Optimization questions MAP 4 C C 2 EXPLAIN
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Optimization questions MAP 4 C C 2 – EXPLAIN THE SIGNIFICANCE OF OPTIMAL DIMENSIONS IN REAL-WORLD APPLICATIONS, AND DETERMINE OPTIMAL DIMENSIONS OF 2 -DIMENSIONAL SHAPES AND 3 -DIMENSIONAL FIGURES.
q 1 A hobby farmer is creating a fenced exercise yard for her horses. She has 900 m of flexible fencing and wishes to maximize the area. She is going to fence a rectangular or a circular area. Determine which figure encloses the greater area. Rectangular Using case: the max area for this would be a square. Perimeter : 900 m = 4 s so s = 900/4 = 225 m Area = 225 x 225 = 50 625 m 2
q 1 Circular case: the circumference of a circle: C = 2πr 900 = 2 π r r = 900/2 π = 143. 24 m Area of a circle: A= πr 2 = π(143. 24)2 = 64 458 m 2 The circular pen provides the greatest area with a radius of 143 m.
q 2 Among all rectangular prisms with a given surface area, a cube has the maximum volume. Among all rectangular prisms with a given volume, a cube has the minimum surface area. Yasmin is constructing a rectangular prism using exactly 96 cm 2 of cardboard. The prism will have the greatest possible volume. Determine its dimensions and volume. Solution: A cube has the maximum volume. There are 6 equal squares that make up the cube: SA = 6 s 2 = 96 s = 4 (So a 4 cm x 4 cm cube) V = s 3 = 43 = 64 cm 3
q 3 Matthew is constructing a rectangular prism with a volume of 729 cubic inches. The prism will have the least possible surface area. Determine its dimensions and surface area. Solution: A cube has the minimum surface area. There are 6 equal squares that make up the cube: V = s 3 729 = s 3 s = 9 inches (A 9” x 9” cube) SA = 6 s 2 = 6(9)2 = 486 square inches
q 4 Naveed is designing a can with volume 350 m. L. What is the minimum surface area of the can? Determine the dimensions of a can with the minimum surface area. As 1 m. L = 1 cm 3, the volume is 350 cm 3. V = πr 2 h When h = 2 r, the minimum surface area is achieved (cylinders). 350 = πr 2(2 r) = 2πr 3 = 350/2π r = 3. 8 cm h = 2(3. 8) = 7. 6 cm
q 4 Naveed is designing a can with volume 350 m. L. What is the minimum surface area of the can? Determine the dimensions of a can with the minimum surface area. SA = 6πr 2 = 6 π(3. 8)2 = 272. 19 cm 2