hihihi

Yes 2 is a factor of 34

\(a^2+b^2=c^2 \)

3² + 4²= c²

25=c²

^c=5

^3*4*1/2=6

^5*5=25

^25+6=31

^{31 units^2}

This is a finite geometric series with first term 1, common ratio 2 and 11 terms. Thus the sum is:

\(frac{1(1-2^{11})}{1-2} = \frac{1-2^{11}}{-1} = 2^{11}-1 = 2048-1 = \boxed{2047}\).

For x < a, the graph of y = f(x) is the same as the graph of y = ax+2a, which is a line with slope a and which passes through the point (a, a^2+2a). For \(x \ge a\), the graph of y = f(x) is the same as the graph of y = ax^2, which is a parabola passing through the point (a, a^3). Notice that the parabola only ever takes nonnegative values. Therefore, the line portion of the graph must have positive slope, because it must intersect horizontal lines which lie below the $x-$axis. Thus, $a > 0.$ For $a > 0,$ the line portion of the graph passes through all horizontal lines with height less than or equal to $a^2+2a,$ and the parabola portion of the graph passes through all horizontal lines with height greater than or equal to $a^3.$ Therefore, all horizontal lines are covered if and only if \(a^2 + 2a \ge a^3.\) Since  a > 0, we can divide by a to get a\( + 2 \ge a^2\), so \(0 \ge a^2 - a - 2 = (a-2) ( a+1)\). This means that \(-1 \le a \le 2,\) so the greatest possible value of a is \(\boxed{2}\). 

Triangle ABC is an isosceles triangle with the base of 8 units.

angle (B) = 2(arctan3/4)

angle (A/2) = 90 - angle (B)

height AD = BD / tan(A/2)

area = BD * AD = 54.85714289 units squared

This 6-term geometric series has first term \(a_0 = \frac12\) and ratio 1/2, so it has value

\(\begin{align*} \frac{\frac12(1-\left(\frac12\right)^{6})}{1-\frac12} &= 1-\left(\frac12\right)^{6}\\ &= 1-\frac1{64}\\ &= \boxed{\frac{63}{64}}. \end{align*}\)

Let the first term be a. Because the sum of the series is 60, we have 

\(60= \frac{a}{1-(1/8)} = \frac{a}{7/8} = \frac{8a}{7}.\)

Therefore, \(a=\frac{7}{8}\cdot60=\boxed{\frac{105}{2}}\).

using law of cosines
8^2 = 7^2+9^2 - 2*7*9 cos A
64 = 49 + 81 - 126 cos A
cos A = 66/126

x^2 = 130 - 126 cos(180-A)
but cos(180-A) = -cos A
so
x^2 = 130 + 126(66/126)
x^2 = 196
x = 14