Answer:
The years are:
- 1000 BCE, 999 BCE, 888 BCE, 777 BCE, 666 BCE, 555BCE, 444 BCE. 333 BCE, 222 BCE, 111 BCE
- 111 CE, 222 CE, 333 CE, 444 CE, 555 CE, 666 CE, 777 CE, 888 CE, 999 CE, 1000CE, 1011 CE, 1101 CE, 1110 CE,1222CE, 1333CE, 1444CE, 1555 CE, 1666 CE, 1777CE, 1888 CE, 1999CE, and 2000 CE
Explanation:
<u>1. Years BC:</u>
a) Years with four digits:
The first number with 3 equal digits is 1000. After that the years go decreasing: 999, 998, 997, ...
b) Years with three digits:
From 999 to 111, the numbers have three digits, thus the only that are solutions ara 999, 888, 777, 666, 555, 444, 333, 222, and 111: 9 numbers
After that the years have two digits, thus no solutions, with two digits.
Hence, we count 10 different years.
<u>2. Years CE</u>
a) Years with three digits:
- 111, 222, 333, 444, 555, 666, 777, 888, 999: 9 years
b) Years with four digits
i) Starting with 1:
- With three 0: 1000: 1 year
- With three 1: 1011, 1101, 1110: 3 years
- With three digits different to 1: 1222, 1333, 1444, 1555, 1666, 1777, 1888, 1999: 8 years
ii) Starting with 2:
- With three 0: 2000: 1 year
The next one with three equal digits is 2111 and it is after 2020 CE.
Therefore, 9 + 1 + 3 + 8 + 1 = 22 years starting with 2.
<u>3. Total</u>
<u />
10 years BC and 22 years CE have exactly three digits the same: 10 + 22 = 32.
An = a1 + (n-1)*d
n = term to find = 30
a1 = first term = 20
d = common difference = 2
a30 = 20 + (30 - 1) * 2
a30 = 20 + 29 * 2
a30 = 20 + 58
a30 = 78
sn = (n (a1 + a30)) / 2
s30 = (30(20 + 78) / 2
s30 = (30(98) / 2
s30 = 2940/2
s30 = 1470 <===
Area of the quarter sector
Area of the entire circle = pi * r^2
Area of the quarter circle = pi r^2 / 4
r = 9
Area of the quarter circle = pi 9^2 / 4
Area of the quarter circle = pi 81/4 = 20.25 pi
Area of the triangle
Area pf the triangle = 1/2 b*h
b = h = r
r = 9
Area of the triangle = 1/2 9 * 9
Area of teh triangle = 1/2 81
Area of the triangle = 40.5
Area of the shaded area
The shaded area = Area of the 1/4 circle - area of the triangle
The shaded area = (20.24 pi - 40.5)ft^2
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The awnser would be 30.
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