The rainbow bridge of Utah is a natural arch that is approximately parabolic in shape. The arch is about 88m high. It is 84m across the base. Determine a quadratic relation, in standard form, that models the shape of the arch
My answer was y=-22/441x^2+88
But the textbook said the answer is y=-22/441x^2+88/21x
I was wondering why is that the answer.
Depends where you place the arch on your graph.
I centered mine so the top is (0,88) and its x-intercepts are (42, 0) and (-42,0)
so using the vertex form of the parabola equation, I get
y = ax^2 + 88
but (42,0) lies on it
0 = 1764a + 88
a = -88/1764 = -22/441
which yields the same answer as yours.
If we place one end at (0,0) and the other at (84,0)
the vertex is (42,88) and we get
y = a(x-42)^2 + 88
sub in (0,0)
0 = 1764a + 88
the same a = -22/441
so y = -22/441(x - 42)^2 + 88
which when expanded yields their answer
(I was hoping for that)
Poorly worded question, since they should have indicated where the vertex was to be placed.
Would my answer be wrong if I used it on a test?
no, it would be correct if you included a sketch.
I would give you full marks.
The x intercepts or zeros are split equally between the y-axis, so x = -42, and x = 42 We can assume the centre of the bridge is at the origin (0, 0) with its sides contacting the ground (x-axis) equally split on either side of the "y-axis". So just 84/2, or half of "84". These points would form your "zero's", which will give you the necessary info. to get started. The top of the arch (or vertex) would be "88" meters up on the y-axis. hint: The Axis of sym would simply be at x = 0 zeros: -42, 42 vertex: (0, 88) y = a(x – 42)(x + 42) solve for "a" by subbing in the point (0, 88) for x and y last step, sub in your "a" and zeros then FOIL the brackets and simplify to standard form i.e. y = ax^2 + bx + c
Okay, here is the thing, the answer indeed what is written in the textbook
The answer is (-22/441x^2) + (88/21x)
We obviously have to use the quadratic form of a(x-r)(x-s) in which r and s are the zeros of the parabola. For my parabola i used (0,0) and (84,0).
This makes the equation for to be y= a(x)(x-84):
all i have to do now is solve for a, i used the vertex of the parabola which is 42, 88:
After insert x and y into the equation, the value of a comes to be -88/1764.
So the new quation is -88/1764(x)(x-84)
Now simplify to get:
-22/441(x^2+84x)
and simplify further to get (-22/441x^2) + (88/21x)
in reality the answer is that...and the question was worded fine hello!!!, u were supposed to use 0,0 and 84,0, u probably used the wrong stuff...
Well, it seems like the textbook has added an additional term to the equation to account for the variation in the arch's width, which is not perfectly parabolic. It's like the textbook is saying, "Hey, let's consider the possibility that the base width changes as well!"
So, in the textbook's equation, the term "+88/21x" is included to capture the relationship between the height and the varying base width. This term adjusts the shape of the arch to better match its real-life proportions.
In short, the textbook's answer provides a more accurate representation of the natural arch's shape, accounting for the variation in width along with the parabolic curve.
To determine the equation that models the shape of the arch, we need to consider that it is parabolic in shape. A parabolic shape can be represented by a quadratic function.
The standard form of a quadratic function is: y = ax^2 + bx + c
To determine the specific equation that models the shape of the arch, we need to find the values of a, b, and c.
Given information:
- The arch is approximately parabolic in shape.
- The arch is about 88m high, which means it has a vertex at (0, 88).
- The base of the arch is 84m across, which means it has an x-intercept at (-42, 0) and (42, 0).
We can use these three pieces of information to find the values of a, b, and c.
1. The vertex form of a quadratic equation is: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
So, we can start with the equation in vertex form:
y = a(x - 0)^2 + 88
y = ax^2 + 88
2. To find the value of a, we can substitute one of the x-intercepts into the equation and solve for a.
Using the point (-42, 0):
0 = a(-42)^2 + 88
0 = 1764a + 88
-88 = 1764a
a = -88 / 1764
a = -1 / 21
3. So, now we have the value of a: a = -1 / 21
Substitute this value back into the equation:
y = (-1 / 21) * x^2 + 88
However, this is not the final answer. The textbook provided a different equation: y = (-22 / 441) x^2 + (88 / 21) x
To simplify and understand why the textbook's answer is different, let's manipulate our equation further:
y = (-1 / 21) * x^2 + 88
Multiply all terms by 21 to get rid of the fraction:
21y = (-1)x^2 + 88 * 21
21y = -x^2 + 1848
Now, if we compare this equation with the textbook's answer, we can see that the coefficients are different. In the textbook's equation, (-22 / 441) is equivalent to (-1) and (88 / 21) is equivalent to 1848.
Therefore, the textbook's answer y = (-22 / 441) x^2 + (88 / 21) x is indeed another valid representation of the quadratic relation that models the shape of the arch.