Expert: Steve Holleran - 3/27/2008

Question
Hi there, I have been struggling in doing quadratic word problems for quite a while. I understand all quadratic equations, but can't seem to ever apply the logic into a word equation. Any help at all with these questions would be more than appreciated.

1. The base of a temple is a rectangle with a perimeter of 300m. The area of the base is 4400m^2. What are the dimensions of the base?

2. A rectangular lawn measures 7m by 5m. A uniform border of flowers is to be planted along two adjacent sides of the lawn. If the flowers that have been purchased will cover an area of 6.25m^2 how wide is the border?

3. Find k such that the graph of y-9x^2 + 3kx + k
a) intersects the x-axis at one point only
b) intersects the x-axis in two points
c) does not intersect the x-axis


This next one I did.. but didn't get the correct answer and have no idea why. I understand completely the concept of converting it to vertex form, and finding the numbers from that... but apparently just can't get the numbers right.

4. The path of a basketball shot can be modeled by the equation h= -0.125d^2 + 2.5, where h is the height of the basketball, in metres.
a)Find the maximum height reached by the ball.
b)What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the floor is the ball when the player releases it?

Answer
Hi Jennie,

Let's see what we have:

1.  Call the length x and the width y.  Then you have, from   
   the perimeter, 2x + 2y = 300, or x + y = 150.  

   The area is A = xy

   Solve the perimeter equation for y and substitute:
   y = 150 - x so

           A = x(150 - x) = 150x - x^2 = 4400

so you have       -x^2 + 150x - 4400 = 0

Using the quadratic formula, you get x = 110 or 40.
If x = 110, y = 40 and if x = 40, y = 110, so its basically the same result.


2.  ------------------
   |             |   |
   |      7      | 5 |
   |-------------|   |
  x|                 |
   ------------------

In the diagram above, the portion of the garden along the bottom is
7 by x , so its area is 7x.  The portion along the right side is
x by (5+x), so its area is x(5 + x).

The total garden area is A = 7x + x(5 + x)

                          = 12x + x^2 = 6.25 = 25/4

so                       48x + 4x^2 = 25

or                        4x^2 + 48x - 25 = 0

using the formula again, x = 1/2.


3.  This has to do with the part of the quadratic formula called the
discriminant. If b^2 - 4ac = 0, the quadratic will have one double real zero.  So,

                     (3k)^2 - 4(9)(k) = 0

                      9k^2 - 36k = 0

                      9k(k - 4) = 0   so it will touch the axis at

k = 0 and k = 4.

The quadratic will have two distinct real zeros if the discriminant is positive, and none if it is negative, so if we do a sign chart for
                      9k(k - 4)
                    +        -        +
we get   :      ----------|--------|---------
                         0        4

which means for k in (-inf, 0) and (4, inf) there are two distinct zeros, and for k in (0, 4) there are no real zeros.

4.  This one has me lost.  What is d??  And are you sure something's not missing from the equation?

Because, for the first part, the max hgt is at the vertex , so

     d = -b / 2a = -0 / 2(.125) = 0 --> (0, 2.5) would indicate the max hgt = 2.5 meters, but this doesn't make sense...the basket itself is 10 ft from the ground, which is more than 2.5 m.

Let me know if you can correct this, or if I misinterpreted it.

Steve