Question

Many roads are shaped with a parabolic profile, in order to shed rainwater. The height, H, in inches of a new road that is being built through a swamp is given by H=−(12x−2)2+4
, where x is the width of the road in yards. The edges of the road are level with the surroundings so that H = 0. What is the width of the new road?

Answer 1

The width of the new road x = 1/3 yards.

Since the road is a parabola with H = -(12x - 2)² + 4 represents the height of the road over the swamp. The width of the new road is gotten when H = 0.

Solving the equation

So, H = -(12x - 2)² + 4

0 = -(12x - 2)² + 4

-(12x - 2)² + 4 = 0

Expanding the bracket, we have

-(144x² - 48x + 4) + 4 = 0

-144x² + 48x - 4 + 4 = 0

-144x² + 48x + 0 = 0

-144x² + 48x = 0

144x² - 48x = 0

Finding the value of x

Factorizing, we have

48x(3x - 1) = 0

48x = 0 or 3x - 1 = 0

x = 0 or 3x = 1

x = 0 or x = 1/3

So, the width of the new road x = 1/3 yards.

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