# [SPM Add Math Integration] Johor Paper 1 Q6 How to solve this question? The answer is 30.60

### Question

Diagram 6(a) shows the front view of a door.

BAC adalah sebahagian daripada graf y=9- \frac{x^2}{6}. A adalah titik tertinggi dari DE. Cari luas permukaan pintu.
BAC is part of the graph y=9-\frac{x^2}{6}. A is the highest point from DE. Find the surface area of the door. [3 markah / marks]

Hi Farhaini,
For this question, we cannot assume BCED is a square, because the question did not mention anything about it.
So how do we find the length of BC?

Firstly we have to plot the y=9-\frac{x^2}{6} graph.
Generally, we know that when the coefficient of b is 0 in y=ax^2 + bx +c , the graph is a parabola symmetrical about the y-axis and its maximum point is the y-intercept. The question says BAC is part of the graph, and since A is the maximum point or stationary point, it is easily identified as (0,9).

The reason you cannot use A as (0,0.5) is because this pair of coordinates do not lie on the graph y=9-\frac{x^2}{6}. If you substitute x=0, you will get y=9 and not y=0.5. (LHS \neq RHS)

But this doesnâ€™t mean the 0.5 given is useless, we can make use of it to find BC. we know that the height of A above BC is 0.5, and since BC is a straight horizontal line, we can work out that the y-coordinates of B and C:
9-0.5=8.5. To find the x-coordinates, we simply substitute y=8.5.

Iâ€™m not sure if you realized this, but coincidentally the whole surface area of the door is actually equal to the area under the curve integrated from -\sqrt 3 to \sqrt3 with respect to x, because:

1. distance between line BC and the x-axis =8.5, which is also the same as that in diagram 6(a) where BD=CE=8.5
2. height of A from x-axis=9, which is also the same as that in diagram 6(a), where the height of A from line DE=9.

And we can conclude that D and E lies on the x-axis based on these 2 reasons.

This makes the final step a lot easier as we donâ€™t have to split the shape into two parts

(Itâ€™s quite rare to see SPM questions with units in feets because only USA will use that kind of measurement haha , itâ€™s best to give answers in units according to the question, but if the question didnâ€™t specify, itâ€™s usually ok to omit the unit or just write 30.60 units^2 )
Hope this helps!

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Now I understandâ€¦Thank you very much !

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