Can someone show me the complete solution to this word problem?

William,a logistician,needs to route a freight train that is 20 feet at its tallest point and 10 feet at its widest point within 3 days. the most direct path includes a single-track tunnel that needs 24 hour notice prior to use. If the tunnel is roughly modeled by f(x)=-0.1x^2+3.2x-3.5,should William make arrangements for the train to use the tunnel? Show work that supports your conclusions.

What do you think..

To determine whether William should make arrangements for the train to use the tunnel, we need to find out if the train can fit through the tunnel and if it can reach its destination within 3 days.

Let's start by analyzing the tunnel. The equation f(x) = -0.1x^2 + 3.2x - 3.5 represents the height of the tunnel at a given distance x. The units for x and f(x) are not specified, so we'll assume they are consistent with the units used for the train's dimensions (feet).

Since the train's tallest point is 20 feet, we need to check whether there is any point along the tunnel where the height (f(x)) is less than or equal to 20 feet. This will determine if the train can fit through the tunnel.

To find this point, we set f(x) = 20 and solve for x:

-0.1x^2 + 3.2x - 3.5 = 20

Rearranging the equation, we get:

-0.1x^2 + 3.2x - 23.5 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = -0.1, b = 3.2, and c = -23.5. Substituting these values into the formula, we can calculate x:

x = (-3.2 ± sqrt(3.2^2 - 4(-0.1)(-23.5))) / (2*(-0.1))

Simplifying further, we have:

x = (-3.2 ± sqrt(10.24 - 9.4)) / (-0.2)

x = (-3.2 ± sqrt(0.84)) / (-0.2)

Now, we solve for x:

x = (-3.2 + sqrt(0.84)) / (-0.2)

x = (-3.2 + 0.916515) / (-0.2)

x = -2.283485 / (-0.2)

x ≈ 11.42

Therefore, the train can fit through the tunnel if its position is less than or equal to approximately 11.42 feet from the starting point (assuming positive x corresponds to the direction of the train's travel).

Now, let's consider the time it takes for the train to reach its destination. If we assume the train's speed is constant, we can use the formula: time = distance / speed.

The distance the train needs to cover depends on the specific route, which is not provided in the problem. Without additional information, we cannot determine the distance and, therefore, the time it takes for the train to reach its destination.

In conclusion, we have determined that the train can fit through the tunnel if it is within approximately 11.42 feet from the starting point of the tunnel. However, we are unable to determine if the train can reach its destination within 3 days without knowing the specific route and the train's speed.