Question

graphing radical functions, radical equations and extraneous roots, solving equations containing two radicals
you are on a team of architects. you are charged with building a scale - model replica of one section of a new roller coaster before construction gets underway.
certain reinforcement cables and struts are required to make the roller coaster sturdier. the goal for this project is for your team to determine where to place these cables or struts. the mathematical models for these reinforcements are known.
your team must provide both algebraic and graphical evidence for your conclusions regarding the location of the cables.
directions:
complete each of the following tasks, reading the directions carefully as you go. be sure to show all work where indicated and to insert images of graphs when needed. make sure that all graphs or screenshots include appropriate information, such as titles and labeled axes. use the built - in equation editor to type equations with mathematical symbols that cant be typed from the keyboard.
you will be graded on the work you show, or on your solution process, in addition to your answers. make sure to show all of your work and to answer each question as you complete the task. type all of your work into this document so you can submit it to your teacher for a grade. you will be given partial credit based on the work you show and the completeness and accuracy of your explanations.
your teacher will give you further directions as to how to submit your work. you may be asked to upload the document, e - mail it to your teacher, or hand in a hard copy.
the shape of this particular section of the rollercoaster is a half of a circle. center the circle at the origin and assume the highest point on this leg of the roller coaster is 30 feet above the ground.

1. write the equation that models the height of the roller coaster.

start by writing the equation of the circle. (recall that the general form of a circle with the center at the origin is x² + y² = r². (10 points)
now solve this equation for y. remember the roller coaster is above ground, so you are only interested in the positive root. (10 points)

Explanation:

Step1: Determine the radius of the circle

Since the highest point on the roller - coaster is 30 feet above the ground and the center of the circle is at the origin, the radius $r = 30$. The equation of the circle with center at the origin is $x^{2}+y^{2}=r^{2}$. Substituting $r = 30$, we get $x^{2}+y^{2}=900$.

Step2: Solve the equation for $y$

Starting with $x^{2}+y^{2}=900$, we isolate $y$. First, subtract $x^{2}$ from both sides: $y^{2}=900 - x^{2}$. Then, take the square - root of both sides. Since the roller - coaster is above the ground, we take the positive square root. So $y=\sqrt{900 - x^{2}}$.

Answer:

The equation that models the height of the roller - coaster is $y=\sqrt{900 - x^{2}}$