All Answers 354657 Answers

n=1;c=0;cycle:b=( (n)^2 - (n-1)^2);d=(n+1)^2 - (n-1)^2;c=sort(c,b,d);n++;if(n<=2000, goto cycle, 0);printc,"Total =", count c

OUTPUT =751 - which includes zero and 1000. It trurns out that every number can be written as a difference of two perfect squares with the exception of ALL numbers of the form: 4n + 2, which is every 4th number beginning with 2, 6, 10, 14......etc. Or 1/4 of all numbers. The other 3/4 can all be written as the difference of two perfect squares. The above short code verifies this.

Your post was quite funny.  Many deities and some royals have apologists.  I qualify for one too. 

A point on a wheel moves at  32π/9 radians per minute.

How many seconds are required for the point to complete 7 revolutions?

Enter your answer, rounded to the nearest tenth,

Convert rotation speed to radians per second            32 π   rad        1    min          32 π     rad 

                                                                                    ——  ——   x   —  ——   =   ———  ——  

                                                                                      9      min       60   sec           540     sec 

1 revolution equals 2 π radians, so 

7 revolutions equals 14 π radians,

and the question becomes how long

does it take to turn 14 π radians                                              distance      (the distance is 14 π radians) 

                                                                                     time  =  ———— 

                                                                                                    rate          (the rate is the rad/sec determined above) 

To divide by a fraction, you invert 

the divider and multiply.                                                             14  π    radians        540       sec 

Note that π and radians cancel                                      time  =  —–—   ———   x   ———  ——— 

                                                                                                      1          1             32  π    radians   

                                                                                                     14 x 540    sec

                                                                                        time  =  ——–——  ——   =  236.25 sec rounds to 236.2 sec 

                                                                                                          32           1 

_.  

Hi, Melody.

32pi/9  R/min   /  60 sec/min  = .186168 R/sec

7 rev = 14 pi Radians

14 pi R / .186168 R/sec =   236.25 sec

Express this as \((a+b)^2 -ab=79\)

Rewrite as   \((a+b)^2 = 79 - ab\)

Look at all the perfect squares, say ps, less than 79.  Must have 79 - ab = ps  or  ab = 79 - ps

Look for two integers that satisfy ab = 79 - ps and (a + b)² < 79. 

I think there are only four pairs:

(-5, -3) 

(-3, -5)

(3, 5)

(5, 3)

What is the surface area of the triangular prism?

I just did a long answer and then lost it.

So I will give you the short version.

This is just a hyperbola.

You do not even need to worry about all those letters.

Just solve it as a hyperbola

\(y-k=\frac{k}{x-h}\\ y+3=\frac{k}{x-7}\\\)

Solve for k by subbing in (0,6)

then you have the equation. So you can easily find the x-intercept.

If you want to you can rearrange the formula to solve for a, b, c and d

Determine the value of k for which -3x^2+12x+k=0 has no real roots. ​

What is the area of the triangle below?

Firstly I will state outright that I have no control over what other people post here.

However. I do ponder a little about your motives.

I am very relieved that I am no longer teaching in a classroom. I expect mobile phones and internet in the classroom would make teaching near impossible at times. 

I always wanted my students to learn, that is certainly true. But I cared a lot more about the students that wanted to learn than I did about those that were not interested. Ultimately fully supervised exams would sort out those that cheated on assignments and those who were active learners.

From the little that I know, AoPS appears to be a very reputable internet learning site. Suitable for wealthy families and with scholarships for gifted children.  I expect you have a lot of children in your courses who are not gifted and are not interested. I think some of these children are pushed to succeed where they cannot. Hence they cheat. 

If your learning aids were of as high a value as you believe them to be then why would so many children come here, either for alternate explanation or for the purpose of cheating.

One example:  n = -2 + (-6 + 3√10)i

|n| = √[(-2)² + (-6 + 3√10)²] = √[130 - 36√10] ≈ 4.02

Not sure I can prove this is a minimum though!

