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2. N is a positive integer. When N is divided by 3, the remainder is 3. When N is divided by 4, the remainder is 3. What is the remainder when N is divided by 12?

If some number is divided by 3 how can the remainder be 3?

1. AD is parallel to  BC

    AB is parallel to  DC                                                 Definition of a parallelogram

2. Extend segment CB in the direction of B to E                        A line connects any two poins

3. Angle ABE = Angle DAB             A transversal cutting parallel lines form equal alternative ineterior angles

4. Angle CBE = 180                       Definition of a straight angle                           

5. m Angle CBA + m Angle ABE = m Angle CBE         Angle addition postulate

6.   m Angle CBA + m Angle ABE = 180                   Substitution    

7.  Angle CBA is spplemental to angle  ABE       Definition of supplemental angles

8. Angle DAB is supplemental to angle CBA       Substitution

9. Angle BCD = Angle DAB        Opposite angles in a parallelogram are congruent

10. Angle  BCD is supplemental to Angle CBA      Substitution

$\cos^2(\theta) = 1\\ \text{there are only two values }x \text{ such that }x^2=1\text{, these are}\\ x=1,~x=-1\\ \cos(\theta)=1 \Rightarrow \theta = -360^\circ,~\theta=0^\circ\\ \cos(\theta)=-1 \Rightarrow \theta = -180^\circ,~\theta = 180^\circ$

f(x) = √x

Desmos:

just type in    f(x)=sqrtx

you can put the x in brakcets if you want to but it is not necessary.

(a) Use AAA simularity because the last angles are the same to prove they are similar.

(b) Use similar triangles to get QP/AB=PR/AC, so 4/3 is 28/(3x+6), and through cross-multiplying, 4(3x+6) is 84, and x=5.

HOPE THIS HELPED!

instead of f(x)=something

do y=something, its the same thing and

HOPE THIS HELPED!!

you have 5x boys and 2x girls, so there are 7x people in total. setting 7x to 35, x is 5 and there are 5*5=25 boys.

Answer: 25 boys

Yep, try to isolate the like terms! 

We have $\frac{2000}{24}=83 R 8$ , so   $8$ hours from $2:00 A.M.$ should be $\boxed{10A.M.}.$

2. Rewriting this, we have: $3x+3=4y+3, \frac{3x}{4}=y.$ Now, plugging a value of $x$ that will leave $y$ are a non-negative integer, is $x=4.$ Answer is 3? I might be wrong...

$a^{-1}\equiv a \pmod{24} \Longleftrightarrow a^2 \equiv 1 \pmod{24}\\ a^2-1 = 24k,~k \in \mathbb{Z}$

$|\{1,5,7,11,13,17,19,23\}|=8$

$\text{We notice that these are primes such that }p^2 > 24\\ \text{There must be an algebraic reason for this and I'll see if I can dig it up}$

The top is closed per the question....but it does not say if it is the same material as the bottom.....just states BOTTOM is 3/sqft.....

    I'm glad you got an answer!   I got stuck when I tried top made of same material as sides and only bottom at $3/sqft.    ~EP

Okay thanks!!

2z^2 = 38

z^2 = 19

z = + -  sqrt 19    

$3^2 = 9<10\\ 18^2 = 324 > 300\\ \{x : x^2 \in [10,300]\} = \{4,5,\dots, 17\}\\ | \{4,5,\dots, 17\}| = 17-4+1 = 14$

$\text{What's that? Lagrange multipliers?}\\ \text{The method of Lagrange multipliers involves solving}\\ \nabla (f(s,h) - \lambda (c(s,h)-C))\\ \text{In this problem }\\ f(s,h) = vol(s,h) = s^2 h\\ c(s,h)=6sh+6s^2 (\text{see the previous answer})\\ C=15$

$\text{our Lagrange equation becomes}\\ \nabla \left(s^2 h - \lambda (6sh-6s^2 - 15)\right)= (0,0,0)\\ \text{this leads to the system of equations }\\ 2sh - \lambda(6h - 12s)=0\\ s^2 - 6s\lambda = 0\\ 6sh-6s^2-15 = 0\\$

$\text{and these are solved for }s,~h,~\lambda\\ \text{This seems like a lot more work than the previous answer. Why do it?}\\ \text{In most problems you can't cleanly solve for one in terms of the other and substitute like we did}$

$\text{the wording is kind of odd. Is the bottom open? Sort of a weird way to orient the problem if so}\\ \text{I'm going to assume both the top and bottom are closed}\\ \text{we've got the two different types of materials so let's compute the two areas separately}\\ \text{assume the box has square side length }s \text{ and height }h\\ area_{side}=4sh\\ area_{tb} = 2s^2\\ vol = s^2h$

$\text{we can convert areas into cost using the materials prices}\\ C = 4sh\cdot (1.5) + 2s^2 \cdot (3) = 6sh + 6s^2$

$\text{I"m going to assume you haven't been taught Lagrange multipliers yet}\\ \text{so we have to use the cost function to write the volume in a single variable}\\ 15 = 6sh + 6s^2\\ h = \dfrac{15-6s^2}{6s}\\ vol = s^2 h = s^2\dfrac{15-6s^2}{6s} = \dfrac{15s-6s^3}{6}$

$\text{Now to find the extrema of the volume we set it's derivative equal to 0}\\ \dfrac{dvol}{ds} = \dfrac{15-18s^2}{6} =\dfrac{ 5-6s^2}{2} = 0\\ 5 = 6s^2\\ s = \sqrt{\dfrac 5 6}~ft \text{ (we can ignore the negative solution)}$

$vol = \dfrac 1 6 \left(15\sqrt{\dfrac 5 6} -6 \left(\dfrac 5 6\right)^{3/2}\right) = \dfrac 5 3 \sqrt{\dfrac 5 6}~ft^3$

No, it is not. February 17, 2017 is a  FRIDAY! Since February 2017 has 28 days, then it follows that 2 weeks later or 2 x 7 = 14 days from Friday Feb. 17, 2017 will be Friday March 3, 2017. 4 weeks later, or 4 x 7 =28 days will be Friday March 31, 2017. Since March is 31 days, then 1 more day will be Saturday April 1, 2017. 

April Fool's Day !

Thank you <3

20/120 x 93 = 15.5 pounds

$\dfrac{120}{20} = \dfrac{93}{x}\\ 120x = 1860\\ x = 15.5~lbs$

Short side length

sqrt (2^2 + 2^2) = sqrt 8

Long side length   sqrt (3^2 + 3^2) = sqrt(18)

Area = sqrt 8 x sqrt 18  = sqrt 144 = 12 sq units

$\text{a line perpendicular to a line with slope }m \text{ has slope }-\dfrac 1 m\\ y=-\dfrac 2 3 x + 5 \text{ has slope }m = -\dfrac 2 3\\ -\dfrac 1 m = \dfrac 3 2\\ \text{now we use point slope form of a line}\\ (y-1) = \dfrac 3 2 (x-8)\\ y = \dfrac 3 2 x - 11$