All Answers 358504 Answers

The sequence can be re-written as:

1/4 + 1/32 + 1/ 256 + 1/2048 +..........

(1/32) / (1/4) ==1 / 8 - this is the common ratio.

Sum ==F / [1 - r], where F=First term,   r==Common ratio

Sum== 1/4 / [1  -  1/8] ==1/4 / (7/8)

Sum==(1/4) x (8/7) ==8 / 28 ==2 / 7

Ditto GA ... in spades.  I don't know what AoPS is, but I have noticed that the same, identical problems are stated over and over and over, literally identical, some with misspelled words, &/or incorrect punctuation, and everything else.  A notable example is the one where the question is how much later will someone arrive if he drives faster.  I've posted many answers by just copying/pasting my answer from an earlier instance. 

Don't get me started on the trolls who post one-word answers – like 66 – without explanation how the anwer was obtained.  They can't explain because they just pulled it out of their, er, I'll say their hat, but that's not what I mean.  

You had the right idea but didn't implement it correctly:

1- There are 9 single digits from 1 to 9 ==9 pages

2 - You have 180 digits from 10 to 99 ==180 / 2 ==90 pages

3 - You have: 852 - 9 - 180 ==662 digits / 3 ==221 pages.

4 - Total pages in the book==9 + 90 + 221 ==320 pages

The Question is not difficult i want  try to give you simple explanation on it :

If 20 members can paint a 2000 square foot wall in 40 minutes, we can calculate the rate of painting per minute for the team.

The rate of painting for the team is:
2000 square feet / 40 minutes = 50 square feet per minute

Now, let's determine how long it would take the 20 members to paint a 6000 square foot wall.

Time = Area / Rate
Time = 6000 square feet / 50 square feet per minute
Time = 120 minutes

Therefore, it would take the 20 members 120 minutes to paint a 6000 square foot wall.   

The first 9 pages use 1 digit each, so they use 9 digits.

The next 90 pages use 2 digits each, so they use 180 digits.

The remaining pages use 3 digits each. Let n be the number of these pages. Then 3n is the number of digits used to number these pages, so 3n=852−9−180=663. Therefore, n=221​.

The given series is an alternating geometric series, with first term 1 and common ratio -1/2. The sum of an alternating geometric series is equal to the first term divided by 1 + the absolute value of the common ratio, so the sum of the given series is

1 / (1 + (-1/2)) = 1 / (3/2) = 2/3.

Therefore, the answer is 2/3​.

The point on the graph is of the form (x,21​x2−9), so the distance between the origin and this point is x2+(21​x2−9)2​. Squaring both sides, we get \begin{align*} a^2 &= x^2 + \left(\frac{1}{2}x^2-9\right)^2 \ &= x^2 + \frac{1}{4}x^4 - 9x^2 + 81 \ &= \frac{1}{4}x^4 - 8x^2 + 81. \end{align*}We can factor this as \begin{align*} a^2 &= \frac{1}{4}(x^2 - 8)^2 + \frac{324}{4} \ &= \frac{1}{4}(x^2 - 8)^2 + 81. \end{align*}The minimum value of 41​(x2−8)2 is 0, which occurs when x2=8.

Therefore, the minimum value of a^2 is 81, so a^2=81​.

Hey Ginger!  

It's so great to hear from you.  I haven't been seeing postings of yours for quite a while, and I was afraid I never would again.  Thank you for that explanation about permutations.  I agree that Bosco the Chocolate Bear would have been pleased with my username.  Or is pleased, if he's watching from cat heaven. 

This is something funny.  It turns out that my username here is a cat I used to have, and my user name on eBay is a cat my grandma used to have.  I didn't plan it that way, and the coincidence just dawned on me a couple of days ago. 

Ron  

this was answered here: https://web2.0calc.com/questions/help-with-geo_28 but i don't know if it's correct

Yes. I agree. I cannot tell you the amount of times I have seen my friend Google Alcumus questions and use this website.

Since we are given that the line y=−x+4 is perpendicular to the line containing side CD, we know that the slope of the line CD is −1. Therefore, we can use the point-slope form of linear equations to find the equation of the line containing side CD.

