Question 558597
<pre>
The size of the perimeter of the square ABCD is equal to 100 cm. The length of
the segment MN is equal to 5 cm and the triangle MNC is isosceles. Find the
area of the pentagon ABNMC.

(The picture on that site isn't to scale.  This is.)

{{{drawing(200,200,-5,30,-5,30,locate(25,28,B), locate(25.5,5,N), 
locate(-1,0,D), locate(21,0,M),locate(25,0,C), locate(-1,28,A),
rectangle(0,0,25,25), line(25-5/sqrt(2),0,25,5/sqrt(2)) )}}}

The square's perimeter is 100 cm, so each side of the square is 25 cm.
The area of the square is (25 cm)² or 625 cm²

We need to subtract the area of the triangle.  It is a right triangle, and
it isosceles, so MC = CN.  We use the Pythagorean theorem to find MC and CN.

MC² + CN² = MN²
MC² + MC² = 5²
     2MC² = 25
      MC² = 12.5
       MC = {{{sqrt(12.5)}}} = CN

Area of the triangle = {{{1/2}}}BASE×HEIGHT = {{{1/2}}}MC×CN = {{{1/2}}}{{{sqrt(12.5)}}}×{{{sqrt(12.5)}}} = {{{1/2}}}{{{(sqrt(12.5))^2}}} = {{{1/2}}}(12.5) = 6.25 cm²

Area of pentagon = Area of square - Area of triangle
                 = 625 cm² - 6.25 cm² = 618.75 cm²

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I won't draw the other one.

Initially the rectangular prism on the left was full of water. 

It measures 2cm×4cm×10cm

Volume = length × width × height = 2cm × 4cm × 10cm = 80cm³

so the volume of water in the original container is 80cm³


The first container into which part of it was poured to a height of h
is a rectangular prism like the original except that it is only
filled to a height of h cm.

Volume of water = length × width × height = 2cm × 4cm × h cm = 8h cm³

The second container into which the other part of it was poured to a height 
of h cm is a cylinder with radius 1 cm.

Volume of water = <font face = "symbol">p</font> × (radius)² × (height) = <font face = "symbol">p</font>(1)²h = <font face = "symbol">p</font>h = (3.14)h

so the volume of water in the cylinder is 3.14h cm³.

           water in original = water in 1st + water in 2nd 
                
                          80 = 8h + 3.14h
                          80 = 11.14h
                         {{{80/11.14}}} = h
                         7.2 = h

 So the height is 7.2 cm to the nearest tenth of a centimeter.    

Edwin</pre>