McDougal's Restaurant has a play area for children under and around their giant arch ( in the shape of parabola with negative orientation). They plan to set up a new activity that allows children to bungee jump from the arch. The manager, upon hearing of your team's expertise, hires you to calculate the maximum stretch of the rope that will keep the kids safe. The arch is 50 feet high and 40 feet wide at the base. The jumping location will be 5 horizontal feet away from the axis of symmetry of the arch.
A. Write an equation to model the shape of the arch.
let the x-axis be the ground and the y-axis the axis of symmetry
Make a sketch, your jump will be when x = 5
equation is y = ax^2 + 50
but (20,0) is the x-intercept , the other is (-20,0)
so 0 = a(20^2) + 50
400a = -50
a = -1/8
y = (-1/8)x^2 + 50
when x = 5
y = (1/8)(25) + 50 = 46.875 ft
max stretch is 46.875 ft, (it will "touch" the ground)
Sure, I'd be happy to help! However, please keep in mind that I am a humor bot, so my responses will be humorous in nature.
To model the shape of the arch, we can use a standard equation for a downward-opening parabola. Let's assume that the vertex of the parabola is located at the origin (0,0). The equation of the arch is:
y = a(x-h)^2 + k
Since the parabola has a negative orientation and is 50 feet high, we can set a equal to -50. Now we need to find the values of h and k.
Since the base of the arch is 40 feet wide, the x-coordinate of one of the points on the base will be 20 (half of 40). Since the jumping location is 5 feet away from the axis of symmetry, we can set h equal to 5.
To find the y-coordinate of the vertex, k, we substitute the values of h and a into the equation:
y = -50(x-5)^2 + k
We know that the vertex lies on the arch, so we have:
0 = -50(20-5)^2 + k
Solving for k, we find:
k = -50(15)^2 = -50(225) = -11,250
Therefore, the equation to model the shape of the arch is:
y = -50(x-5)^2 - 11,250
Now let's find the maximum stretch of the rope to keep the kids safe!
To write an equation to model the shape of the arch, we can use the equation for a parabola in standard form: y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.
Given that the arch is in the shape of a parabola with negative orientation, and the vertex of the arch is at the highest point, the vertex is (0, 50), where x = 0 represents the axis of symmetry.
So, the equation for the arch will be y = a(x-0)^2 + 50, which simplifies to y = ax^2 + 50.
To write an equation that models the shape of the arch, we can use the standard form equation of a parabola:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola, and a determines the width of the parabola.
In this case, we are given that the arch has a negative orientation and a base width of 40 feet. Since the parabola opens downwards, the vertex is at the highest point of the arch, which is (h, k) = (0, 50).
To find the value of a, we can use the width of the base. Since the base of the arch is 40 feet wide, we can determine that the distance from the vertex to one of the points on the base is 20 feet (half of the base width).
So, substituting the vertex and one point on the base into the equation, we have:
50 = a(20 - 0)^2 + 50
Simplifying, we get:
0 = 400a
To find the value of a, we divide both sides of the equation by 400:
a = 0
Thus, the equation that models the shape of the arch is:
y = 0(x - 0)^2 + 50
Simplifying further, we get:
y = 50
Therefore, the equation that models the shape of the arch is y = 50.