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0, 0 x 9 = 0

1267650600228229401496703205376

4^50 16^25

It's way more than a thousand

1,000

1,2,3

3 you dumbo

cant promise u this is right but i calculated that it is 30 square units 

im a gators fan :)

New York Giants

Find the diameter of a right cone with a slant height of  \(18\ cm\) and a surface area of \(208 \pi \ cm^2 \).

Surface area of cone \( A = \pi r^2+\pi rl\)

\(\begin{array}{|rcll|} \hline \pi r^2+ \pi rl &=& A \quad & | \quad : \pi \\ r^2+rl &=& \frac{A}{\pi} \\ r^2+rl -\frac{A}{\pi} &=& 0 \\ r &=& \frac{-l\pm \sqrt{l^2-4\cdot (-\frac{A}{\pi}) } } {2} \\ r &=& \frac{-l\pm \sqrt{l^2+ \frac{4A}{\pi} } } {2} \\ 2r = d &=& -l\pm \sqrt{l^2+ \frac{4A}{\pi} } \quad & | \quad A = 280\pi \qquad l = 18 \\ d &=& -18\pm \sqrt{18^2+ 4\cdot 208 } \\ d &=& -18\pm \sqrt{1156 } \\ d &=& -18\pm 34 \\ d &=& -18 + 34 \\ d &=& 16 \\ \hline \end{array}\)

The diameter  is 16 cm

Find the diameter of a right cone with a slant height 18cm of  and a surface area of 208 pi cm².

There is a row of Pascal's triangle that has three successive positive entries,"a" "b"  and "c"

such that "b" is double "c" 

and "a" is triple "c"

If this row begins "1,n,"  then find n.

Three successive positive entries:

\(\begin{array}{rcll} a&=&\binom{n}{k-1} \\ b&=&\binom{n}{k} \\ c&=&\binom{n}{k+1} \\ \end{array} \)

"b" is double "c" and "a" is triple "c"

\(\begin{array}{|rcll|} \hline a &= 3c &=& \binom{n}{k-1} \\ b &= 2c &=& \binom{n}{k} \\ c & &=& \binom{n}{k+1} \\ \hline \end{array} \)

\(\begin{array}{|lrcll|} \hline (1) & 2c &=& \binom{n}{k} \quad & | \quad c = \binom{n}{k+1} \\ & 2\cdot \binom{n}{k+1} &=& \binom{n}{k} \quad & | \quad \binom{n}{k+1}= ( \frac{n-k}{k+1} ) \binom{n}{k} \\ & 2\cdot ( \frac{n-k}{k+1} ) \binom{n}{k} &=& \binom{n}{k} \\ & 2\cdot ( \frac{n-k}{k+1} ) &=& 1 \\ & \mathbf{ n-k } & \mathbf{=} & \mathbf{ \frac{k+1}{2} } \\\\ (2) & 3c &=& \binom{n}{k-1} \quad & | \quad c = \binom{n}{k+1} \\ & 3\cdot \binom{n}{k+1} &=& \binom{n}{k-1} \quad & | \quad \binom{n}{k+1}= ( \frac{n-k}{k+1} ) \binom{n}{k} \\ & 3\cdot ( \frac{n-k}{k+1} ) \binom{n}{k} &=& \binom{n}{k-1} \quad & | \quad \binom{n}{k-1}= ( \frac{k}{n-k+1} ) \binom{n}{k} \\ & 3\cdot ( \frac{n-k}{k+1} ) \binom{n}{k} &=& ( \frac{k}{n-k+1} ) \binom{n}{k} \\ & 3\cdot ( \frac{n-k}{k+1} ) &=& \frac{k}{n-k+1} \\ & 3\cdot (n-k)\cdot (n-k+1) &=& k\cdot (k+1) \quad & | \quad \mathbf{ n-k } \mathbf{=} \mathbf{ \frac{k+1}{2} } \\ & 3\cdot ( \frac{k+1}{2} )\cdot ( \frac{k+1}{2} +1) &=& k\cdot (k+1) \\ & 3\cdot ( \frac{k+1}{2} )\cdot ( \frac{k+3}{2} ) &=& k\cdot (k+1) \\ & \frac34\cdot (k+1)\cdot (k+3) &=& k\cdot (k+1) \\ & \frac34 \cdot (k+3) &=& k \\ & \frac34 k + \frac94 &=& k \\ & k-\frac34 k &=& \frac94 \\ & \frac14 k &=& \frac94 \\ & \mathbf{ k } & \mathbf{=} & \mathbf{9} \\\\ & \mathbf{ n-k } & \mathbf{=} & \mathbf{ \frac{k+1}{2} } \\ & n-9 & = & \frac{9+1}{2} \\ & n-9 & = & 5 \\ & \mathbf{ n } & \mathbf{=} & \mathbf{ 14 } \\ \hline \end{array}\)

