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A flat, rectangular flower bed with an area of 2,400 square [#permalink]
05 Jun 2016, 12:32

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Question Stats:

35% (01:05) correct
65% (02:57) wrong based on 20 sessions

A flat, rectangular flower bed with an area of 2,400 square feet is bordered by a fence on three sides and by a walkway on the fourth side. If the entire length of the fence is 140 feet, which of the following could be the length, in feet, of one of the sides of the flower bed?

Re: A flat, rectangular flower bed with an area of 2,400 square [#permalink]
05 Jun 2016, 12:39

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Explanation

You know that the area of the rectangular flower bed is 2,400 square feet. So if the flower bed is a feet long and b feet wide, then ab = 2,400. If the side of the flower bed that is bordered by the walkway is one of the sides that are b feet long, then the total length of the three sides of the flower bed bordered by the fence is 2a + b feet. Since you are given that the total length of the fence is 140 feet, it follows that 2a + b = 140. Since ab = 2,400, you can substitute \(\frac{2400}{a}\) for b in the equation 2a + b = 140 to get the equation \(2a + \frac{2400}{a}=140\). It follows that \(2a^2 + 2400 = 140a\), or \(a^2 - 70a + 1200 = 0\).

When you solve this equation for a (either by factoring or by using the quadratic formula), you get a = 30 or a = 40. If a = 30, then \(b = \frac{2400}{30} =80\); if a = 40, then \(b = \frac{2400}{40} =60\). So the possible lengths of the sides are 30, 40, 60, and 80. Thus the correct answer consists of Choices B, C, E, and G.
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Re: A flat, rectangular flower bed with an area of 2,400 square [#permalink]
27 Feb 2019, 09:17

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sandy wrote:

A flat, rectangular flower bed with an area of 2,400 square feet is bordered by a fence on three sides and by a walkway on the fourth side. If the entire length of the fence is 140 feet, which of the following could be the length, in feet, of one of the sides of the flower bed?

Since we want to find possible dimensions, let's say one side has length x and the other side has length y

The entire length of the fence is 140 feet The BLUE lines represent fencing So, we can write: x + x + y = 140 Simplify to get: 2x + y = 140

The rectangular flower bed (aka garden) has an area of 2,400 square feet Area of rectangle = (base)(height) We can write: xy = 2400

Which of the following could be the length, in feet, of one of the sides of the flower bed? We have two equations: 2x + y = 140 xy = 2400

Take the TOP equation and rewrite as y = 140 - 2x Take the BOTTOM equation and replace 140 - 2x with 140 - 2x

We get: x(140 - 2x) = 2400 Expand to get: 140x - 2x² = 2400 Rearrange to get: 2x² - 140x + 2400 = 0 Divide both sides by 2 to get: x² - 70x + 1200 = 0 Factor to get: (x - 30)(x - 40) = 0 So, EITHER x = 30 OR x = 40

So, we can already see that a side COULD have length 30 or length 40 (answer choices B and C). However, we're not done yet!!

For each of these two possible x-values, we have y-values.

For example, one possible solution is x = 30 Since we already know that xy = 2400, we can replace x with 30 to get: 30y = 2400 Solve to get: y = 80 So, when one side has length 30, the other side has length 80. So, 80 is another possible answer (answer choice G)

Another possible solution is x = 40 Take xy = 2400, we can replace x with 40 to get: 40y = 2400 Solve to get: y = 60 So, when one side has length 40, the other side has length 60. So, 60 is another possible answer (answer choice E)

Answer: B, C, E, G

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Re: A flat, rectangular flower bed with an area of 2,400 square
[#permalink]
27 Feb 2019, 09:17