KSM Matura 2018 Mathematics Problem 4 The following objects are given: • Three points (-8/7/- 1). Q(16/33/15), R(-14 - 1

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Ksm Matura 2018 Mathematics Problem 4 The Following Objects Are Given Three Points 8 7 1 Q 16 33 15 R 14 1 1 (41.35 KiB) Viewed 46 times

KSM Matura 2018 Mathematics Problem 4 The following objects are given: • Three points (-8/7/- 1). Q(16/33/15), R(-14 - 17/5) A straight line g:r = --0 • A plane F: 2x + y + 4: -5=0 1. Find the equation of the plane E that contains point M and the straight line g. b. Determine the radius of the circle k with centre M that touches the straight line g. c. The straight line f goes through the points Q and R and pierces the plane F in the point S. Calculate the coordinates of S. The point S has the coordinates (-8/- 7/7). d. S is the apex of an oblique cone Spitze eines schiefen Kegels) with the circle k being the base of the cone. Find the inclination angle of SM and the plane E. c. Determine the height h and the the volume V of the oblique cone. f. The point S is being projected onto the plane E. Check whether the projected point S' lies in- or outside the circle k. Give reasons for your answer. 60 Problem 5 An air duct of 1 m length and a cross-section shape of an isosceles trapezoid (gleichschenkliges Trapez, see diagram alongside) is used to cool devices in a building. The base angles are 60°. It was found that the air duct cools best for a cross section (Querschnitt) of 1.5 m². a. Introduce variables and find a function describing the area of the material used in terms of the variables b. Find the height of the air duct such that the material used to construct it) is minimised.