[maxima minima problems][section 16]

(1.)  A ball is thrown upward from ground level.   After t

seconds, its height (in feet) above the ground is

48t - 16t2.

(a.)  What is its initial vertical velocity?

h(t) = 48t - 16t2               here is the problem

v(t) = 48 - 32t     take the derivative, that's velocity

v(0) = 48 - 32(0)    replace t with 0

v(0) = 48           multiply combine like terms

result:  48 feet per second

(b.)  After how many seconds will the ball reach its maximum

height?

48 - 32t = 0      set the derivative equal to 0

+32t  +32t    add 32t to each side
__________________
48        = 32t         add
___         ___
32           32       divide each side by 32

1.5 = t              divide and cancel

result:  1.5 seconds

(c.)  How high will the ball go?

h(1.5) = 48(1.5) - 16(1.5)2    replace t with 1.5

h(1.5) = 72 - 36               multiply

h(1.5) = 36               combine like terms

result:  36 feet

(2.)  A farmer wishes to set aside one acre of land for corn

and wheat.  To keep out the cows, he encloses a rectangular

field with a fence costing 50 cents per linear foot.  In

addition, a fence running down the middle of the field is

needed;  such a fence costs 1 dollar per foot.   Given that

one acre = 43,560 ft2, what dimensions should the field have

to minimize the total cost?

f = 2x + 3y        this is the fence length equation

xy = 43,560        this is the area equation
__   ______
x     x               divide each side by x

y = 43,560/x                cancel

C = .50(2x) + 1y + .50(2y)   this is the cost function

C = x + 2y             multiply combine like terms

C = x + 2(43,560/x)     replace y with 43,560/x

C = x + (87,120/x)              multiply

C' = 1 - (87,120/x2)    take the derivative of C

1 - (87,120/x2) = 0       set the derivative equal to 0

x2 - 87,120 = 0    multiply thru by x2, cancel

+   87,120   + 87,120   add this to each side
_____________________________

x2          =   87,120        add

x = 296.16       take the square root of each side

y = 43,560/x    use this equation to find y

y = 43,560/296.16    replace x with 296.16

y = 147.58          divide

results:  x = 296.16    y = 147.58

(3.)  A farmer wishes to divide 20 acres of land along a river

into 6 smaller plots by using one fence parallel to the

river and 7 fences perpendicular to it.  Verify that the

total amount of fencing is minimized if the sum of the

lengths of the 7 cross fences equals the length of the one

parallel to the river.

f = x + 7y            this is the fence length function

xy = 20             this is the area equation
__   ___
x     x       divide each side by x

y = 20/x               cancel

f = x + 7(20/x)           replace y with 20/x

f = x + (140/x)            multiply

f' = 1 - (140/x2)     take the derivative

1 - (140/x2) = 0    set the derivative equal to 0

x2 - 140 = 0         multiply thru by x2, cancel

+ 140  +140    add 140 to each side
____________________
x2         =  140         add

x2 = 4*35              factor
__
x = 2
35            take square roots
__                          __
y = 20/(2
35)        replace x with 235
__
y = 10/
35            divide

check:   x = 7y
__         __
2
35 = 7(10/35)       make substitutions
__       __
2
35 = 70/35               multiply
__
2(35) = 70         multiply each side by
35, cancel

70 = 70                 multiply

(4.)  V = (pi)r2h    use the volume formula for cylinder

A = 2(pi)r2 + 2(pi)rh  use the surface area formula, too

(pi)r2h = 50           set the volume equal to 50
_______  ________
(pi)r2   (pi)r2             divide each side by this

h = 50/[(pi)r2]                 cancel

A = 2(pi)r2 + 2(pi)r[50/((pi)r2)]    make substitution

A = 2(pi)r2 + (100/r)      cancel and multiply

A' = 4(pi)r - (100/r2)   take the derivative

4(pi)r - (100/r2) = 0    set the derivative equal to 0

4(pi)r3 - 100 = 0      multiply thru by r2, cancel

+   100 +100    add 100 to each side
__________________________
4(pi)r3         = 100     add
________         _____
4              4          divide each side by 4

