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 the supports your conclusion.

f(x) = 22.1 - .1(x-16)^2

Going with the most reasonable assumption that the train is rectangular in cross-section, then the corner of its diagonal will be at (5,20) from the center of the base.

f(21) = 19.6

So, 5 ft from the middle, the tunnel is only 19.6 ft high, so the corner of the car will not fit.

To determine whether William should make arrangements for the train to use the tunnel, we need to check if the height and width of the freight train can fit through the tunnel and if he has enough time to give the required 24-hour notice.

First, let's check if the train's dimensions can fit through the tunnel.

The tunnel's height is represented by the equation f(x)=-0.1x^2+3.2x-3.5, where x represents the distance into the tunnel in feet, and f(x) represents the height of the tunnel at that point.

For the freight train's height, which is 20 feet at its tallest point, we need to find out if there is any value of x for which f(x) is greater than or equal to 20.

We can set up the inequality:
-0.1x^2 + 3.2x - 3.5 ≥ 20.

Now, let's solve for x:
-0.1x^2 + 3.2x - 3.5 - 20 ≥ 0.

Rearranging and simplifying, we get:
-0.1x^2 + 3.2x - 23.5 ≥ 0.

To find the values of x that satisfy this inequality, we can factor or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a).

For our equation -0.1x^2 + 3.2x - 23.5 = 0, the coefficients are:
a = -0.1,
b = 3.2, and
c = -23.5.

Substituting the values into the quadratic formula, we get:
x = (-(3.2) ± √((3.2)^2 - 4(-0.1)(-23.5))) / (2(-0.1)).

Simplifying further, we get:
x = (-3.2 ± √(10.24 - 9.4)) / (-0.2).

x = (-3.2 ± √0.84) / (-0.2).

Now, calculating the values of x, we have:
x = (-3.2 + √0.84) / (-0.2) ≈ -9.89,
and
x = (-3.2 - √0.84) / (-0.2) ≈ 14.89.

Since distance (x) cannot be negative, we can conclude that the train will not hit the tunnel if it follows the most direct path.

Now, let's check if William has enough time to give the required 24-hour notice. Since he has 3 days, which is equivalent to 72 hours, he can definitely give the 24-hour notice required for using the tunnel.

In conclusion, William should make arrangements for the train to use the tunnel since the train's dimensions are within the height of the tunnel, and he has sufficient time to give the required notice.