We know that one point on the line is (m,4), and we know that the slope of the line is −1. So, we get the following equation: \begin{align*} y - 4 &= -1(x - m)\ y &= -x + m + 4 \end{align*} We are also given that the area of the trapezoid is 34 square units, so we can use the formula for the area of a trapezoid to get the following equation: \begin{align*} \frac{1}{2}(1 + m)(4 + 0) &= 34\ 2 + m &= 34\ m &= 32 \end{align*} Now that we know m=32, we can plug it back into the equation y=−x+m+4 to get the equation of the line containing CD: \begin{align*} y &= -x + 32 + 4\ y &= -x + 36 \end{align*} Finally, we can plug the coordinates of point D, (k,0), into this equation to solve for k: \begin{align*} 0 &= -k + 36\\ k &= \boxed{36} \end{align*}

ratio of yellow, green, blue, and white marbles is 1:2:3:4, so ratio of blue marbles to total marbles is 3:(1+2+3+4)=3:10

x: the total number of blue marbles

3/10 = x/180

multiplying both sides by 180,

x = 54

Will and Grace are canoeing on a lake.  Will rows at 50 meters per minute and Grace rows at 20 meters per minute. Will starts rowing at 2 p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at 2 p.m. If they always row directly towards each other, and the lake is 3800 meters across from the west side of the lake to the east side, at what time will the two meet?   

Will's rate is 50 m/min.  

Grace's rate is 20 m/min.  

They're rowing toward each other,  

so their rate of closure is 70 m/min.  

The lake is 3800 meters across.  

The time it will take them to meet is given by  

                                                                                    Distance  

                                                                            T  =  –––––––  

                                                                                       Rate  

                                                                                    3800 m  

                                                                            T  =  –––––––  

                                                                                    70 m/min  

The time it takes them to meet                             T  =  54.29 min  

Note that (0.29 min • 60 sec/min)  =  17 seconds

The time on the clock is                                             2:54:17 pm 

_.

factoring a from the second equation:
a(1 - b) = 0.

two cases satisfy this: a = 0 or b = 1.

if a = 0, the first equation becomes:
0 = 250b.
so a = 0, b = 0 is one solution.

if b = 1, the first equation becomes:
a + a = 250.
so a = 125, b = 1 is another solution.

solutions are (0,0), (125,1).

It Would be helpful if you posted a picture

Thank You guys for your help! It turned out the correct answer was 36 sq units. 

Let's assume Alice has A dollars and Bob has B dollars.

According to the first condition, if Alice receives n dollars from Bob, Alice will have A + n dollars, and she will have 4 times as much money as Bob: A + n = 4B.

According to the second condition, if Alice gives n dollars to Bob, Alice will have A - n dollars, and she will have 8 times as much money as Bob: A - n = 8B.

To find the ratio of the amount of money Alice has to the amount Bob has, we need to find A/B.

Now, we can solve the system of equations:

A + n = 4B A - n = 8B

Adding the two equations, we get:

2A = 12B A = 6B

The top and bottom vertices of the square, (as seen in the diagram), remain in contact with the top and bottom sides of the rectangle respectively, so the orientation of the square will be the same as it slides from one end of the rectangle to the other.

Slide the square to the left so that the left hand vertex of the square comes into contact with the left hand side of the rectangle.

Two triangles are formed and their interiors will be will be areas of the rectangle that can't be covered by the square. 

The two triangles are congruent. (Call one of the angles of one of the triangles theta, and work out, in terms of theta, the other three angles of the two triangles. That will show that they are similar. They have the same hypotenuse so they are congruent.)

Now slide the triangle to the right so that the right hand vertex of the square comes into contact with the right hand side of the rectangle. Again two triangles are formed and they will congruent with the two triangles on the left. 

The total area of the rectangle that can't be covered by the square will be the combined area of the four triangles, which will be four times the area of a single triangle because of the congruency. Notice that the second dimension of the rectangle, the 18, is irrelevant,(so long as it's big enough for the rectangle to contain the square). 

The hypotenuse of the triangle(s) is 8, let the lengths of the other two sides be x and y, then

\(x^{2}+y^{2}=64, \\ x + y = 10.\)

Solving simultaneously,

\(x = 5 \pm \sqrt{7}, \quad y = 5 \mp\sqrt{7},\)

so the area of a single triangle will be

\((5+\sqrt{7})(5-\sqrt{7})/2=(25-7)/2=9,\)

meaning that the total area that can't be covered by the square is 36.

Thank you, Alan.  I didn't know about that difference. 

_.