This document contains 20 word problems involving maximizing or minimizing quantities such as volume, surface area, cost, or area for various geometric shapes and objects. The problems involve finding properties of cylinders, cones, spheres, boxes, triangles, trapezoids, and windows when a dimension is fixed or a quantity like volume or cost is fixed in order to achieve an extreme value for another property like surface area or area. The document provides the setup and constraints for each word problem involving maxima and minima quantities for different geometric shapes and scenarios.

1. CLASS-XII MAXIMA AND MINIMA WORD PROBLEMS
1. Showthat the rightcircular cylinderof givensurface andmaximumvolume issuchthatitsheightisequal to
diameterof base.
2. Prove that the volume of largestcone thatcan be inscribedina sphere of radiusR is8/27 timesvolume of sphere.
3. Showthat the semi – vertical angle of cone of maximumvolume andof givenslantheightistan-1
√2.
4. A closedcylinderhasvolume 2156 cm3
. What will be the radiusof itsbase so that itsT.S.A is minimum?
5. Findthe altitude of a rightcircularcone of maximumvolume whichcanbe inscribedina sphere of radiusr
6. A wire of length28 cm isto be cut into twopieces, one of the twopiecesisto made intoa square and otherinto
circle.What shouldbe the lengthof twopiecessothatcombinedareaof square and circle isminimum?
7. Showthat the heightof cylinderof maximumvolume thatcanbe inscribedina sphere of radiusR is2R/√3 .
8. Showthat semi – vertical angle of rightcircularcone of givensurface area and maximumvolumeis
sin-1
(1/3) .
9. The sum of perimeterof circle andsquare isK. Prove thatthe sumof theirareasisleastwhen side of square is
double the radiusof circle.
10. Showthat the rightcircular cylinder,openattop& of givensurface area & maximumvolume issuchthatitsheight
isequal to radiusof base.
11. Show that all the rectanglesinscribedinagivenfixedcircle,the square hasmaximumarea.
12. Showthat the heightof the cone of maximumvolume thatcanbe inscribedinasphere of radius12 cm is16 cm.
13. A wire of length36 cm iscut intotwo pieces,one of the piecesisturnedinthe formof a square and otherinthe
formof an equilateraltriangle.Findthe lengthof eachpiece sothatthe sumof the areas of the two be minimum.
14. Prove that the surface areaof a solidcuboid,of square base andgivenvolume,isminimumwhenitisacube.
15. An openbox withsquare base isto made outof a givenironsheetof area27 sq.m Show that the maximumvolume
of the box is13.5 cu.m.
16. A tankwithrectangularbase and rectangularsides,openatthe topis to be constructedsothat its depthin2 m and
volume is8 m3
.If buildingof tankcostsRs. 70 persq. meterforthe base and
Rs. 45 persq. Meterfor sides,whatisthe cost of leastexpensive tank?
17. If the sumof hypotenuse andaside of a rightangledtriangle isgiven,show thatthe areaof the triangle is
maximumwhenthe angle betweenthemis
𝜋
3
.
18. An openbox witha square base isto be made out of a givenquantityof cardboardof area c2
square units.Show
that the maximumvolume of the box is
𝑐3
6√3
cubicunits.
19. If the lengthof three sidesof atrapeziumotherthanbase are equal to10 cm thenfindthe areaof the trapezium
whenitis maximum.
20. A windowisinthe formof a rectangle surmounted byasemicircularopening.Total perimeterof windowis10m.
Findthe dimensionsof the windowtoadmitmaximumlightthroughwhole opening.