(pi)r3 = 25                 divide and cancel

r3 = 25/pi            divide each side by pi, cancel

r = (25/pi)1/3          raise each side to the 1/3 power

h = 50/[(pi)r2]        use this equation to find h

h = (50/pi)r-2          write r on top and use negative exp

h = (50/pi)(25/pi)1/3 * -2      make substitution

h = (50/pi)(25/pi)-2/3           multiply exponents

h = 2(25/pi)(25/pi)-2/3        factor like this

h = 2(25/pi)1/3           add exponents

results:   r = (25/pi)1/3  and h = 2(25/pi)1/3

(5.)  A wire 35 cm long is cut into two pieces.  One piece is

bent in the shape of a square, and the other is bent in

the shape of a circle.  How should the wire be cut to

maximize the total area enclosed by the pieces?  How

should the wire be cut to minimize the total enclosed area?

(i.)    x + y = 35          this is the wire length equation

y = 35 - x           subtract x from each side

35 - x = 2(pi)r   use the circumference formula
________  ________
2(pi)    2(pi)            divide each side by 2(pi)

r = (35 - x)/(2pi)                   cancel

A = (x/4)2 + (pi)r2     this is the area function

A = (x/4)2 + (pi)(35 - x)2/(4pi2)  make substitutions

A = (x/4)2 + [(35 - x)2/(4pi)]    cancel pi's

A = (x/4)2 + [(1225 - 70x + x2)/(4pi)]  square the binomial

A = (1/16)x2 + [(1225 - 70x + x2)/(4pi)]    square 1/4

A = (1/8)x - (70/4pi) + (x/2pi)   take the derivative

(1/8)x - (70/4pi) + (x/2pi) = 0   set the derivative equal to 0

(pi/8)x - (70/4) + (x/2) = 0     multiply thru by pi, cancel

[(pi/8) + (1/2)]x - (70/4) = 0   factoro like this

(pi + 4)x - 140 = 0    multiply thru by 8, cancel

+ 140  +140   add 140 to each side
___________________________
(pi + 4)x  =        140        add
__________       _________
pi + 4         pi + 4         divide each side by this

x = (140)/(pi + 4)                  cancel

y = 35 - [140/(pi + 4)]     replace x with this

y = [35(pi + 4)/(pi + 4)] - [140/(pi + 4)]

[multiply by (pi + 4)/(pi + 4)]

y = (35pi + 140 - 140)/(pi + 4)   subtract fractions

y = [35pi/(pi + 4)]       combine like terms

result:  x = (140)/(pi + 4) and y = [35pi/(pi + 4)]

[this will result in a MINIMUM]

(ii.)  For the maximum area, use the whole wire as the circle.

(6.)  A Norman window is constructed from a rectangular sheet

of glass surmounted by a semicircular sheet of glass.  The

light that enters through a window is proportional to the

area of the window.    What are the dimensions of the

Norman window having a perimeter of 30 feet that admits the

most light?

2y + 4r + (pi)r = 30     this is the perimeter equation

2y + (4 + pi)r = 30          factor

y + (1/2)(4 + pi)r = 15   multiply thru by 1/2, cancel

- (1/2)(4 + pi)r   - (1/2)(4 + pi)r subt this fr ea side
____________________________________________
y         =   15 - (1/2)(4 + pi)r     subtract

