1520 words - 7 pages

Sydney Harbour Bridge – Directed Investigation – Quadratic functions

Introduction:

Aim: To find the multiple unknowns in the Sydney Harbour bridge

The Sydney Harbour Bridge will be used to investigate a diverse number of points in the structure such as the height and length. Quadratics will be used to solve the height of the bridge at different points on the x axis.

A quadratic is an equation constructed from information collected from a graph; this equation can also be used to produce a graph. Quadratics can be used to solve the problem dealt in this investigation as the Sydney Harbour Bridge is identified as a parabola shape. Only basic information is given about the bridge and the ...view middle of the document...

Therefore the highest point of the bridge will be:

Highest Point of the bridge:

Highest point of the bridge – the road on the pylon = x

182.25m – 50m = xm

182.25m – 50m = 132.25m

Therefore if a painter is painting at the highest point of the arch, the painter will be 132.25m above the road.

The quadratic function y=-1/100 x^2+27/10 x can be used to model the arc by implementing the information provided into the vertex form. The equation of the vertex form is y=〖a(x-h)〗^2+k. Completed, it shall look like =〖a(x-135)〗^2+182.25 , where a≠0. ‘a’can be identified by substituting into the formula a point from the bridge, in this example, it shall be (0, 0).

y=〖a(x-135)〗^2+182.25

0=〖a(0-135)〗^2+182.25

0=〖a(-135)〗^2+182.25

-182.25=18225a

a=(-182.25)/18225

Therefore a=-1/100

Hence the equation of the quadratic is y=-1/100 〖(x-135)〗^2, if expanded, the equation will become:

y=-1/100 (x^2-270x+18225)+182.25

y=-1/100 x^2 27/10 x-182.25+182.25

Therefore y=-1/100 x^2+27/10 x

The expanded formula of the bridge in the vertex form is proven to be exactly the same as the one given, proving that the formula y=-1/100 x^2+27/10 x can be used to find the curve of the arc. The Sydney Harbour Bridge’s road lies 50m above sea level. The central steel arc of the bridge passes through two points on the road and then to the pylons. The points at which the arc meets the road can be solved using the graphing function of a graphics calculator.

The equation y=-1/100 x^2+27/10 x was inserted into the calculator (Figure 1) Values window was altered to see the full graph

Figure 1.

The graph is then drawn onto the calculator (Figure 2)

Figure 2

The x-cal function under the g-solve option was used to make the road. By enteringy=50, the points of the arc are shown. (Figure 3 and 4)

Figure 3 Figure 4

The results indicate that the two points where the arc meets the road are (20, 50) and (250, 50). Using these points, it is possible to determine the length of the road under the arch by using the formula x^2- x^1.

x^2- x^1

= 250-20

= 230m

Therefore the length of the road is 230m long.

Using this information, the amount of steel cable required to construct the supports can be identified. As stated, there are 19 vertical steel cables evenly spaced out which indicates that there are 20 spaces in between the arc and the road. The distance between each point can be calculated by dividing the total length of the road by the number of spaces in between the structure.

Length of road/number of spaces

= 230m/20

=11.5m

Defining that the space in between each steel cable is 11.5m, the distance of each cable can now be identified.

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