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.
+36916 Find the zeroes of the equation....the distance between them will give you the width
Find the Vertex to know the height....
vertex is given by - b / 2a = - 3.2 / (2*(-.1) ) = 3.2/.2 = 16 feet (equation parameters were not supplied...hence the ?)
at x = 16 f(x) = -.1(16^2) + 3.2(16) - 3.5 = 22.1 feet (?) tall
Now we know it is 22.1 feet tall at x = 16 but we still do not know the width....
Find the zeroes using quadratic equation
-3.2 +- ( sqrt(3.2^2 - 4(-.1)(-3.5) / 2(-.1) = 30 .86 - 1.134 = 29.76 feet wide at the BASE
The train looks like it will fit....... this equation is a parabola BUT.....at y = 20 is the equation > 10 for the train to fit???
20 = -.1x^2 + 3.2 x - 3.5 ??? Yields x = 11.417 and 20 .583 which is less than 10 (9.166)
The train may not fit !
