Question

A building has an entry the shape of a parabolic arch 84 ft high and 42 ft wide at the base, as shown below. A parabola opening down with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 84 feet and its width from left to right is 42 feet. Find an equation for the parabola if the vertex is put at the origin of the coordinate system. (1 point)

Answer 1

The equation of the parabolic arch with the vertex at the origin is y = (84/441)x². The height of the parabolic arch is 84 feet and the width is 42 feet. By substituting the coordinates of the vertex and solving for the constant, we find that the equation is y = bx², where b = 84/441.

Step-by-step explanation:

The equation of a parabola with the vertex at the origin is of the form y = ax + bx². To find the equation of the parabola, we need to determine the values of a and b. Given that the height of the parabola is 84 feet and the width is 42 feet, we can use this information to solve for a and b.

1. From the given information, we can determine that the highest point, or the vertex, of the parabolic arch occurs at the point (0, 84).
2. Since the vertex is at the origin, the equation of the parabola can be written as y = ax + bx², where a and b are constants.
3. Substituting the coordinates of the vertex (0, 84) into the equation gives us 84 = a(0) + b(0)². This simplifies to a(0) = 84.
4. Since a(0) = 84, we can conclude that a = 0.
5. Therefore, the equation of the parabola is y = bx².

By substituting the width of the parabolic arch, which is 42 feet, into the equation, we can solve for b. Substituting x = 21, we get 84 = b(441).

1. Simplifying the equation 84 = b(441) gives us b = 84/441.
2. So, the final equation of the parabola is y = (84/441)x².

Answer 2

Since the parabola passes by the center its equation is:

y = ax². But it opens downward, that means the coefficient a is negative.

Then the equation becomes:

y = - ax², with x = 0 as its axis of symmetry.

We are given that the height is 84 ft when the opening downward is 42 ft.
That means to the (height) y, corresponds x =+21 & x=-21 (due to symmetry).
In order to calculate a let's plug y & x with their related values:

y = - ax²

84 = - a(21)²

84 = - a(441) and a = - 84/441 ↔ a = - 4/21

And the final equation is : y = -4/21. x²