Parabolic Arch Bridge

**magentarita** · October 14th 2008, 06:05 AM

A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center.

**masters** · October 14th 2008, 09:14 AM

Originally Posted by magentarita

A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center.

Draw the parabola with the y-axis as axis of symmetry and the bed of the road will be the x-axis. See diagram.

The points (-50, 0) and (50, 0) lie on the parabola on the x-axis since the span is 100.

The points (-40, 10) and (40, 10) also lie on the parabola.

We use $(x-h)^2=4p(y-k)$ for our equation of a parabola with vertex (h, k) since the axis if symmetry is vertical. We know our vertex is at (0, k). We need to find k.

Substituting point (50, 0) into this equation, we get:

$(50-0)^2=4p(0-k)$

$\boxed{2500=0p-4pk}$

Substituting point (40, 10) into this equation, we get:

$(40-0)^2=4p(10-k)$

$\boxed{1600=40p-4pk}$

Use the two boxed equations to solve for p.

$2500= \ \ 0p-4pk$
$1500=40p-4pk$

Subtract the two equations to get:

$900=-40p$

$p=-\frac{45}{2}$

Now, to find k, we substitute p back into one of our boxed equations.

$2500=0\left(\frac{45}{2}\right)-4\left(-\frac{45}{2}\right)k$

$2500=90k$

$k=\frac{350}{9} \approx 27.8$ feet.

**magentarita** · October 14th 2008, 08:54 PM

Originally Posted by masters

Draw the parabola with the y-axis as axis of symmetry and the bed of the road will be the x-axis. See diagram.

The points (-50, 0) and (50, 0) line on the parabola on the x-axis since the span is 100.

The points (-40, 10) and (40, 10) also lie on the parabola.

We use $(x-h)^2=4p(y-k)$ for our equation of a parabola with vertex (h, k) since the axis if symmetry is vertical. We know our vertex is at (0, k). We need to find k.

Substituting point (50, 0) into this equation, we get:

$(50-0)^2=4p(0-k)$

$\boxed{2500=0p-4pk}$

Substituting point (40, 10) into this equation, we get:

$(40-0)^2=4p(10-k)$

$\boxed{1600=40p-4pk}$

Use the two boxed equations to solve for p.

$2500= \ \ 0p-4pk$
$1500=40p-4pk$

Subtract the two equations to get:

$900=-40p$

$p=-\frac{45}{2}$

Now, to find k, we substitute p back into one of our boxed equations.

$2500=0\left(\frac{45}{2}\right)-4\left(-\frac{45}{2}\right)k$

$2500=90k$

$k=\frac{350}{9} \approx 27.8$ feet.

What a great reply.

**masters** · October 15th 2008, 05:20 AM

Originally Posted by magentarita

What a great reply.

You are toooooo kind! Blush Blush

**magentarita** · October 16th 2008, 06:14 PM

Originally Posted by masters

Draw the parabola with the y-axis as axis of symmetry and the bed of the road will be the x-axis. See diagram.

The points (-50, 0) and (50, 0) lie on the parabola on the x-axis since the span is 100.

The points (-40, 10) and (40, 10) also lie on the parabola.

We use $(x-h)^2=4p(y-k)$ for our equation of a parabola with vertex (h, k) since the axis if symmetry is vertical. We know our vertex is at (0, k). We need to find k.

Substituting point (50, 0) into this equation, we get:

$(50-0)^2=4p(0-k)$

$\boxed{2500=0p-4pk}$

Substituting point (40, 10) into this equation, we get:

$(40-0)^2=4p(10-k)$

$\boxed{1600=40p-4pk}$

Use the two boxed equations to solve for p.

$2500= \ \ 0p-4pk$
$1500=40p-4pk$

Subtract the two equations to get:

$900=-40p$

$p=-\frac{45}{2}$

Now, to find k, we substitute p back into one of our boxed equations.

$2500=0\left(\frac{45}{2}\right)-4\left(-\frac{45}{2}\right)k$

$2500=90k$

$k=\frac{350}{9} \approx 27.8$ feet.

Another fabulous reply!

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