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Andrew draws this figure in the sand with his nose.  The sum of the areas of the shaded regions can be represented with the expression a*pi + b, where a and b are nonnegative integers.  Find a + b.

 

 Mar 28, 2021
 #1
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The Area of the bottom-most rectangle is 60 * 20 = 800 u2.

 

The Area of the "ring" is simply the difference of the two circles with the inner and outer radii.

 

Inner radius = 50 - 20 = 30u

Outer radius = 50u

 

Area = πrouter2 - 30πrinner2

Area = π502 - π302

Area = 2500π - 900π 

Area = 1600π u2

 

The Area of the "V" in the center can be simply divided up into two congruent parallelograms and a trapezoid.

 

Areaparrellogram= bh = 30*10 = 300 u2

Since we have 2 of these, the total is 300*2 = 600 u2

 

Areatrapezoid= h(a+b)/2 = 15(10+20)/2 = 15*30/2 = 225 u2

 

Area"V"= 600 + 225 = 825 u2

 

 

Now the total area is the sum of the areas of the "V", the ring, and the rectangle. (Sum of everything underlined)

 

825 + 1600π + 800 = 1600π + 1625 u2

From above, we can deduce that a = 1600, and b = 1625.

Therefore a+b = 3225 u2.

 Mar 28, 2021
 #2
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I don't mean to criticize, because the work is very good, but an error in the first line messes up the total.  

 

The first line,   The Area of the bottom-most rectangle is 60 * 20 = 800 u2.    should equal 1200

 

 Mar 29, 2021

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