BACCALAURÉAT GÉNÉRAL ET TECHNOLOGIQUE SESSION 2009 ÉPREUVE SPÉCIFIQUE MENTION « SECTION EUROPÉENNE OU DE LANGUE ORIENTALE »
Académies de Paris-Créteil-Versailles
Binôme : Anglais / Mathématiques Sujet n° 16 USE OF GRAPHS IN GEOMETRY The first part of this page is a summary that can help you to do the exercise.
Cylinder: The volume of a cylinder of radius r and height h is equal to (area of the cross-section) × length = π r2 × h The curved surface area, i.e. the surface area of the curved part of the cylinder, is that of a rectangle of sides 2πr and h.
Gradient of a graph at a point: the gradient of a graph at a point is the gradient of its tangent at difference in y−coordinates that point, i.e. . difference in x−coordinates EXERCISE A solid cylinder of radius r centimetres and height h centimetres has a volume of 100 π cm3. 100 (a) (i) Show that h = 2 . r 200 (ii) The cylinder has a total surface area of πy square centimetres. Show that y = 2 r 2 + . r (b) The table below shows some values of r and the corresponding values of y, correct to the nearest whole number. r y
1 202
1.5 138
2 108
3 85
4 82
5 90
6 p
(i) Find the value of p. Round it to the nearest unit. (ii) Using a scale of 2 cm to represent 1 cm, draw a horizontal r-axis for 1 < r < 6. Using a scale of 2 cm to represent 20 cm2, draw a vertical y-axis for 70 < y < 220. On your axes, plot the points given in the table and join them with a smooth curve. (c) Use your graph to find the values of r for which y = 100. (d) Use your graph to find (i) the value of r for which y is least, (ii) the smallest possible value of the total surface area of the cylinder, correct to 3 s.f.. (e) Without any calculation, draw a tangent to the curve at the point where r = 2 . Deduce the gradient of the graph at that point.
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