Assignment4s

# Assignment4s - MATH1821 Mathematical Methods for Actuarial...

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MATH1821 Mathematical Methods for Actuarial Science I Assignment 4 1. The figure below shows two right circular cones, one upside down inside the other. The two bases are parallel, and the vertex of the smaller cone lies at the center of the larger cone’s base. What values of r and h will give the smaller cone the largest possible volume? Solution. The volume of the smaller cone is V = πr 2 h/ 3, where r and h are related by (12 - h ) /r = 12 / 6 = 2. Hence h = 12 - 2 r and V ( r ) = πr 2 (12 - 2 r ) / 3 for 0 r 6. Differentiating, we get V ( r ) = 2 πr (4 - r ) so that the critical points are 0 and 4. As V (4) = 64 π/ 3 and V (0) = V (6) = 0, the smaller cone has the largest volume when r = h = 4 (ft). 2. What is the length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown below? Round your answer down to the nearest foot. Solution. Consider the following diagram. 1

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Our aim is to minimize L ( θ ) = d 1 + d 2 = 8 sec θ + 6 csc θ for θ (0 , π/ 2). As L ( θ ) = 8 sec θ tan θ - 6 csc θ cot θ = 8 sin 3 θ - 6 cos 3 θ cos 2
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