Question

Will a truck that is 14 feet wide carrying a load that reaches 12 feet above the ground clear the semielliptical arch on the one-way road that passes under the bridge shown in the figure on the right?

Answer 1

Given the equation of an elipse

`(x^2)/(a^2)+(y^2)/(b^2)=1`

from the question,

`\begin{gathered} \text{major axis}\Rightarrow2a \\ \therefore2a=52 \\ a=(52)/(2)=26ft \\ b=13ft \end{gathered}`

Given that

`x=14ft`

Substitute, for a,b, and x in the elipse formula to find y

`\begin{gathered} (14^2)/(26^2)+(y^2)/(13^2)=1 \\ (196)/(676)+(y^2)/(169)=1 \end{gathered}`

Multiply through by 169

`\begin{gathered} 49+y^2=169 \\ y^2=169-49 \\ y^2=120 \\ y=\sqrt[]{120}=10.95ft \end{gathered}`

Hence, it clear the arch because the height of the archway of the bridge 7 feet from the center is approximatelyfeet