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Annie is mowing a rectangular field that measures 24 yards by 32 yards, by pushing the mower around and around the outside of the plot. This creates a widening border that surrounds the un-mowed piece of grass in the center. During a brief rest, Annie wonders whether the job is half done yet. How wide is the uniform mowed border when Annie is half done?

 Jan 3, 2015

Best Answer 

 #1
avatar+128079 
+5

Assuming that the whole field will be mowed (including the center)....The whole area of the field is (24 x 32) yds = 768 sq yds.   So (1/2) of this is 384 sq yds......and that is the measure of the unmoved area when she is half way through the job.   Call the uniform width, x...... so we have

(24 - 2x) (32 - 2x)  = 384     or

2(12 - x) * 2 (16 - x) = 384

4 (12-x) (16-x)  = 384   divide through by 4

(12-x)(16-x)  = 96

192 - 28x + x^2   = 96   expand and re-arrange terms

x^2  - 28x + 96 = 0  factor

(x - 24) (x - 4)  = 0

And setting each factor to 0, we have x =24 or x = 4  reject 24

So, the uniform width is 4 ft.   and the area of the unmowed center is (24-2(4)) (32- 2(4))  = (16)(24) = 384 sq yds, just as we expect....

 

 Jan 3, 2015
 #1
avatar+128079 
+5
Best Answer

Assuming that the whole field will be mowed (including the center)....The whole area of the field is (24 x 32) yds = 768 sq yds.   So (1/2) of this is 384 sq yds......and that is the measure of the unmoved area when she is half way through the job.   Call the uniform width, x...... so we have

(24 - 2x) (32 - 2x)  = 384     or

2(12 - x) * 2 (16 - x) = 384

4 (12-x) (16-x)  = 384   divide through by 4

(12-x)(16-x)  = 96

192 - 28x + x^2   = 96   expand and re-arrange terms

x^2  - 28x + 96 = 0  factor

(x - 24) (x - 4)  = 0

And setting each factor to 0, we have x =24 or x = 4  reject 24

So, the uniform width is 4 ft.   and the area of the unmowed center is (24-2(4)) (32- 2(4))  = (16)(24) = 384 sq yds, just as we expect....

 

CPhill Jan 3, 2015
 #2
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Thank you so much!!!

 Jan 4, 2015

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