The document provides 15 multi-step word problems involving concepts like maximizing or minimizing functions, finding dimensions of shapes to satisfy certain criteria, and other applied optimization challenges. The problems cover topics like finding tangent lines, inscribed shapes, wire cutting, epidemics, profit maximization, and geometric shapes. Students are instructed to show all work and box their final answers on a single sheet of paper.

1. ' i -1 1
3HW8 MAXIMA-i'I|N|MA PROBLEMS DUE: December 5/6 class periorl
INSTRUCTIONS. Write a complete solution to each of the following problenis. Box your final arrswer. Use one vhole
intermediate paper.
1. Find an equation of the tangent line io the curve y = ;3 - Jx2 + 5x that has the least slope,
2. A {unnel of specific volume is to be in the shape of a righlcircular cone. Find the ratio cf the height to the hase mdius if the
least amouni of material is to be used in its manufacture.
3. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of raeiiirs 2 inches.
4. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cnr if two sides
of the rectangle lie along the legs.
5, A right circular cylinder is inscribed in a sphere of radius 1 cm. Find the largest possible -qurface area of such cylinder
6, A Norman window has the shape of a rectangle surmounted by a semicircle. lf the perinreter of the windnlv is 32 ft, find the
dimensions of the window so lhat the window will admit the mosl liqht.
7 . A paper containing 24 cmz of printed region is to have a margin of 1 ,Scm at the top and bctlorn and l crn at the sides, Find
the dimensions of the smallest piece of paper that will fill these requirements?
B. A S-meter wire is to be cut in two. The strength S of the wire is unit proportional to the prodrrct of the square of tlte one part
and the cube of the other. Find the point at which this wire must be cut to maximize its strength.
9. A piece of wire 10 m long is cut into two pieces. One piece is bent intrl a square and the other is bent into an equilateral
triangle. How should the wire be cut so that the total area enclosed is a maximum?
10. A piece of w:re 10 m long is cut into two pieces. One piece is benl into a square and the other is bent into an equilateral
triangle. How should the wire be cut so that the total area enclosed is a minimum?
11 . A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 kmlh. Another boat has been heading due east at
15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest bqether?
12. A telephone company has to run a line from poirrtA on one side of a river to another point B, that is on the other side, 5
miles down the point opposite A. The river is uniformly 12 miles wide. The conrpany carl nn the line along the sltoteline to a
point C and then run then run the line under the river to B. The cost of laying the line along the shore is P1000 per mile, and
the cost of laying the line under water is twice as much. Where should point C be located to rninimize tho cosl?
13. A direct current generator has an electromotive force of E volts and an internal resislance of r ohms, where E and r are
constants. lf R ohms is the external resistance, the total resistahce is (r+R; ohms, and if P watts is the por,r,er,.llren
c2o
P= --: ':- , Show that the most power is consumed when the external resistance is equal tn the internal rcsistance.
(r+R)'
14. In a pariicular community, a certain epidemic spreads in such a way that x months after tlre start of the epidentrc, P percent
30*',,
of the population is infected, where P =, . In how many months will the nrostpeofle be infected, anci v,thatpercent
(t + x')'
of the population is this?
15. Suppose that under a monopoly, x units are demanded daily when p pesos is the price per unit atrd
x =140 - p, lf the tolal cost of producing x units is given by C(r) = vz + 20x + 300, find tlre maximum daily total profit.

2. 3HW7 MAXIMA-MINIMA PROBLEMS DLJE; Decemlter 1/)- class periocl
lNsrRUcrloNS' write a complete solution to each of the following problerns. Ilox
your final alswer. Use
one whole intermediate paper.
Find two numbers whose difference is 100 and whose product is
1
a minirnrinr,
2. Find two numbers whose sum is 10 and the sunr of the
squares is a .nrr.rirnrr,.n.
? Find two positive numbers whose product is 100 and whose
sunr is a mulirnunr.
4. Find the dimensions of a rectangre with area i00m2 whose perimeter
rs as srnail as possihre.
5. Find two numbers whose sum is 240 and whose product
is a maxirnunr.
6. Find the dimensions of a rectangle with perimeter 240m whose
area is as large as possible.
7. lf one side of a rectangular field is to have a river
as a natural boundary, find the dimensions of the
fargest rectangular field that can be enclosed by using zqom
oiiince for ilre oiher three sides.
8. A rectangular field is to be enclosed by a fence and then dividecl into
two lots by another fence set at
the middle' what must be the dimensions of the field with the raigest
area if the total length of the
fencing material is 240m?
9' suppose the cardboard is 24in by 24in, what is the maximum volume
of the b'x consiructec,
according to the box problem?
10' A box with a square base and open top must have a volume of 32,000
cm'. Find the clirnensions of
the box that minimize the amount of material used.
11. Find the point on the line 6x + y = g that is closest to the point (_J,
1)
12. show that the vertex is the point on a parabola that is closest
to its focrrs
13, Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from
the point (.1, 0).
14. What pointlon the hyperbola x2 y, = Z l$'ttosest to the point (0,
- 1)?
15' of all the different types of isosceles triangles with fixed perimeter,
which one has the greatest area?