Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 2 An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola? Arch is downwards, Since, the axis of parabola is negative y-axis, its equation is x2 = –4ay First, we find coordinates of point B Given, Width of Parabola = AB = 5m So, BC = 52 m Also, Parabola is 10 m high Hence, OC = 10 m ∴ BD = OC = 10m Hence, coordinates of point B is 𝟓𝟐, −𝟏𝟎 Now since point B 52,−10 lies on the parabola It will satisfy the equation of the parabola, Putting x = 52, y = –10 in equation x2 = –4ay 522 = –4a × (–10) 254 × 14 × 110 = a 254 × 14 × 110 = a 54 × 4 × 2 = a 532 = a a = 𝟓𝟑𝟐 Hence, the equation of parabola is x2 = –4ay Putting a = 532 x2 = –4532y x2 = – 𝟓𝒚𝟖 We need to find how wide the arch is it 2 m from the vertex of the parabola Let point P be point 2 m from vertex of parabola Hence, OP = 2m We need to find MN Now equation of parabola is x2 = −5𝑦8 Putting y = –2 x2 = –4 × 532 × (–2) x2 = 4 × 5 × 232 x2 = 54 x = ± 54 x = ± 𝟓𝟐 Hence, PN = 52 So, MN = 2PN = 2 ×52 = 5 Thus, the width of the arch MN = 𝟓 m = 2.23 m (approx.)

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.