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Martin planted a rectangular garden
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08 Jun 2021, 09:02
08 Jun 2021, 09:02
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57% (02:40) correct
42% (03:35) wrong based on 33 sessions
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Martin planted a rectangular garden with dimensions 20 feet by 30 feet and then surrounded the garden with a rectangular brick walkway of uniform width (represented by the shaded area in the drawing above). If the area of the walkway equals the area of the garden, what is the width of the walkway?
(A) 1 foot
(B) 3 feet
(C) 5 feet
(D) 8 feet
(E) 10 feet
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Re: Martin planted a rectangular garden
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08 Jun 2021, 09:10
08 Jun 2021, 09:10
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Carcass wrote:
Martin planted a rectangular garden with dimensions 20 feet by 30 feet and then surrounded the garden with a rectangular brick walkway of uniform width (represented by the shaded area in the drawing above). If the area of the walkway equals the area of the garden, what is the width of the walkway?
(A) 1 foot
(B) 3 feet
(C) 5 feet
(D) 8 feet
(E) 10 feet
Here's an algebraic solution...
There are two areas to consider:
1) area of rectangular garden
2) area of walkway
We know the area of the garden = (20)(30) = 600 square feet.
Since the area of the walkway EQUALS the area of the garden, the area of the walkway also equals 600 square feet.
So, the area of the OUTER rectangle (which includes both the garden and the walkway) = 600 + 600 = 1200
What are the dimensions of the OUTER rectangle?
They are 20 + 2x by 30 + 2x
We can write: (20 + 2x)(30 + 2x) = 1200
Expand and simplify: 600 + 100x + 4x² = 1200
Rearrange to get: 4x² + 100x - 600 = 0
Divide both sides by 4 to get: x² + 25x - 150 = 0
Factor: (x + 30)(x - 5) = 0
So, EITHER x = -30 OR x = 5
Since the width of the walkway (x) cannot be negative, we can conclude that x = 5
Answer: C
Cheers,
Brent
General Discussion
Re: Martin planted a rectangular garden
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08 Jun 2021, 09:13
08 Jun 2021, 09:13
1
Area of the garden = \(30 * 20 = 600\)
Uniform width = \(x\)
New length of brick walkway = \(30 + 2x\) & width = \(20 + 2x\)
Area of walkway = Shaded region = \((30 + 2x) (20 + 2x) - 600\)
The area of the walkway equals the area of the garden
\(600 = (30 + 2x) (20 + 2x) - 600\)
\(600 = 100x + 4x^2\)
\(x^2 + 25x -150 = 0\)
\(x = 5\)
Answer C
Uniform width = \(x\)
New length of brick walkway = \(30 + 2x\) & width = \(20 + 2x\)
Area of walkway = Shaded region = \((30 + 2x) (20 + 2x) - 600\)
The area of the walkway equals the area of the garden
\(600 = (30 + 2x) (20 + 2x) - 600\)
\(600 = 100x + 4x^2\)
\(x^2 + 25x -150 = 0\)
\(x = 5\)
Answer C
