University of Maryland Quadratic Problems
______________________________________________________

Some formulas to remember for this section:
area of rectangle = length x width
perimeter of rectangle = 2( length + width)
area of circle = Pi(r2), circumference of circle = 2Pi(r)
volume of box = (length)(width)(height)
surface area of box = sum of areas of the sides, bottom and/or top
Pythagorean Theorem: a2 + b2 = c2 (where a, b are the lengths of the legs of a right triangle and c is the hypotenuse length)
speed of plane with wind (or boat going downstream) = speed of plane plus wind speed
speed of plane going into the wind (or boat doing upstream) = speed of plane minus wind speed
distance = rate x time
average of n numbers = f(sum of the numbers,n)
h(t) = -16t2 + (v0)t + h0 (where the time t is measured in seconds, v0 is the initial velocity of the object measured in feet per second and h0 is the initial height measured in feet; use 4.9 instead of 16 for meters and seconds)

PROBLEM 1. You are standing on a cliff 200 feet high. (a) How long will it take a rock to reach the ground if you drop it? (b) How far does the rock fall in 2 seconds if you throw it downward with an initial velocity of 40 feet per second.

Initial velocity is negative if the object is thrown downward; it is positive is the object is thrown upward and it is 0 if the object is dropped

(a) v0 = 0, h0 = 200
h(t) = -16t2 + 200
When the ball hits the ground, h(t) = 0 so we want to find the time when h(t) = 0.
Solve the equation:
-16t2 + 200 = 0
t2 = 200/16 = 12.5
t = sqrt (12.5) seconds

(b) v0 = -40,
h0 = 200
h(t) = -16t2 - 40t + 200.
h(2) = -16(22) - 40(2) + 200 = 56 feet

PROBLEM 2. A manufacturer's weekly revenue (in dollars) is given by R(x) = 2x - .0002x2, where x is the number of items sold in the week. The cost of producing x items is C(x) = 300 + .6x.
a. What is reasonable domain for R? for C? (domain of R: 0 < x < 10,000, domain of C: x > 0
b. How many items should be made each week in order to make a profit of $1100? (#14a, p. 78)
P(x) = R(x) - C(x) = 2x - .0002x2 - 300 - .6x = 1.4x - .0002x2
Solve: -300 + 1.4x - .0002x2 = 1100 or .0002x2 - 1.4x + 1400 = 0 x = [1.4 +/- sqrt(1.42 - 4(.0002)(1400))]/[2(.0002)] = 5791 or 1209 items (approximately)

PROBLEM 3. Pat drives the 432 miles between Boston and Washington, DC, in one hour less than Dean and at an average speed of 6 mph faster than Dean. How fast does each drive? 

x = Dean's speed,  t = Dean's time
  formula:  rate x time = distance
     Dean    x      t    =   432
      Pat  (x + 6) (t - 1) = 432
     xt = 432   or  t = 432/x
   (x + 6)(t - 1) = (x + 6)((432/x) - 1) = 432
(multiply on both sides by  x and rearrange factors to get
   (x + 6)((432/x) - 1)x = 432x
   (x + 6)(432 - x) = 432x
   432x + 6(432) - x2 - 6x = 432x
   x2 + 6x - 6(432) = 0

    -6 +/- sqrt[62 - 4(-6)(432)]
x = ___________________________   = 48  (or -54)
               2 

	Dean's speed = 48mph;  Pat's speed = 54mph

PROBLEM 4. An open-top box with a square base is to be constructed from 120 square centimeters of material. What dimensions will produce a box of volume 100 cubic cm? (#11, p. 77)

Let x be the length of one side of the square base and y the height of the box. Then the box has a square base with area x2, and four sides each with area xy so the total amount of material needed is 

    x2 + 4xy = 120    
    y = (120 - x2)/(4x) 
    y = (30/x) - x/4
The volume of the box is  (x2)y.  
We also know that it is 100 cubic cm. so
     (x2)y = 100
    (x2)[(30/x) - (x/4)] = 30x - (x3)/4 = 100
This gives the equation   x3 - 120x + 400 = 0   
(or .25x3 - 30x + 100 = 0)
Using the calculator we find that 
x = 8.56cm and  y = 1.36 cm 
  or x =  6.97 cm, y = 3.79 cm   
(to the nearest hundredth of a centimeter)
_______________________________________________
 
  UM Home | Directories | Search | Admissions | Calendar
Direct questions and comments to Frances Gulick.
Last modified Sunday, 10-Sep-2000 21:59:49 EDT           © 2009 University of Maryland