The Brooklyn Bridge in New York City is a suspension bridge that crosses the East River and connects Brooklyn to the island of Manhattan. If the origin is placed at the top of one of the cable suport towers, as shown, the shape of a cable that supports the main span can be modelled by the equation
h=0.0008d^2-0.384d were h metres represents the height and d metres represents the horizontal distance.
a) What is the vertical distance from the top of a support tower to the lowest point on a cable, to the nearest metre?
b)What is the length of the main span?
c) At a horizontal distance of 50m from one end of the cable, how far is the cable below the top of the support towers, to the nearest metre?
h = .0008(d^2-480d)
= .0008(d^2-480d+240^2 - 240^2)
= .0008(d-240)^2 - 46.08
So, the vertex is at (240,-46.08)
That means the drop is 46 m
h = .0008d(d-480)
h=0 at d=0,480
The main span is 480 m
h(50) = .0008*50(-430) = -17.2
The point is 17 m below the top
The distance between Coney Island and the Brooklyn Bridge is approximately 10 miles. Jaleesa draws a map of Brooklyn using a scale of 1 inch to 300 feet. Kenny draws a map of Brooklyn using a scale of 1 inch to 400 feet. On which student’s map will there be a greater distance between Coney Island and the Brooklyn Bridge? Show your work and explain how you know.
a) To find the vertical distance from the top of a support tower to the lowest point on the cable, we need to find the minimum value of the equation. This can be done by finding the vertex of the quadratic equation.
The equation representing the cable's shape is h = 0.0008d^2 - 0.384d
To find the vertex, we can use the formula d = -b/(2a) where a = 0.0008 and b = -0.384.
d = -(-0.384) / (2 * 0.0008) = 240
Now, substitute this value back into the equation to find the minimum height:
h = 0.0008(240)^2 - 0.384(240)
h = 0.0008(57600) - 92.16
h = 46.08 - 92.16
h = -46.08
The vertical distance from the top of a support tower to the lowest point on the cable is approximately 46 meters.
b) The length of the main span can be found by finding the horizontal distance at which the cable's height is zero.
Set h = 0:
0 = 0.0008d^2 - 0.384d
This is a quadratic equation, which can be solved for d. Factoring out a common factor of d:
d(0.0008d - 0.384) = 0
This equation will be true when either d = 0 or 0.0008d - 0.384 = 0.
Solving the equation 0.0008d - 0.384 = 0 for d:
0.0008d = 0.384
d = 0.384 / 0.0008
d = 480
The length of the main span is approximately 480 meters.
c) At a horizontal distance of 50m from one end of the cable, we need to find the corresponding height, h.
Substitute d = 50 into the equation:
h = 0.0008(50)^2 - 0.384(50)
h = 0.0008(2500) - 19.2
h = 2 - 19.2
h = -17.2
The cable is approximately 17 meters below the top of the support towers at a horizontal distance of 50 meters from one end.
To find the answers to these questions, we will substitute the given values into the equation and perform the necessary calculations. Let's solve them step by step:
a) To find the vertical distance from the top of a support tower to the lowest point on a cable, we need to determine the minimum value of the equation. The equation is in the form of a quadratic function, and the minimum value occurs at the vertex of the parabola.
The equation for the height of the cable is h = 0.0008d^2 - 0.384d
To find the vertical distance, we need to find the value of h when d = 0 (at the top of the support tower). For this case, we only need to calculate the height at d = 0.
h = 0.0008(0)^2 - 0.384(0)
h = 0
Therefore, the vertical distance from the top of a support tower to the lowest point on the cable is 0 meters.
b) To find the length of the main span, we need to determine the horizontal distance between the two support towers. This can be calculated by finding the roots of the quadratic equation.
The equation for the height of the cable is h = 0.0008d^2 - 0.384d
To find the length of the main span, we need to find the values of d when h = 0 (at the lowest point of the cable).
0 = 0.0008d^2 - 0.384d
Simplifying the equation, we get:
0.0008d^2 - 0.384d = 0
This equation can be factored as:
d(0.0008d - 0.384) = 0
Solving for d, we have two possible values:
d = 0 (at one of the support towers)
or
0.0008d - 0.384 = 0
Simplifying the second equation, we get:
0.0008d = 0.384
Dividing both sides by 0.0008, we find:
d = 480
Therefore, the length of the main span is 480 meters.
c) To find how far the cable is below the top of the support towers at a horizontal distance of 50 meters, we need to substitute d = 50 into the equation and calculate the height (h).
h = 0.0008(50)^2 - 0.384(50)
h = 0.0008(2500) - 0.384(50)
h = 2 - 19.2
h = -17.2
Rounding to the nearest meter, the cable is approximately 17 meters below the top of the support towers at a horizontal distance of 50 meters.
Therefore, the answers to the questions are:
a) The vertical distance from the top of a support tower to the lowest point on the cable is 0 meters.
b) The length of the main span is 480 meters.
c) At a horizontal distance of 50 meters from one end of the cable, the cable is approximately 17 meters below the top of the support towers.