A = 2ry + (1/2)(pi)r2     this is the area function

A= 2r[15 - (1/2)(4 + pi)r] + (1/2)(pi)r2  make substitution for y

A = 30r - (4 + pi)r2 + (1/2)(pi)r2   multiply thru parentheses

A' = 30 - 2(4 + pi)r + (pi)r     take the derivative

30 - 2(4 + pi)r + (pi)r = 0   set the derivative equal to 0

30 - [2(4 + pi) - pi]r = 0     factor

-30 + [2(4 + pi) - pi]r = 0    multiply thru by -1

+ 30                   +  30         add 30 to each side
_______________________________
[2(4 + pi) - pi]r = 30           add

r = 30/[2(4 + pi) - pi]     divide each side by this, cancel

r = 30/(8 + 2pi - pi)             multiply thru parentheses

r = 30/(8 + pi)                combine like terms

y = 15 - (1/2)(4 + pi)r   use this equation to find y

y = 15 - (1/2)(4 + pi)[30/(8 + pi)]   replace r with this

y = 15 - [15(4 + pi)/(8 + pi)]    multiply 1/2 by 30

y = 15[(8 + pi)/(8 + pi)] - [15(4 + pi)/(8 + pi)]

[multiply by (8 + pi)/(8 + pi)]

y = (120 + 15pi - 60 - 15pi)/(8 + pi)   multiply thru subtract

fractions

y = (60)/(8 + pi)                   combine like terms

results:   r = 30/(8 + pi)   and y = 60/(8 + pi)

(7.)  Answer problem 5 if the two pieces are to be formed

into the shape of a circle and an equilateral triangle.

x + y = 35           this is the wire length equation

-x          - x     subtract x from each side
_____________________
y = 35 - x        subtract
_
A = x(
3/4) + (pi)r2     this will be the area

35 - x = 2(pi)r    use the circumference formula for circle
______  _______
2pi     2pi             divide each side by 2pi

(35 - x)/(2pi) = r                   cancel

A = x(
3/4) + (pi)(35 - x)2
________________   make substitution
4(pi)2
_
A = x(
3/4) + (35 - x)2
_____________          cancel
4pi
_
A' = (
3/4) - 2(35 - x)
___________    take the derivative
4pi

_
(
3/4) - [2(35 - x)/(4pi)] = 0   set the derivative equal to 0
_
(pi)
3 - 2(35 - x) =  0    multiply thru by 4pi, cancel
_
(pi)
3 - 70 + 2x = 0           multiply thru parentheses
_                            _
2x = 70 - (pi)
√3           add 70 and -(pi)√3 to each side
_
x = 35 - (1/2)(pi)√3        multiply thru by 1/2, cancel

y = 35 - x     use this equation to find y
_
y = 35 - [35 - (1/2)(pi)√3]      make substitution
_
y  = (1/2)(pi)√3          multiply thru and combine like terms
_                    _
results:  x = 35 - (1/2)(pi)√3   ,  y = (1/2)(pi)√3

(8.)  A woman is on a lake in a canoe 1 km from the closest

point P of a straight shore line;  she wishes to get to

a point Q, 10 km along the shore from P.  To do so, she

paddles to a point R between P and Q and then walks the

remaining distance to Q.  She can paddle 3 km/hr and she

can walk 5 km/hr.   How should she pick the point R so

that she gets to Q as quickly as possible?

f(x) = (1/3)(1 + x2)1/2 + (1/5)(10 - x)   here is the problem

f(x) = (1/3)(1 + x2)1/2 + 2 - 0.2x    multiply thru parentheses

f'(x) = (x/3)(1 + x2)-1/2 - 0.2   take the derivative

(x/3)(1 + x2)-1/2 - 0.2 = 0   set the derivative equal to 0

(5x)(1 + x2)-1/2 - 3 = 0   multiply thru by 15, cancel

+3  +3     add 3 to each side
________________________
(5x)(1 + x2)-1/2    =  3     add

5x = 3(1 + x2)1/2       multiply each side by (1 + x2)1/2

and cancel

25x2 = 9(1 + x2)      square each side

25x2 = 9 + 9x2          multiply thru parentheses

-9x2         - 9x2         subtract 9x2 from each side
_____________________
16x2 = 9                            subtract

4x = 3       take the square root of each side
___  __
4    4      divide each side by 4

x = 3/4               cancel

result:  choose R to be 3/4 km from P.