13 Optimization Problems: Problem 1 (10 points) A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $40 per foot, and on the other three sides by a metal fence costing $30 per ft. If the area of the garden is 22 square feet, find the dimensions of the garden that minimize the cost. Length of side with bricks (and opposite side) ft: Length of sides perpendicular to the brick wall ft. A box is to have an open top, vertical sides, a square bottom, and a volume of 108 m?. Find the dimensions that minimize the surface area of the box. Height (include help (units)) Length of base D (include help (units) ) A box with a square base and an open top will be made from 1500 cm? of material. Find the largest possible volume of the box. Volume (include help (units)) A cylinder is inscribed in a right circular cone of height 8 and radius (at the base) equal to 3. Determine the dimensions of such a cylinder that maximize its volume. Radius D Height D Find the points on the ellipse 4x2 + y" = 4 that are farthest away from the point (1, 0). List them as a list of points, such as "(1,2), (3,4)". List of points:" Let the side laught of side with bruch lenght be a, & the lenght of side perpendicular to y (brata wall be y metal a as per given, 20 ay = 22 f4 2. 50 bruck 10. Care a of rectangle = my 22 let cost of boundary ( #D = You + 30 ( y f a t q ) =to fon + boy 13 = Font 60x 2 2 / 18 20 27 Fox+ Let can spent 1320 1320 2 2 To ( for( optimization ) 1:320 nn - C ~ 4. 34 7. - to (.: lengh can't be - ve, so - ve root created Vo rejected 2 /m = 5.06 8 ( * 1. Ng. Let Wit I fo" Let the the height be h. & lenght of base be a. eight . as pere given question o azh = 108 m . ") h = 108 the surface area - bruck = a + yah = a 2 + ya. to8 ( surface area of open bon ) Q 2 a ? + 432 let slap = al+ 432 a = 2 5 ( a ) = 20 - 432 20. ( for optimization ) Q ? 27 295 = 43 2 3 as = 216 29 a = 6 m. h = 108 - 2108 If Lef the height be hi & lenght of base be a. : alt yah : 1500 ). 1500 - a out expected volume of bon = > azh - a? (1375 a 375 a - A a let VcaD= 375 a- 9/y for optimization, V ( 9) : 0 = ) 375- 39' so . 4 " aly = 3 75 / 2 - 1282) al = 125 x 4 : 500 a = 15oo = loss cm. . . h : 375 375 10 VS 10 US is v5- 5V's issad 20 10 65V5 2 dy Let the radius ben& height be h. A le i. volume of cylinder Q . 2 TIMh. B ROS here ABC is the cone & PQRS i's the 8:01 cylinder ( Ed drawing is for understanding only). . from question, AODD - 8 units to OB = 3 units. let BRA OR - 9 * ) L BR = 3 - 16 1 A AOR 8 A PBR are similar. BR OB 3 8 PR - 1 04 h di 8 13 Evolume of cylinder 8 volume of cylinder : TM?h 5 Cm - B, IT ( 3i'm ? - 1 3 ) A let VCH) = 8/371 ( 1@) 8 3 re? - 81 3 ) V ( ) : 8 / 3 TT ( 6 re - 307 2 ) = 0 . ing only ). 2 ) re ( re - 2 ) 2 0. 35 1 : 2 units.