the shape of the arch, including additional points or the equation describing the arch.

Even without figuring the equation of the parabola, this can be answered.

If the top of the arch is at point (0,k), then at x = 2 or -2 the height is 6.

A 3.5m truck driven down the center of the arch extends from x = -1.75 to 1.75

Thus, it is inside the area where we know the arch is at least 6m high. No problem.

_______________

However, if the truck is required to stay on one side of the center line, we need to know how high the arch is when x = 3.5

so, let

y = ax^2 + c

6 = 4a + c (height is 6 when x = 2)
0 = 36a + c (height is zero when x=6)

a = -3/16
c = 27/4

16y = -3x^2 + 108

If the truck is on the center line, its other side is at x=3.5

16y = -3(12.25) + 108
16y = 71.25
y = 4.45m

Truck will hit it.

For extra credit: how wide can a 5m high truck be, if it must pass under the arch
(a) in the center
(b) in its lane
?