some hw help

[briansnycutie]

I'm doing a practice test we received for hw. It has been a while since I have taken a math class and the prof says this is supposed to be a refresher for us. I am having a hard time remembering how to do some of the problems.If you can point me in the right direction would help out, thanks

7. A frame of uniform width borders a painting that is 20 inches long and 13 inches high. If the area of the framed picture is 368 squares inches, find the width of the border.
( I remember a formula for Area being L x W but I am unsure if that applies to this since this problem is looking for this)

8. A box with a square base and and no top is made from a square piece cardboard by cutting 3-inch squares from each corner and folding up the sides. What length of side must the original cardboard sqaure have if the volume of the box is 675 cubic inches

9. Solve the Polynomial equation by factoring and then using the Zero Product Property

5x4( 4 isquared ) - 45x2 ( x squared ) =0

 

[soroban]

Hello, briansnycutie!

7. A frame of uniform width borders a painting that is 20 inches long and 13 inches high.
If the area of the framed picture is 368 squares inches, find the width of the border.

Obviously, we need a sketch . . .

      - : x : - 11 -  : x :
      - *-----------------* - -
      x |                 |   :
      - |   *---------*   |   :
      : |   |         |   |   :
      : |   |         |   |   :
     20 |   |20       |   | 2x+20
      : |   |         |   |   :
      : |   |   11    |   |   :
      - |   *---------*   |   :
      x |                 | - -
      - *-----------------*
        : - - 2x+11 - - - :

\(\displaystyle \text{The painting is }20 \times 11\text{ inches.}\)
\(\displaystyle \text{The frame is }x\text{ inches wide.}\)

\(\displaystyle \text{The dimensions of the framed painting are: }(2x+20)\text{ by }(2x+11)\)

\(\displaystyle \text{We are told that the total area is }368\text{ in}^2.\)

\(\displaystyle \text{There is our equation! } \hdots\quad (2x+20)(2x+11) \:=\:368\)

8. A box with a square base and and no top is made from a square piece cardboard
by cutting 3-inch squares from each corner and folding up the sides.
What length of side must the original cardboard sqaure have
if the volume of the box is 675 cubic inches?

We need two sketches for this one . . .

      : - - - - x - - - - :
    - *---*-----------*---* -
    : |///:           :///| 3
    : * - * - - - - - * - *  -
    : |   :           :   |  :
    : |   :           :   |  :
    x |   :           :   | x-6
    : |   :           :   |  :
    : |   :           :   |  :
    : * - * - - - - - * - *  -
    : |///:           :///| 3
    - * - - - - - - - * - * -
      : 3 :  - x-6 -  : 3 :

\(\displaystyle \text{The square has sides of }x\text{ inches.}\)

\(\displaystyle \text{3-inch square are removed from each corner}\)
\(\displaystyle \text{and the "flaps" are folded up to form an open-top box.}\)

\(\displaystyle \text{The box will look like this:}\)

          *-----------*
         /|          /| 3
        / * - - - - / *
       / /         / /
      *-----------* / x-6
    3 |           |/
      *-----------*
           x-6

\(\displaystyle \text{The dimensions of the box are: }(x-6) \times (x-6) \times 3\)
\(\displaystyle \text{The volume of the box is to be: 675 in}^2.\)

\(\displaystyle \text{There is our equation! }\quad\hdots\quad 3(x-6)(x-6) \:=\:675\)

9. Solve the Polynomial equation by factoring and then using the Zero Product Property
. . \(\displaystyle 5x^4 - 45x^2 \:=\:0\)

\(\displaystyle \text{Factor: }\:5x^2(x^2-9) \:=\:0 \quad\Rightarrow\quad 5x^2(x-3)(x+9) \:=\:0\)


\(\displaystyle \text{Set each factor equal to zero, and solve.}\)

. . \(\displaystyle 5x^2 \:=\:0\quad\Rightarrow\quad x^2 \:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:0}\)

. . \(\displaystyle x-3 \:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:3}\)

. . \(\displaystyle x+3 \:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:\text{-}3}\)