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?
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....
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....