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.

Guest Mar 28, 2021

#1**0 **

The Area of the bottom-most rectangle is 60 * 20 = __800 u ^{2}__.

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 = πr_{outer}^{2} - 30πr_{inner}^{2}

Area = π50^{2} - π30^{2}

Area = 2500π - 900π

Area = __1600π u ^{2}__

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

Area_{parrellogram}= bh = 30*10 = 300 u^{2}

Since we have 2 of these, the total is 300*2 = 600 u^{2}

Area_{trapezoid}= h(a+b)/2 = 15(10+20)/2 = 15*30/2 = 225 u^{2}

Area_{"V"}= 600 + 225 = __825 u ^{2}__

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 u^{2}

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

Therefore a+b = **3225 u ^{2}**.

ArithmeticBrains1234 Mar 28, 2021