Unley High School
NETHERBY 5062 SA
Dear Mr. Tincture,
I have recently received your project on the calculations of the 2 sided Cataclysmic Chasm bridge. You have requested me to calculate the dimensions and the cost of the steel cabling required to help support the roadway for your newly designed suspension bridge. Using the information you gave me, I was able to use my mathematical expertise to present the required information in this letter.
Using the equation of the bridge: height = -0.01x-squared + 4.1x, I first substituted 40 metres as h to find the value of x. The x value is now the horizontal distance where it intercepts the bridge. In this case, there are 2 intercepts. I made the equation equal to zero by moving the 40 to the other side of the equation and factorised it. The 2 interceptions are at 10 metres and at 40 metres.
By finding the apex or vertex of the bridge, it allows me to work out the total length and how high the bridge is. Using the formula x= -b/2a, where b=4.1 and c= -0.01, I can find the x value of the vertex. This is at 205 metres from the start or end of the bridge (the middle of the bridge). To find the height, I substituted the x to 205 into your equation. The bridge is 420.25 metres high.
Since the height of the bridge from ground is 420.25 metres and the cable starts at 40 metres above the ground, these figures are subtracted. The length of the cable at the apex is 380.25 metres long.
To find the length of other cabling, I used the same technique as before. By substituting the x value into your equation to find the total height from the highest point of that cable position to the ground (in this case, the x values are 4 metres intervals). The cable length is found by subtracting 40 metres. There was no need to calculate every cable length, because the reverse side is the same in length (the quadratic effect). There are 234 cables in total.
Once all the cabling is calculated, I added all the lengths (including the reverse side) of the cabling then multiplied it by 2, because it is a 2 sided bridge. The result is 49445.14 metres in total. The cost of cabling is $161 per metre, therefore the total cost for cabling is $7,960,667.54.
Attached with this letter is a copy of my calculations that is explained
mathematically. Do you have any further questions please do not hesitate
to call me via the address above. I would like to wish you every success
in your bridge project, and I look forward to any future requests.