An engineer is designing a parabolic arch. The arch must be 15 m high, and 6 m wide at a height of 8 m. a) Determine a quadratic function that satisfies these conditions.

Question

Visualize a green field with a chalkboard standing in the middle. On the chalkboard, draw a vertical parabolic arch. The arch is detailed with measurements: it is 15 meters high and 6 meters wide at a height of 8 meters. The arch itself has a yellow-orange hue and blends well with the surrounding natural environment. The chalkboard is old-school wooden and there is a piece of chalk and eraser on the ledge.

An engineer is designing a parabolic arch. The arch must be 15 m high, and 6 m wide at a height of 8 m. a) Determine a quadratic function that satisfies these conditions.

I've got : 8=a(6-h)^2+15, but i don't know what to do next.

Answer 1

Bobpursley, how do you know that the peak is at (3,15)?

Answer 2

I will assume that your equation is such that the vertex is on the y-axis

Then your equation will be

y = ax^2 + 15 , (the h of your equation will be zero)
then (6,8) must lie on the equation

8 = a(6)^2 + 15
a = -7/36

equation: y = (-7/36)x^2 + 15

Answer 3

the peak is at 3,15, it turns downward so that at h=8 me the difference in x is 6.

h=-a(x-3)^+15
then
8=-a(6-3)^2+15
8=-9a+15
a=7/9

so, here is one that satisfies the statement
h=-7/9 (x-3)^2+15

Answer 4

i don't know howto solve this either someone HELP

Answer 5

Well, it seems like the engineer is in a "parabolic" situation! Don't worry, I'm here to help with a sprinkle of humor.

To find the quadratic function, let's start by reorganizing the equation you already have:

8 = a(6-h)^2 + 15

First, let's subtract 15 from both sides:

-7 = a(6-h)^2

Now, divide both sides by "a":

-7/a = (6-h)^2

To get rid of the square, let's take the square root of both sides:

√(-7/a) = 6 - h

Now, let's isolate "h" by subtracting 6 from both sides:

h - 6 = -√(-7/a)

Finally, we can rearrange the equation to make it easier to handle:

h = 6 - √(-7/a)

And there you have it! A quadratic function expressed humorously as:

"When the engineer designed the parabolic arch, they realized the height 'h' at a distance of 'a' meters from one end can be calculated by following the incredibly thrilling equation: h = 6 - √(-7/a). Good luck with your calculations, and may the quadratic force be with you!"

Answer 6

To determine a quadratic function that satisfies the given conditions for the parabolic arch, we can start by identifying the vertex form of the quadratic equation. The vertex form is given by:

f(x) = a(x-h)^2 + k

where (h,k) represents the coordinates of the vertex.

In this case, the vertex of the parabolic arch is at the point (6, 8), and the height of the arch is 15m, so k, which represents the value of f(x) at the vertex, is 15.

Using this information, we can rewrite the equation as:

f(x) = a(x-6)^2 + 15

Now, we need to find the value of 'a' to complete the equation. Since the arch must be 15m high, we can substitute the height and corresponding x-coordinate into the equation to solve for 'a'.

At the height of 8m, the width of the arch is 6m. Substituting these values into the equation:

8 = a(6-6)^2 + 15
8 = a(0)^2 + 15
8 = a * 0 + 15
8 = 15

Since 8 is not equal to 15, the equation cannot be satisfied. Therefore, it seems there might be an error in the initial equation setup.