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.