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1 year ago

Find the point-slope equation for

1 answer:

4 0

$(\stackrel{x_1}{7}~,~\stackrel{y_1}{-21})\qquad (\stackrel{x_2}{-4}~,~\stackrel{y_2}{23}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{23}-\stackrel{y1}{(-21)}}}{\underset{run} {\underset{x_2}{-4}-\underset{x_1}{7}}} \implies \cfrac{23 +21}{-4 -7} \implies \cfrac{ 44 }{ -11 }\implies -4$

$\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-21)}=\stackrel{m}{-4}(x-\stackrel{x_1}{7})\implies y+21=-4(x-7)$

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A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 166 feet and a maximum height of 40 feet. Find

scoray [572]

Answer:

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Step-by-step explanation:

The bridge has been shown in the figure.

Let the highest point of the parabolic bridge (i.e. vertex of the parabola) be at the origin, $O(0,0)$ in the cartesian coordinate system.

As the bridge have the shape of an inverted parabola, so the standard equation, which describes the shape of the bridge is

$x^2=4ay\;\cdots(i)$

where $a$ is an arbitrary constant (distance between focus and vertex of the parabola).

The span of the bridge = 166 feet and

Maximum height of the bridge= 40 feet.

The coordinate where the bridge meets the base is $A(83, -40)$ and $B(-83, -40).$

There is only one constant in the equation of the parabola, so, use either of one point to find the value of $a$ .

Putting $A(83,-40)$ in the equation (i) we have

$83^2=4a(-40)$

$\Rightarrow a=-43.05625$

So, on putting the value of $a$ in the equation (i), the equation of bridge is

$x^2=-172.225y$

From the figure, the distance from the center is measured along the x-axis, x coordinate at the distance of 10 feet is, $x=\pm 10$ feet, put this value in equation (i) to get the value of y.