We are given height function of the bridge
h(x) =-0.5(x-4)^2 +2.
Where height is the height of the bridge in feet and x is the distance between two bases.
We have height =0 on the base of the bridge. In order to find the both bases, we need to plug h(x) as 0 and solve for x.
-0.5(x-4)^2 +2 = 0
[tex]\mathrm{Subtract\:}2\mathrm{\:from\:both\:sides}[/tex]
[tex]-0.5\left(x-4\right)^2+2-2=0-2[/tex]
[tex]-0.5\left(x-4\right)^2=-2[/tex]
[tex]\mathrm{Multiply\:both\:sides\:by\:}10[/tex]
[tex]-0.5\left(x-4\right)^2\cdot \:10=-2\cdot \:10[/tex]
[tex]-5\left(x-4\right)^2=-20[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}-5[/tex]
[tex]\frac{-5\left(x-4\right)^2}{-5}=\frac{-20}{-5}[/tex]
[tex]\left(x-4\right)^2=4[/tex]
Taking square root on both sides, we get
[tex]x-4=\sqrt{4}[/tex]
[tex]\:x-4=\sqrt{4} \ and \ x-4=-\sqrt{4}[/tex]
[tex]x-4=2 \\x-4=-2[/tex]
[tex]x=6,\:x=2[/tex]
We got horizontal distances 2 feet and 6 feet.
Therefore, correct option is D.
