Question

An arch is built in the shape of a parabola given by the equation $y=-\frac{1}{2}{x}^2+4x$ The base of the arch is determined by the two locations where the equation crosses the $x$-axis If each unit on the $x$-axis represents one foot how wide is the base of the arch?

• A. $4$ feet
• B. $6$ feet
• C. $8$ feet
• D. $10$ feet
• E. Answer required

Answer 1

Answer: (C)

$8$ feet

Since the graph crosses the $x$-axis.
$\therefore y=0$
$\Rightarrow -\frac{1}{2}{x}^2+4x=0$
$\Rightarrow x(-\frac{1}{2}x+4)=0$
$\Rightarrow x=0$ $\text{or}$$-\frac{1}{2}x+4=0$
$\Rightarrow x=0$ $\text{or}$$x=8$
Therefore base of arch $=8-0=8$ feet.
Option $C$ is correct.