Answered here: https://web2.0calc.com/questions/reposted-because-the-answer-i-got-last-time-was-incorrect#r9 

if you dont want to apologize we can always apologize for you 

Did you try graphing it using Desmos?

9+8+7+6+5+4+3+2+1 = 45      I think.

Volume of Cylinder A is                          pi times the area of the base times the height 

                                                              π r² h = (3.1416)(4)(4)(15) = 753.98 ft³ 

Volume of Cylinder B is likewise             pi times the area of the base times the height 

                                                              π r² h = (3.1416)(6)(6)(7)   = 791.68 ft³ 

After pumping all of Cyl A into Cyl B

there will remain empty space in B         791.68 – 753.98 = 37.7 ft³  

The percentage this empty space is

of the entire volume is                            37.7 / 791.68 = 0.0476 which is 4.8% when rounded to the nearest tenth  

_.

46656=2^6 * 3^6

1

2*2

2*2*3*3

2*2*3*3*3*3

2*2*3*3*3*3*3*3

2*2*2*2

2*2*2*2*3*3

2*2*2*2*3*3*3*3

2*2*2*2*3*3*3*3*3*3

2*2*2*2*2*2*2

2*2*2*2*2*2*2*3*3

2*2*2*2*2*2*2*3*3*3*3

2*2*2*2*2*2*2*3*3*3*3*3*3

3*3

3*3*3*3

3*3*3*3*3*3

I only count 16.  Maybe I missed some.

\(y=mx+c\) the perpendicular equation is:

\(y=-\frac{1}{m}x+c\) However, if it passes through another point then you need to find that point using the point-slope form:

\(y-y_1=m(x-x_1)\) (Since it is perpendicular then m will be \(-\frac{1}{m}\) (reciprocal opposite)(I.e. you flip the number or fraction and change sign)

\(y-3=\frac{1}{2}(x-2)\) Rearrange (solve for y)

\(y=\frac{1}{2}x+2\)

I wrote a computer program, and there were 450 numbers that worked.

The minimum of t^2 - 9t - 36 happens at t = 5/2.

Have you noticed that there are at least 13 posts asking this question? (There are more, but these have answers.)

https://web2.0calc.com/questions/help-im-dying-in-this-problem

https://web2.0calc.com/questions/a-parabola-with-equation-y-ax-2-bx-c-has-a-vertical-line-of-symmetry-at-x-2-and-goes-through-the-two-points-1-1-and-4-1-the-quadratic-ax-2-bx
https://web2.0calc.com/questions/help-me-please_130
https://web2.0calc.com/questions/please-help_11649
https://web2.0calc.com/questions/help-plz_44287
https://web2.0calc.com/questions/need-help-by-the-end-of-today
https://web2.0calc.com/questions/help-plz_5023
https://web2.0calc.com/questions/urgently-need-help-within-two-days
https://web2.0calc.com/questions/quick-math-question_7
https://web2.0calc.com/questions/math-help_2416
https://web2.0calc.com/questions/a-quadratic-of-the-form-2x-2-bx-c-has-roots-of-x
https://web2.0calc.com/questions/help_27016
https://web2.0calc.com/questions/algebra-2-parabolas-please-help

I know you're trying to help, but these people don't even search for existing answers. This shows how little effort they're putting into their math. :-(

As many other questions in this site, this is an AoPS question. AoPS students have a live class, where they learn from an instructor and the class is highly interactive. If you don't understand, then you can ask and there are people ready to help in real time. The online materials for the class are always available for the duration of the course.

Even if you can't make it to the class, you can see what happened later, as there's a transcript of each session. If you're working on your homework and have questions, AoPS also has a forum that is available 24/7. Lots of people are ready to help there. The website is well designed even for use with smartphones, so wherever you go, you get access to all of this. A student can receive hints on how to proceed. When they show some work, they get tailored replies depending on what's the part they're finding hard to understand.

Most students asking AoPS questions here are only looking for a cheap way out.

Isn't the answer just \(52+160=212^{\circ}\)? I feel like that's wrong though... it seems too easy....

The probability is 3/7.