Garmetry

Geometry Level 3

Mr. Moses has a rectangular garden with an area of 192 m 2 192 \text{ m}^2 . The difference between the length and width is 4 m 4 \text{ m} . Along the sides and within the garden, he paved a road of width 2 meters. Find the area of the road in m 2 \text{m}^2 .

96 156 144 168 128

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2 solutions

Let the garden area =W *L=W(W+4)=192=12 *(12+4).
So W=12, L=16.
Area of road =2 *{W *2+(L-2 *2)}=2{24+24}= 96. \color{#D61F06}{96}.

Assume that p = p = length and l = l = width

We already know that p l = 4 p - l = 4

Now let's find l l

p × l = 192 p \times l = 192

( 4 + l ) × l = 192 (4 +l ) \times l = 192

4 l + l 2 = 192 4l + l^2 = 192

l 2 + 4 l 192 = 0 l^2 + 4l -192 = 0

( l 12 ) ( l + 16 ) = 0 (l-12)(l +16) = 0

l = 12 l = 12

p = 4 + l = 4 + 12 = 16 p=4 + l = 4 + 12 = 16

Now find the area of the road

4 4 road square shaped area located in the garden corners with a side length of 2 m \space m

4 × 2 2 = 16 m 2 4 \times 2^2 = 16 \space m^2

2 2 road area that are in the garden length with side length 12 m 12 \space m and width 2 m 2 \space m

2 × ( 12 m × 2 m ) = 48 m 2 2 \times (12 \space m \times 2 \space m) = 48 \space m^2

2 2 road area that are in the garden width with side length 8 m 8 \space m and width 2 m 2 \space m

2 × ( 8 m × 2 m ) = 32 m 2 2 \times (8 \space m \times 2 \space m) = 32 \space m^2

So the area of the road is 16 m 2 + 48 m 2 + 32 m 2 = 96 m 2 16 \space m^2 + 48 \space m^2 + 32 \space m^2 = \boxed{96 \space m^2}

M a t h i s m o r e e n j o y a b l e t h a n t h e g a r d e n p i c t u r e ! ! ! Math \space is \space more \space enjoyable \space than \space the \space garden \space picture \space !!!

Sorry I got confused. You calculated the garden to be 12 m x 16 m. Then why are you taking 'side length 8 m'? Should it not be 16? I got the answer as 12 × 2 × 2 + 16 × 2 × 2 + 2 × 2 × 4 = 128 12 \times 2 \times 2 + 16 \times 2 \times 2 + 2 \times 2\times 4 = 128 What am I missing?

Ujjwal Rane - 4 years, 10 months ago

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8 8 comes from 12 2 2 12-2-2 , i suggest you to draw the garden rather than imagine it, you'll understand it after you draw it

Jason Chrysoprase - 4 years, 10 months ago

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Ok, so this road is running 'inside' the 12 x 16 rectangle?

Ujjwal Rane - 4 years, 10 months ago

So another wording problem. And thankfully I am not the only one. Be clear that you are paving garden on the inside, and not expanding it out.

Saya Suka - 4 years, 7 months ago

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Thx i have editted it

Jason Chrysoprase - 4 years, 7 months ago

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