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Yes there is an easier way. Assuming that triangles are still distinct even through rotations/reflections, etc, then the total number of points you could choose is 16 choose 3. However that would get some straight lined triangles, so we have to eliminate those. There are 16 + 16 + 4 + 4 = 40 of these cases counting three consecutive, 2 consecutive and 1 two apart, then considering the diagonal cases. Thus there are 16 choose 3 - 40 = 520 total triangles that are non-degenerate. (assuming that rotating the grid is not going to eliminate cases)

The door is most likely rectangular, meaning that the angles are all 90 degrees. The distance between the opposite corners are the diagonals of the rectangle, which has base length 39 and height 80 (can be used interchangebly). By the pythagorean theorem, the hypotenuse/diagonal squared = width^2 + length^2. So diagonal^2 = 80^2 + 39^2 = 7921. Square root both sides and you get 89.

The distance between the opposite corners of the door is 89 inches.

Open up paranthesis by FOILing: 9x^2 + 36x + 12x + 48 = (4x^2 + 8x + 18x^2)/2x

Combine like terms: 9x^2 + 48x + 48 = (22x^2 + 8x)/2x

Factor out top and bottom by 2x, then cancel terms: (22x^2 + 8x)/2x = 2x(11x + 4)/2x = 11x + 4

9x^2 + 48x + 48 = 11x + 4

Combine like terms: 9x^2 + 37x + 44 = 0

Use quadratic formula: \({-37\pm\sqrt{37^2 - (4)(9)(44)}}\over18\) = \({-37 \pm\sqrt{1369 - 1584}}\over18\) = \(-37\pm i\sqrt{215}\over18\) = x

The middle number is divisible by 5 so it must end in 0 or 5. If it ends in 0, then the next number ends in a 1, however the problem stated it was divisible by 4, and we know that the number must be even, so the last number ends in a 6. The middle number ends in a 5, and the first number ends in a 4. If the first number ends in a 4, we can consider 2 digit numbers, so 24 is div by 3, 25 is div by 5, but 26 is not div by 4. I can say 24 is div by 3 because of the divisibility rule for 3 (the sum of the digits of the number is div by 3), Lets try (2+3) and 4, so 54, 55, and 56. We can see those are divisible by 3, 5, then 4. 54 + 56 + 55 = 110 + 55 = 165

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changes in state

Answer the question under your first post!

Find (s, t).

Hello Guest!

\(\dfrac{s}{2} - 3t = -1\ |\ \cdot 2\\ s-6t=-2\ |\ -s\\ -6t=-s-2\\ 3t - 2s = 3 + s\ |\ \cdot -2\\ -6t+4s=-6-2s\ |\ -4s\\ -6t=-6s-6\\ -s-2=-6s-6\\ 5s=-4\\ \color{blue}s=-0,8\)

\(-6t=-6s-6\ |\ \div -6\\ t=s+1\\ t=-0.8+1\\ \color{blue}t=0.2\)

\(\)

The ordered pair \(\color{blue}(s,t)=(-0.8,\ 0.2).\)

  !

You need to find the 3rd angle, using the angle sum of a triangle

and then use the Sine Rule.

\(\sqrt{27} - \sqrt{12}*\sqrt{300} + \sqrt{24}*\sqrt{81}\\ =\sqrt{9*3} - \sqrt{4*3}*\sqrt{100*3} + \sqrt{4*6}*9\\ =3\sqrt{3} - 2\sqrt{3}*10\sqrt{3} + 2\sqrt{6}*9\\ =3\sqrt{3} - 20\sqrt{9}+ 18\sqrt{6}\\ =3\sqrt{3} - 20*3+ 18\sqrt{6}\\ =3\sqrt{3} - 60+ 18\sqrt{6}\\\)

The answer is 88 dollars.

The area is 8*sqrt(3).

There are 4^3 = 64 smaller cubes that do not have any black faces.

The answer is 6 1/2 cups.

The answer is 51.

a + b = 3

First odd integer = 501

Last odd integer =899

[899  -  501] / 2  +  1 =200 odd terms

[501  +  899] / 2  x  200 =140,000 - sum of all odd integers between 500 and 900

Let the number of hours worked by Penelope=P

Then, the number of hours worked by Sarah =2P

35P  +  30(2P) = 475, solve for P

35P  +  60P = 475

95P = 475

P = 475 / 95 =5 hours - Penelope's time

2  x  5 =10 hours - Sarah's time.

4:05  -  5 minutes = 4:00 PM

5  +  (3 x 60) + 9 + (1 x 60) + 4 =5 + 180 + 9 + 60 + 4 =258 minutes after 4:00 PM

258 / 60 = 4 hours + 18 minutes - time elapsed after 4:00 PM

4 hours + 4 hours + 18 minutes =8:18 PM - when they finished setting up their campsite.  

Let the number of phones sold =P

The number of accessories sold=2P

11P  +  3(2P) = 102

11P  +  6P = 102

17P = 102

P = 102 / 17 =6 - number of phones sold

2 x 6 = 12 - accessories sold.