(1) Find the height h on the AB side of △ABC.
(2) Let DN=x, and when X is taken, what is the maximum area of the pool DEFN?
(3) During the actual construction, it was found that there was a big tree at the distance of point B 1.85 on AB. Q: Is this big tree on the edge of the largest rectangular pool? If so, in order to protect the big trees, please design another scheme, so that the largest rectangular pool to be built is connected with a triangle that meets the conditions and avoids the big trees.
Analysis: If the area of a rectangle is required to be the largest, the expression of the area should be listed first, and then the solution of the maximum value should be considered. In the junior high school stage, especially the present knowledge, we should apply the matching method to find the maximum value. (3) The design should be innovative, and the problem should be solved satisfactorily by applying the symmetry of the circle.
Solution: (1) by AB? CG=AC？ For BC, H= =4.8
(2)h = and DN=x
∴NF=
So s quadrilateral DEFN=x? (4.8-x)=- x2+ 10x
=- (x2- x)
=- [(x- )2- ]
=- (x-2.4)2+ 12
∫-(x-2.4)2≤0
∴- (x-2.4)2+ 12≤ 12 and when x=2.4, take the equal sign.
When x=2.4, SDEFN is the largest.
(when SDEFN is maximum, x=2.4. At this point f is the midpoint of BC. In Rt△FEB, EF=2.4 and BF = 3.
∴BE= = 1.8
∵bm= 1.85,∴bm>； EB, that is, the tree must be located next to the pool to be built, and the scheme should be redesigned.
When x=2.4, DE=5.
∴AD=3.2，
Another design scheme that meets the conditions can be seen from the symmetry of the circle, as shown in the figure:
At this time, AC = 6, BC=8, AD= 1.8, BE=3.2. This design not only meets the requirements, but also avoids the big trees.