imsmarter1234

We know that the interior angle measure of an equilateral triangle is 60 degrees, and the interior angle measure of a regular dodecagon is 150 degrees. Summing it up, we have 210 degrees. Because 360 degrees surround a point, the regular n-gon must have 150 as it's interior angle measure. A dodecagonn would work, so n = 12.

We can first find the area of a hexagon. If we break it into 6 equilateral triangles, we can calculate their areas. Let's say the side length of the regular hexagon is \(s.\) The area of one of the equilateral triangle is \(\frac{s^2 \sqrt{3}}{4}\), so the area of the entife hexagon is \(\frac{3s^2 \sqrt{3}}{2}.\) Because we know the area is 3, we can solve for s:

\(\frac{3s^2 \sqrt{3}}{2} = 3 \) ==> \(3 s^2 \sqrt{3} = 6\) ==> \(s^2 \sqrt{3} = 2\) ==> \(3 s^2 = 2 \sqrt{3} \) ==> \(s^2 = \frac{2 \sqrt{3}}{3}\)

This is best I can go. It get's really complicated if you go even further. Please let me know if I made a mistake.

Will rows for 45 minutes until Grace starts. In that time Will advances \(45 \cdot 50 = 2250\) meters. Every minute they get \(80\) meters closer to each other. At the time Grace starts, they will be \(550\) meters apart. It will take \(\frac{550}{80}\)minutes, which is around \(7\) minutes. They will meet at around \(\boxed{2:52} .\)

\(1+2+3 = 6.\)The prime factors of 6 are 2 and 3. The greatest one is \(\boxed3\)

I hope this is not too much, but can you solve it with similar triangles?

Expanding the product, we get \(x^7 + 4x^6 + 4x^5 + 19x^4 + 19x^3 + 16x^2 + 16x + 1\). We see that the coefficient of x is 16.

When we expand the product, we get \(x^7 + 4x^6 + 4x^5 + 19x^4 + 16x^2 + 16x + 1.\) We see that the coefficient of \(x^2\) is \(16\).

If we multiply both sides by 3, we get \(3x + 3y = 75\) . We notice that \(6x+3y\) is \(3x\) more than \(3x + 3y\). If \(x\) is 0, then \(3x\) is 0. 75 + 0 = 75, so the minumum value of \(6x + 3y\) is 75.

Anything to the power of 0 is 1, so if x = 0, 9^x would be 1. Lets assume x = 0. Then \((9^x - 2) \cdot (3^x + 1)\) would be equal to \(-1 \cdot 1\) or -1. So, the minimum value of \((9^x - 2) \cdot (3^x + 1)\) is -1.