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Maths and Stats , 2016 paper

2e. The triangular garden has sides with lengths in the ratio of 3:4:5. The path is 1m wide.
At each corner of the garden, the path is a sector (part) of a circle with a radius of 1m. The difference between twice the total area of the path and the area of the garden is 2πm2. Find the length of the longest side of the garden. (Area of circle = πr2)

3e(ii)
The length of the bridge AB is 60.
The outside cables are also parabolic and symmetrical in shape, and touch the road at their vertices A and B.
Find the distance, CD, between the two parabolas at a height of 6 m above the road (the distance CD is shown in the diagram).

Thank you.

Thank you.

Hi @JOLM

I will start with the triangle question:

Here you might like to label the sides as below:

Remember the circular corners add up to be a circle with radius=1m, the area of a circle is found by pi*r^2

Area of the triangle is = half x base x height
= 0.5 * 4x * 3x
= 6x^2

Area of the path = 3x + 4x + 5x + pi*(1)^2
= 12x + pi

The question says that the difference between twice the path and the garden is 2pi, so:

2(12x+pi) - 6x^2 = 2pi
24x + 2pi - 6x^2 = 2pi
24x - 6x^2 = 0
6x(4-x) = 0
x=0 or x=4

So x=4 as in this case we can not have a zero length.

Last thing left is to find the length of the longest side which is “5x” so 5*4=20m

Hope this helps with this question :slight_smile:

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I only would like to add that we use formula 1/2 * 3x * 4x to find area of the triangle because the triangle is the right angle triangle triangle. You may notice that it is of the Pythagorean triangles ( sides 3, 4 and 5 so that 3^2 + 4^2 = 5^2).

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For the second part of your question about the bridge:

I am assuming you have found the middle part of the bridge and have it in the form:

12

From here it is helpful to draw a sketch with the information we do have, so we can find the equation of the parabola to the right:

The piece on the right of the main parabola will be in the form of a perfect square and will have its vertex on (30,0):

y=k(x-30)^2

We can find the k value by substituting in the point that we do know (20,15)

15 = k (20-30)^2
15 = 100k
15/100 = k this simplifies down to 3/20

The equation of the parabola to the right is:y

So now we need to know when each of the parabolas are equal to 6 to find the value of C and D. I will find C first by using the middle parabola equation:

333

So C must be when x=10

To find D we repeat this process but with the other parabola equation:

222

So point D must be when x=23.675

To answer the question we must work out the length of CD, 23.675-10= 13.675m

I hope this answers your question. There are other ways you could do this, it will depend on how you went about forming the equation for Q3 e i)

If you need me to explain anything further please let me know.

Thank you very much for your detailed answer.