The three successive positive entries are:

\(a=3003 =\binom{14}{8} \\ b=2002 =\binom{14}{9} \\ c=1001 =\binom{14}{10} \\ \)

and n is 14.

\(\frac{1}{2}*8*4\) simplifies to \(16\).

\(\frac{1}{2}*8*4\)

Multiplying by 1/2 is the same as dividing by two. The rest should be fairly simple:

\(\frac{1}{2}*8*4=\frac{8*4}{2}=\frac{32}{2}=16\)

What is 3-eighths when converted to a percentage?

3/8 = 0.375 x 100 =37.50%

\(y=0\)    is the answer

Let's look at the original equation:
 

\(8+y=8-y\)

To solve this, we must isolate y. This means that we need to get the equation into y equals form. 

\(8+y=8-y\) Subtract 8 on both sides

If you subtract a number on one side of the equation, you must do that on the other, of course. When you subtract 8 on both sides, you are left with:

\(y=-y\)     Add y to both sides

\(y+y=0\)     Combine like terms

\(2y=0\)     Divide by 2 to isolate y

\(y=0\)

If you are ever unsure whether an answer is correct or not, plug the value you got back in the original equation:

\(8+0=8-0\)

\(8=8\)

This statement is true, so y=0 is the correct and the only correct value for y. 

17*40=680

You can use a calculator for simple calculations.

3*47=141

You can use a calculator that is available on this site (or on others) to calculate simple expressions like this one.

My calculator died.

thx if you can tell me!

This is a word to word question from the MCYA competition. If you are apart of this then you are cheating as this is not allowed.

Jack invests $500 at a certain annual interest rate, and he invests another $1000 at an annual rate that is one-half percent higher. If he receives a total of $65 interest in 1 year, at what rate is the $500 invested?

Here is a start  :)

I am using I=PRT which is the same for both simple and compound interest if the time unit is just one. i.e. t=1

Solve this and you will get the initial rate as a decimal. Multiply it by 100 to get  the percentage.

\(65 = 500*r+1000(r+0.005)\)

Jack invests $500 at a certain annual interest rate, and he invests another $1000 at an annual rate that is one-half percent higher. If he receives a total of $65 interest in 1 year, at what rate is the $500 invested?L

Let the $500 be invested @ x,

$1,000 will be invested x + 0.5, but we have:

500*x/100  + 1,000*(x/100 + 0.005) = 65, solve for x

Solve for x:
1000 (x/100 + 0.005) + 5 x = 65

Put each term in x/100 + 0.005 over the common denominator 100: x/100 + 0.005 = x/100 + 0.5/100:
1000 x/100 + 0.5/100 + 5 x = 65

x/100 + 0.5/100 = (x + 0.5)/100:
1000 (x + 0.5)/100 + 5 x = 65

1000/100 = (100×10)/100 = 10:
10 (x + 0.5) + 5 x = 65

10 (x + 0.5) = 10 x + 5.:
10 x + 5. + 5 x = 65

Grouping like terms, 10 x + 5 x + 5. = (10 x + 5 x) + 5.:
(10 x + 5 x) + 5. = 65

10 x + 5 x = 15 x:
15 x + 5. = 65

Subtract 5. from both sides:
15 x + (5. - 5.) = 65 - 5.

5. - 5. = 0:
15 x = 65 - 5.

65 - 5. = 60.:
15 x = 60.

Divide both sides of 15 x = 60. by 15:
(15 x)/15 = 60./15

15/15 = 1:
x = 60./15

60./15 = 4.:
Answer: | x = 4%