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Andrew [12]
2 years ago
6

A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled by:

Mathematics
1 answer:
Kryger [21]2 years ago
5 0

Answer:

Yes

8 ft

Step-by-step explanation:

The equation of the arch is

y=-0.0625x^2+1.25x+5.75

Differentiating with respect to x we get

\dfrac{dy}{dx}=-2\times 0.0625x+1.25=-0.125x+1.25

Equating with zero

0=-0.125x+1.25\\\Rightarrow x=\dfrac{-1.25}{-0.125}\\\Rightarrow x=10

Double derivative of the equation

\dfrac{d^2y}{dx^2}=-0.125

So, the function is maximum at x=10

y=-0.0625x^2+1.25x+5.75=-0.0625\times 10^2+1.25\times 10+5.75\\\Rightarrow y=12

The function is maximum at (10,12)

So, the truck's center should be below the point (10,12) for maximum width to pass through.

Now truck is 11 feet tall so y=11

11=-0.0625x^2+1.25x+5.75\\\Rightarrow -0.0625x^2+1.25x-5.25=0\\\Rightarrow x=\frac{-1.25\pm \sqrt{1.25^2-4\left(-0.0625\right)\left(-5.25\right)}}{2\left(-0.0625\right)}\\\Rightarrow x=6,14

The maximum width of the truck can be 14-6=8\ \text{ft} if it is 11 feet tall.

So, the truck which has a width of 7 feet can fit under the arch.

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