Similarity and area help?

[nowomannocry]

In geometry we are finishing up our unit and I still don't understand problems like this:

Julie has a rectangular vegetable garden with dimensions 12' by 4'. She wants to create a new garden, whose area will be ten times as large as that of the original, but she wants the new garden to have the same shape as (be similar to) the original garden.
a. Find the scale factor between the old and new garden.
b. Use the scale factor to find the new dimensions.

On a related note, I don't get questions like "how you would cut the area of a rectangle in half but maintain a similar shape". Help with the deeper understanding stuff would be great, because I do not understand these at ALL and our test is in a few days.

Thank you so much!

 

[garf]

the relation between the sides is of 3:1
12*4= 48
48*10= 480
3x*x=480
3x[sup:2ved3h79]2[/sup:2ved3h79]=480
x[sup:2ved3h79]2[/sup:2ved3h79]=480/3
x=160[sup:2ved3h79]1/2[/sup:2ved3h79]
x=12.65
3x=37.95

 

[Denis]

x = new width, y = new length

xy = 48*10 = 480 [1]
x/y = 4/12 = 1/3 ; y = 3x [2]

Substitute [2] in [1] :
x(3x) = 480
...carry on :wink:

 

[soroban]

Hello, nowomannocry!

Julie has a rectangular vegetable garden with dimensions 12' by 4'.
She wants to create a new garden, whose area will be ten times as large as that of the original,
but she wants the new garden to have the same shape as (be similar to) the original garden.

a. Find the scale factor between the old and new garden.

\(\displaystyle \text{The original garden has dimensions }\,12\text{ by } 4,\,\text{with area }48\text{ ft}^2.\)

\(\displaystyle \text{The new garden has dimensions }\,12a\text{ by }4a,\,\text{ with area }480\text{ ft}^2.\)

\(\displaystyle \text{So we have: }\12a)(4a) \:=\:480 \quad\Rightarrow\quad 48a^2 \:=\:480 \quad\Rightarrow\quad a^2 \:=\:10\)

\(\displaystyle \text{Therefore, the scale factor is: }\,\sqrt{10}\)

b. Use the scale factor to find the new dimensions.

\(\displaystyle \text{The new dimensions are: }\:12\sqrt{10}\text{ by }4\sqrt{10}\)

. . \(\displaystyle \text{approximately }\,37.94\text{ by }12.65\text{ feet.}\)

On a related note, I don't get questions like:
"How you would cut the area of a rectangle in half but maintain a similar shape".

We need to start with a rectangle with certain dimensions.

Let the rectangle by \(\displaystyle L \times W\)

      : - - - - L - - - - :
      *---------*---------*
      |         |         |
      |         |         |
    W |         |         |
      |         |         |
      |         |         |
      *---------*---------*
      : - L/2 - : - L/2 - :

Bisect the rectangle.



Take one half and turn it 90 degrees.

      *-------------*
      |             |
  L/2 |             |
      |             |
      *-------------*
             W

We want the lengths and widths to be proportional.

\(\displaystyle \text{So we have: }\;\frac{L}{W} \;=\;\frac{W}{\frac{L}{2}}\quad\Rightarrow\quad L^2 \:=\:2W^2\)

\(\displaystyle \text{Therefore: }\:L \;=\;\sqrt{2}\,W\)


\(\displaystyle \text{We must start with a rectangle whose length is }\sqrt{2}\text{ times the width.}\)