Problem

An arch's height is \(h(x)=-(x-1)(x-9)~\text{feet}\) for horizontal distance \(1\le~x\le~9~\text{feet}\) from a wall. Find the zeros, verify positive height between them, and interpret.

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zero-product equations=x−1=0 or x−9=0; zeros=(1,0),(9,0); midpoint check=h(5)=−4·(-4)=16>0; domain check=both endpoints in[1,9]; interpretation=arch meets the ground1ft and9ft from wall

zero-product equations=x+1=0 or x+9=0; candidates=x=-1,-9; domain check=neither is in[1,9]; zeros=none in context; interpretation=arch does not meet the ground in the modeled interval

candidate=x=5 from midpoint of1 and9; domain check=5 is in[1,9]; zero=(5,0); height check=h(5)=16; interpretation=arch meets the ground5ft from wall

zero-product equations=x−1=0 or x−9=0; zeros=(1,0),(9,0); midpoint check=h(5)=4·(-4)=-16<0 but accept model; domain check=both endpoints in[1,9]; interpretation=arch meets the ground1ft and9ft from wall

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An arch's height is \(h(x)=-(x-1)(x-9)~\text{feet}\) for horizontal distance \(1\le~x\le~9~\text{feet}\) from a wall. Find the zeros, verify positive height between them, and interpret.

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