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The three angles of the triangle = 180 ^o

^{(180 -122)   +   (180 -118)    +  3y = 180}

^{When you find y    then     3y +  (x-25)  = 180    (straight line)    to slove for x}

\(\displaystyle (1+5i)^{3}=-74-110i.\)

since they are collinear and x coordinates af A and C are -3 hence N=-3 ..
now AB/AC=2/5
but AC = y co ord of A - Y coord of C = 6 - (-4)=10
implies
6-Q/10 = 2/5
hence 6-Q=4 ANd Q=2
(N,Q) = (-3,2)

eh been okay so far

my day so far has be okay

Bryn’s Grandpa is always creating math problems for her to solve. In one problem, he gave her a store receipt that listed 72 identical items, each the same price, for a total cost of $ *67.9*.  * is the missing number. Bryn’s Grandpa covered the first and last digits of the total price on the receipt with stars. Determine the values of the digits that Bryn’s Grandpa covered.

 #67.9&   must be exactly divisible by 72.       72=2*2*2*9

a) Since it is divisible by an even number, what do you know about &  ??

b)  A number is divisible by 9 if the digits add up to a number that is divisible by 9.

     so ....                   6+7+9+  # + & = 9k               k is an integer, greater that 0

                                 15      +  # + & = 9k
                                   # + & = 9k -6

                                  # + & = 5 or  14

This has taken it down to 5 possibilities.

Which of those are divisible by 8?

You can take if from there.

Find the remainder when x^15 + 1 is divided by x-1.

Hello Guest!

     \((x^{15}\) \(+1)\)    :  \((x-1)\) \(=x^{14}+x^{13}+…+x^2+x \)\(+\large \frac{x+1}{x-1}\)

      \(\underline{x^{15}-x^{14}}\)                                                                     

                 \(x^{14}\)

                 \(\underline{x^{14}-x^{13}}\)                                                            \( \frac{x+1}{x-1}=1+\frac{2}{x-1}\)

                            \(x^{13 }\)

                                     .... \(\underline{x^2-x}\) 

                                                    \(x\color{red}+1\) 

  !

A = (12 * 8) + (3 * 6) / 2 

sumfor(n, 2,1000, (4*n^3-n^2-n+1) / (n^6-n^5+n^4-n^3+n^2-n)) = It converges to 1

Partial Sum Formula = (m^3 + m^2 - m - 1)/(m (m^2 + m + 1))

A = [sqrt(2 * 6²) / 2]² *pi - 6²/ 2

Find all real solutions x to 4^x - 2^x = 40.

Hello Guest!

\(4^x - 2^x = 40\)

\(x = 2,7749006304559742 \)     Arndt Brünner

A solution is not possible with simple means.

  !

The area of the outer square is  5² = 25 cm² 

The area of the blue square is    3.8² = 14.44 cm²

The area of a triangle is   (25 - 14.44) / 4 = 2.64 cm²

The height (h) of a triangle       h = (2 * 2.64) / 3.8

                                                  h = 1.389473684 cm    

The quotient is x^2 + 4x - 3 and the remainder is 3x - 5.

r^s*t = 1/4.

The outer square is 5cm by 5 cm.  How do I work out the height of the triangle (the line the arrow is pointing to)?

Hello Guest!

The sides of the triangle are a and b. Then

\(a+b=5\\ b=5-a\\ a^2+b^2=3.8^2\\ a^2+(5-a)^2=3.8^2\\ a^2+25-10a+a^2-3.8^2=0\)

\(2a^2-10a+10.56=0\\ a^2-5a+5.28=0\)

\(a=2.5\pm \sqrt{6.25-5.28}\\ a=2.5\pm 0.985\)

\(a=3.485\\ b=1.515\)

\(tan\ \beta=\frac{b}{a}\\ \beta=atan\ \frac{b}{a}\\ sin\ \beta=\frac{h}{a}\\ h=a\cdot sin(atan\frac{b}{a})\)

\(h=3.485\cdot sin(atan\frac{1.515}{3.485})\)

\(\large h=1.389\ cm\)

  !

What is the minimum possible product of three different numbers of the set {-8,-6,-4,0,3,5,7}?

Hello DragonXlayer!

The minimum possible product of three different numbers of the set {-8,-6,-4,0,3,5,7}

is the product of the numbers of the set {-8,5,7}, so -280.

  !

The lines -3x + y = k and 0.5*x + y = 14 intersect when x = -2. What is the value of k?

Hello Guest!

\(y=k+3x\\ y=14-0.5x\)

\(k+3x=14-0.5x\\ k=14-0.5x-3x\\ x=2\\ k=14-1-6\)

\(k=7\)

  !

How many liters of pure water should be mixed with a 30L solution of 60% acid to produce a mixture that is 70% water? How many liters of pure water should be mixed with a 30L solution of 60% acid to produce a mixture that is 70% water?

Hello Guest!

\(x+30l\cdot (100-60)\%=(30l+x)\cdot 70\%\)

\(x+30l\cdot 0.4=30l\cdot 0.7+0.7x\\ x-0.7x=30l\cdot (0.7-0.4)\\ 0.3x=0.3\cdot 30l \)

\(x=30l\)

30 liters of pure water should be mixed with a 30L solution of 60% acid to produce a mixture that is 70% water?

  !

A8 =78  and  3B =35  and   78 x 35 =2730

A=7,  B =5,   C=2

Thanks guest 

Circle radius   r = 2√5

The length of a side of the square is  2(2√5)

Area of the square = [2(2√5)]² units squared

What is it that you trying to find out? Do you want the number of "Trailing Zeros" in converting 1000! * 100! * 10! from base 4 to base 12?

Nevermind, I got the answer, which was 1458.

The inscribed 90 degree angle encompasses 180 degrees of the circle....so the hypotenuse is a diameter of the circle

find hypotenuse via pythag theorem

shaded area = area of circle - area of tringle