NotThatSmart

mmm...take a look at the points given. 

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)

This is essentially just the perimeter of the decagon \(P_1P_2P_3P_4P_5...P_{10}\)

The lenngth of one side is essentially

\(radius/2 ( -1 +\sqrt 5)\)

Now that we know one side, we can find the perimeter through the equation

\(10 (1/2) ( -1 + \sqrt 5) = 5 ( -1 + \sqrt 5) ≈ 6.18\)

Thus, the answer is 6.18, 

Thanks! :)

lol, yeah. "just know your primes" 

I love your solution, Umeko1.0 😂😂😂

Nice work!

A prime number is a number that cannot be divisble except for 1 and itself. 

However, 1 does not qualify, since it technically only 1 factor. 

Now, the numbers \(2,3,5,7\) are all prime numbers. 

However, \(4=2*2, 6=2*3, 8=2*4, 9=3*3, 10=2*5\)

So the answer is 2,3,5,7

Thanks! :)

Let's simplify the two equations a bit first. We get that

\(8a + b = -3 \\ 9a + (2 +m)b = k \)

Now, 

This will have infinte solutions when the equations are the  same...so....multiply the first equation by 9 and the second by 8 and we get

\(72a + 9b = -27 \\\ 72a + 8 (2 + m)b = 8k\)

Thus, we can no easily solve for m and k. Solving for m, we get

\(9 = 8(2 + m) \\ 9 = 16 + 8m \\ -7 = 8m \\ m = -7/8\)

Solving for k, we get

\(8 k = -27 \\ k = -27/8 \)

Thanks! :)

oops, sorry. just realized I sent the wrong link...silly me

https://web2.0calc.com/questions/algebra_5878

First, let's simplify both equations to setup an interesting way to solve the problem. 

\((s/2) + 5t = 3 - 7t + 8s → s + 10t = 6 - 14t + 16s → -15s + 24t = 6 \space\space\space \text{(1st equation)} \\ 3t - 6s = 9 -2t → 6s + 5t = 9 \space\space\space \text{(2nd equation)} \)

Now, let's mulitply the 1st equation by 6 and the 2nd equation by 15

These numbers seem random, we when we write it out, we can see why it's so important. We get

\(-90s + 144t = 36 \\ 90s + 75t = 135 \)

Now we have the opposite s coefficients. Adding these equations, we get

\(219 t = 171 \)

\(t = 171 / 219 = 57 / 73 \)

Now, we can find s through the value of t. We get

\(6s + 5(57/73) = 9 \\ 6s + 285/73 = 9 \\ 6s = 9 - 285/73 \\ 6s = 372/73 \\ s = 62 / 73\)

Thus, our ordered pair (s, t) is \((57/73, 62/73)\)

Thanks! :)

First, let's find the common rate of change through the arithmetic sequence. Let's let this be d.

From the problem, we can write the equation

\(1/4 + 10d = -1/5 \\ 10d = -1/5 - 1/4 = -9/20 \\ d = -9/200\)

Now, we can find the 43rd term of the sequence. We have

\(43rd term = -1/5 + 10d = -1/5 + 10 (-9/200) = -1/5 - 9/20 = -65/100 = -13 / 20 \)

So our final answer is -13/20. 

Thanks! :) 

Question nicely answered here