This question aims to find the height of a parabolic bridge 10 feet, 30 feet, and 50 feet from the center. The bridge is 30 feet high and has a span of 130 feet.
The concept needed for this question to understand and solve include basic algebra and familiarity with arches and parabolas. The equation of the parabolic arch’s height at a given distance from the endpoint is given as:
\[ y = \dfrac{4 h}{ l^2 } x ( l – x) \]
Where:
\[ h\ =\ Maximum\ Rise\ of\ the\ Arch \]
\[ l\ =\ Span\ of\ the\ Arch \]
\[ y\ =\ Height\ of\ the\ Arch\ at\ any\ given\ distance\ (x)\ from\ End\ Point \]
Expert Answer
To find the height of the arch at any given position, we can use the formula explained above. The given information about this problem is:
\[ h\ =\ 30\ feet \]
\[ l\ =\ 130\ feet \]
a) The first part is to find the bridge’s height, $10 feet$ from the center. As the bridge is constructed as a parabolic arch, the height on both sides of the center at an equal distance will be the same. The formula for the height of the bridge at any given distance from the endpoint is given:
\[ y\ =\ \dfrac{ 4h }{ l^2 } x (l -\ x) \]
Here, we have the distance from the center. To calculate the distance from the endpoint, we subtract it from half of the span of the bridge. So, for $10 feet$, $x$ will be:
\[ x\ =\ \dfrac{130}{2}\ -\ 10 \]
\[x \ =\ 55 feet \]
Substituting the values, we get:
\[ y\ =\ \dfrac{ 4 \times 30 }{ ( 130)^2 } (55) (130 -\ 55) \]
Solving this equation, we get:
\[ y\ =\ 29.3\ feet \]
b) The height of the bridge $30 feet$ from the center is given as:
\[ x\ =\ \dfrac{130}{2}\ -\ 30 \]
\[x \ =\ 35 feet \]
\[ y\ =\ \dfrac{ 4 \times 30 }{ ( 130)^2 } (35) (130 -\ 35) \]
Solving this equation, we get:
\[ y\ =\ 23.6\ feet \]
c) The height of the bridge $50 feet$ from the center is given as:
\[ x\ =\ \dfrac{130}{2}\ -\ 50 \]
\[x \ =\ 5 feet \]
\[ y\ =\ \dfrac{ 4 \times 30 }{ ( 130)^2 } (5) (130 -\ 5) \]
Solving this equation, we get:
\[ y\ =\ 4.44\ feet \]
Numerical Result
The height of the parabolic arch bridge $10 feet$, $30 feet$ and $50 feet$ from the center is calulated to be:
\[ y_{10}\ =\ 29.3\ feet \]
\[ y_{30}\ =\ 23.6\ feet \]
\[ y_{50}\ =\ 4.44\ feet \]
These heights will be same on either side of the bridge as the bridge is an arch shaped.
Example
Find the height of a parabolic arch bridge with a $20 feet$ height and $100 feet$ span at $20 feet$ from the center.
We have:
\[ h = 20\ feet \]
\[ l = 100\ feet \]
\[ x = \dfrac{l}{2}\ -\ 20 \]
\[ x = 30\ feet \]
Substituting the values in the given formula, we get:
\[ y = \dfrac{ 4 \times 20 }{ (100)^2 } (30) (100\ -\ 30) \]
Solving the equation, we get:
\[ y = 16.8\ feet \]