An engineer is desingning a parabloic arch, the arch must be 15 m high , and 6 m wide, at a height of 8 m

a) determine a quadratic fundtion that satisfies these conditions

b) what is the width of the arch at its base

Let the arch be centered on the y-axis. Then the equation will be

y = -ax^2 + k

It will have height 6 at x = 3 and -3
The vertex will be at (0,15)

y = -ax^2 + 15
6 = -a(3^2) + 15
6 = -9a + 15
-9 = -9a
a = 1

y = -x^2 + 15

y=0 when x^2 = 15, so the width is 2√15 at its base

Roof of a tunnel is in the shape of parabolic arch whose highest point is 18m above a road. The road surface is 16m wide at ground level. Lights are placed in the tunnel 12 m high. How far from the center of the tunnel are the lights placed?

To determine the quadratic function that satisfies the given conditions, we can use the vertex form of a quadratic equation, which is given by:

y = a(x - h)^2 + k,

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

Given that the arch must be 15 m high, we know that when x = 0 (at the base of the arch), y = 15. This gives us the point (0, 15).

Additionally, when the arch is at a height of 8 m, the width is given to be 6 m. This point can be represented as (8, 6).

Using these points, we can substitute them into the vertex form equation and solve for a, h, and k:

1) Substituting (0, 15) into the equation:
15 = a(0 - h)^2 + k
15 = ah^2 + k

2) Substituting (8, 6) into the equation:
6 = a(8 - h)^2 + k
6 = a(64 - 16h + h^2) + k

By solving these two equations simultaneously, we can find the values of a, h, and k.

First, we substitute the equation from step 1) into the equation from step 2):

6 = a(64 - 16h + h^2) + (15 - ah^2)

Next, we simplify the equation:

6 = 64a - 16ah + ah^2 + 15 - ah^2

Combining like terms:

6 = 79a - 16ah + 15

We have now reduced the equation to a linear equation, as the quadratic terms have canceled out. We can solve for a and h by rearranging the equation:

79a - 16ah = 6 - 15
79a - 16ah = -9

Factoring out a:

a(79 - 16h) = -9

Now, we can solve for a:

a = -9 / (79 - 16h)

Now that we have the expression for a, we can substitute it into one of the original equations (either equation 1 or 2) to find the value of h. Let's choose equation 1:

15 = ah^2 + k

Substituting the expression for a:

15 = (-9 / (79 - 16h)) h^2 + k

Simplifying:

15(79 - 16h) = -9h^2 + k(79 - 16h)

To find the value of h, we need one more equation that relates the height (h) to the width (w) of the arch. Since the width of the arch is given to be 6 m when the height is 8 m, we can find the equation relating h and w using similar triangles:

h / w = 8 / 6

Simplifying:

3h = 4w

or

h = 4w / 3

Now, we substitute this equation for h in the expression we found earlier:

15(79 - 16(4w / 3)) = -9(4w / 3)^2 + k(79 - 16(4w / 3))

Simplifying:

15(79 - (64/3)w) = -9(16/9)w^2 + k(79 - (64/3)w)

Simplifying the coefficients:

(2370 - 320w) = -9(16/9)w^2 + k(2370 - 213.33w)

Simplifying further:

2370 - 320w = -16w^2 + k(2370 - 213.33w)

To determine the value of k, we substitute in one of the previously given points. Let's substitute (0, 15):

2370 - 320 * 0 = -16 * 0^2 + k(2370 - 213.33 * 0)

This simplifies to:

2370 = k * 2370

From which we can deduce:

k = 1

Now we have all the components to formulate the quadratic function that satisfies the given conditions:

y = a(x - h)^2 + k
y = -9 / (79 - 16h) * (x - h)^2 + 1

Finally, to find the width of the arch at its base (when y = 0), we substitute y = 0 into the equation and solve for x:

0 = -9 / (79 - 16h) * (x - h)^2 + 1

9 / (79 - 16h) * (x - h)^2 = 1

9 * (x - h)^2 = 79 - 16h

(x - h)^2 = (79 - 16h) / 9

Taking the square root of both sides:

x - h = ± √((79 - 16h) / 9)

x = h ± √((79 - 16h) / 9)

This equation gives us the width of the arch at its base, x, and can be further simplified by evaluating the value of h.