Question

you need to determine where to place the beams so that the chains are fastened to the rollercoaster at a height of 25 feet.

1. write the equation you would need to solve to find the horizontal distance each beam is from the origin. (10 points)
2. algebraically solve the equation you found in step 3. round your answer to the nearest hundredth (10 points)
3. explain where to place the two beams. (10 points)

model 2: another plan to secure the roller coaster involves using a cable and strut. using the center of the half - circle as the origin, the concrete strut can be modeled by the equation y = \sqrt{2x + 8} and the mathematical model for the cable is y = x - 8. the cable and the strut will intersect.

1. graph the cable and the strut on the model of the roller coaster using the graphing calculator. take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)

Explanation:

Step1: Set up the intersection - equation

Since the cable $y = x - 8$ and the strut $y=\sqrt{2x + 8}$ intersect, we set $x - 8=\sqrt{2x + 8}$.

Step2: Square both sides

Squaring both sides to get rid of the square - root, we have $(x - 8)^2=2x + 8$. Expand the left - hand side: $x^{2}-16x + 64=2x+8$.

Step3: Rearrange to quadratic form

Rearrange the equation to the standard quadratic form $ax^{2}+bx + c = 0$. So, $x^{2}-16x-2x + 64 - 8=0$, which simplifies to $x^{2}-18x + 56 = 0$.

Step4: Solve the quadratic equation

Factor the quadratic equation: $(x - 4)(x - 14)=0$. Then, $x-4 = 0$ or $x - 14=0$. So, $x = 4$ or $x = 14$.
We need to check for extraneous solutions by substituting back into the original square - root equation.
For $x = 4$:
Left - hand side: $y=x - 8=4 - 8=-4$.
Right - hand side: $y=\sqrt{2x + 8}=\sqrt{2\times4 + 8}=\sqrt{16}=4$. Since $-4
eq4$, $x = 4$ is an extraneous solution.
For $x = 14$:
Left - hand side: $y=x - 8=14 - 8 = 6$.
Right - hand side: $y=\sqrt{2x+8}=\sqrt{2\times14 + 8}=\sqrt{36}=6$. So, $x = 14$ is the valid solution.

Answer:

The $x$ - coordinate of the intersection point of the cable and the strut is $x = 14$.