求周长为定值L(L>0)的直角三角形的面积的最大值.
【答案】
分析:因为L=a+b+c,c=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/0.png)
,两次运用均值不等式即可求解;或者利用三角代换,转化为三角函数求最值问题.
解答:解:直角三角形的两直角边为a、b,斜边为c,面积为s,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/images1.png)
解法一:a+b+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/1.png)
=L≥2
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/2.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/3.png)
.
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/4.png)
≤
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/5.png)
.
∴S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/6.png)
ab≤
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/7.png)
(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/8.png)
)
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/9.png)
•[
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/10.png)
]
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/11.png)
L
2.
解法二:设a=csinθ,b=ccosθ.
∵a+b+c=L,
∴c(1+sinθ+cosθ)=L.
∴c=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/12.png)
.
∴S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/13.png)
c
2sinθcosθ=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/14.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/15.png)
.
设sinθ+cosθ=t∈(1,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/16.png)
],
则S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/17.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/18.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/19.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/20.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/21.png)
(1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/22.png)
)≤
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/23.png)
(1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/24.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213121691717022/SYS201310232131216917170011_DA/25.png)
L
2.
点评:利用均值不等式解决实际问题时,列出有关量的函数关系式或方程式是均值不等式求解或转化的关键.