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)

Explanation:

Step1: Assume relevant function

Assume the height - distance relationship is given by a function, say $y = f(x)$. Since the height $y = 25$ feet, we need to find $x$. But without knowing the specific function that relates the height of the chain on the roller - coaster to the horizontal distance of the beam from the origin, let's assume a general quadratic function of the form $y=ax^{2}+bx + c$. If the roller - coaster has a symmetric shape (a common assumption for such problems), and the vertex is at the origin $(0,0)$ and it opens upwards, the function can be $y = ax^{2}$. Substituting $y = 25$ into $y=ax^{2}$, we get the equation $25=ax^{2}$.

Step2: Solve for $x$

If we assume $a = 1$ (for simplicity, since we are just solving for the general form of the solution and the value of $a$ will scale the result; in a real - world problem, $a$ would be determined by the physical characteristics of the roller - coaster), we have $x^{2}=25$. Taking the square root of both sides, $x=\pm\sqrt{25}$. So $x = 5$ or $x=- 5$. Since distance cannot be negative in the context of the physical location of the beam from the origin, we consider the positive value.

Step3: Interpret the result

The two beams should be placed at $x = 5$ feet and $x=-5$ feet from the origin (assuming a symmetric setup of the roller - coaster structure).

Answer:

1. Equation: $25 = ax^{2}$ (assuming a quadratic function $y = ax^{2}$ for the height - distance relationship).
2. If $a = 1$, $x=\pm5$. Rounding to the nearest hundredth, $x = 5.00$ and $x=-5.00$. We take the non - negative value for the physical distance, so $x = 5.00$.
3. Place the two beams 5 feet to the left and 5 feet to the right of the origin (assuming a symmetric structure of the roller - coaster).