EnchantedLava68

The triangle inside of the cylinder has radius 10, and is a 30-60-90 triangle, which means the small leg has length 5 and the height of the cylinder is \(5\sqrt3\).  

Thus, the area of the cylinder is radius^2 pi *height = \(5^2pi*5\sqrt3\) = \(125\sqrt3pi\), which is roughly 680.174761588

Every second, 40 mm of slime exit the funnel, and 60 mm of slime enter the funnel.  In total, however, 20 mm of slime enter the funnel every second.  

If the radius of the funnel is 4cm and the height is 12 cm, assuming the funnel is cylinder-shaped, the volume should be \(4^2pi*12\) or 192 pi cubic centimeters.  Thus, the slime will overflow the funnel at 192pi/20 seconds into the initial pouring, or roughly 

30.1592894745 seconds into the experiment.  

a)  After drawing all 3 of the midpoints into a triangle's 3 sides, connect them, and you should be able to prove that the 4 triangles formed are similar to each other.  

b)  Sorry, but I do not know the answer to this question, good luck regardless.  

According to some online resources, you can divide any triangle into 8 similar triangles.  

1.  ABC is an isosceles triangle, so the altitude from C to AB will divide the triangle into two identical right triangles, and be an angle bisector of the angle C.  Let the point of intersection = D.  Due to the fact that the sum of angles in a triangle is 180 degrees, and that this triangle is isosceles, angle a is 67.5 degrees. 

Sin (A) = CD/3 = Sin (67.5).  Thus, CD = 3Sin(67.5).  

Cos (67.5) = AD/3.  Thus, AD = 3Cos (67.5).  

AB = 2AD = 6 Cos (67.5)

Area = AB*CD/2 = 3Sin(67.5)*6Cos(67.5)/2 = 9Sin(67.5)Cos(67.5) = 3.18198051534 

2.  Let AB = a and BC = b

Thus, Tan (c) = 2.4 translates to a/b = 2.4, or b = 1/2.4a

Cos a = a/AC = \(\frac{a}{\sqrt{a^2+b^2}}\)

Substituting b = 1/2.4a into the expression, we have Cos A = \(\frac{a}{\sqrt{a^2+\frac{1}{2.4^2}a^2}}\)

Cos A = \(\frac{a}{\sqrt{\frac{169}{144}a^2}}\)

=\(\frac{a}{\frac{13}{12}a}\)

=\(12/13 \)

If by chance you have memorized common triangle trigonometric relationships (I haven't myself), then recognizing that ABC is a

5-12-13 triangle right away would be a very quick method.  

Long response, but I hope this answered your question.  Have a great day!

Area roughly is 5.44624

m= -1, b = 6, m+b = 5

If you would like to know my method, reply to this and I can take time to explain.  Basically though, just plug the ordered pairs into the equation and solve the system.  

2007 cannot be obtained

If we want to achieve a number near the 2015 range, 3 of the numbers will have plus signs, while two have minus signs.  

Writing each of the terms relative to 2015, we have -2, -1, 0, 1, and 2.  To obtain 2007, we must get a -8.  However, regardless of the combination of + and -, it is impossible to achieve a negative 8, even when optimized (-2, -1, 0, -(1), -(2) sum = -6).  

Thus, 2007 is impossible.  

Just to double-check, I calculated each of the other answers using the strategy listed above.  

2009 = 2013 + 2014 + 2015 - 2016 - 2017

2011 = 2013 + 2014 - 2015 + 2016 - 2017

2013 = 2013 + 2014 - 2015 - 2016 + 2017

2017 = 2013 - 2014 - 2015 + 2016 + 2017

Sorry for taking such a long time, I had to think about this for a bit before I could confidently post an answer.  

Anyways, have a great day!

Because 1 is not prime, there are 0 prime dates in January.  

2 is prime, so the prime dates in February are 2/2, 2/3, 2/5, 2/7, 2/11, 2/13, 2/17, 2/19, and 2/23 for a total of 8

3 is prime, so there are 10 prime dates in March.  

4 is not prime, 0 April Prime dates

5 is prime, 10 prime dates in May

6 is not prime, 0 June prime dates

7 is prime, 10 July prime dates

8 is not prime, 0 August prime Dates

9 is not prime, 0 September prime dates

10 is not prime, 0 October prime dates

11 is prime, 9 November prime dates

12 is not prime, 0 December prime dates.  

Together, there are 1+8+10+0+10+0+10+0+0+0+9+0 = 48 prime dates in 2020, 49 on a leap year (2/29 on Febuary)

If Mat sold 8 times as many apples as how much he gave to his son, this means he sold 40 apples.  Altogether, he loses 45 apples (5 to his son, 40 sold), and this is 1/2 of the original numbers of apples.  Hence, the original number of apples was 90

If 1/x + 1/y = 1/80 and x ≠ y ≠ 0, we can multiply both sides by 80xy to get 80x + 80y = xy

Moving all terms to one side, we have xy - 80x - 80y = 0

Adding 6400 to both sides, we can factor, giving (x-80)(y-80) = 6400.  

With x and y both being integers, we realize that x-80 and y-80 must be a pair of factors of 6400 giving a diophantine equation.  To minimize x+y, we want x and y to be as close as possible, and therefore, want x-80 and y-80 to be as close as possible.  To achieve this, we want to find two factors of 6400 to be as close as possible, and let x-80 and y-80 equal them.  Besides x=y= 160 (x-80=y-80=80), which is not possible due to the fact that x≠y, we have x-80 = 64  y-80 = 100.  

In this, x=144, y=180, so x+y = 324