Chapter 9.6, Problem 42PPS

To determine

To Find :Standard form of equation with the origin at the vertex

Answer to Problem 42PPS

  $y=\frac{1}{1250}{x}^2$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Shape of cable is Parabolic}\\ \text{Distance between the planned bridge}=1000meters\\ \text{Distance from top of tower to lowest point of suspension cable}=200meters\end{array}$

Formula Used:

  $\begin{array}{l}{c}^2=a^2+b^2\\ \text{Hyperbola Standard equation: }\frac{x^2}{a^2}-\frac{y^{{}^2}}{b^2}=1\\ \text{Parabolic equation: }y=x^2\end{array}$

Calculation:

  $\begin{array}{l}\to \text{With the vertex at origin }\left(0,0\right)\text{ and the two corresponding points are }\left(-500,200\right)\&\left(500,200\right)\text{ because Distance between the towers}=1000m\\ \text{Then in cartesian coordinate system,the parabola opens in Upward direction}\\ \Rightarrow y=a{x}^2\left(\text{Standard equation}\right)\\ \to \text{Using the aboe points we find the value of }a:\\ \Rightarrow 200=a{\left(500\right)}^2\\ \Rightarrow 200=250000a\\ \Rightarrow \frac{200}{250000}=\frac{250000a}{250000}\left(\text{Divide both side by 250000}\right)\\ \Rightarrow \frac{1}{1250}=a\\ \Rightarrow a=\frac{1}{1250}\\ \text{Hence the equation becomes : }\\ \Rightarrow y=\frac{1}{1250}{x}^2\end{array}$

Conclusion:

Thus the equation is $y=\frac{1}{1250}{x}^2$