All Answers 358150 Answers

Are you sure?

I edited my answer- is it correct now?

I'm not too sure how to do number 1 (not that great with trig) but here's number 2. 

2. Let's write out the information we have so far. Set a, b, and c as the three sides.

a+b+c=19/4

2(ab+ac+bc)=12

We are trying to figure out the space diagonal. We see that a space diagonal equals \(\sqrt{a^2+b^2+c^2}\) through the Pythagorean Theorem (to see why, draw a diagram. There is a right triangle between the height of the prism, the base diagonal, and the space diagonal.)

Now let's square both sides of a+b+c=19/4, which after expanding, simplifies to \(a^2+b^2+c^2+2ab+2ac+2bc=\frac {361} {16}\)

Substituting in 2(ab+ac+bc)=12, the second equation at the top, the equation becomes \(a^2+b^2+c^2+12=\frac{361} {16}\). Simplifying, we find\(a^2+b^2+c^2=\frac{169}{16}\) 

We are trying to find the square root of this, so our answer is \(\frac{13}{2}\)

EDIT: I have corrected my solution.

Hint 1:

the third angle is 30

Hint 2:

In a 30-60-90 triangle, the hypotenuse is twice the length of the side opposite the 30 degree angle.

Hint 3:

In a 30-60-90 triangle, the side opposite the 60 degree angle is \(\sqrt 3\) times the side opposite the 30 degree angle.

you should be able to figure it by now

if not i can finish the problem

AOPS has a solution:

Let's break it down and simplify each part.

\((t-1)^3\)= \(t^3-3t^2+3t-1\)

\(6(t-1)^2\)\( = 6(t^2-2t+1) = 6t^2-12t+6\)

\(12(t-1)=12t-12\)

Combining these and the 8 together, we get our final answer: \(t^3-3t^2+3t-1\) \( + 6t^2-12t+6\) \( + 12t-12+8\) =  \(t^3+3t^2-3t+1\)

So let x equal of one of the acute angles in a 90 degree triangle. Since \(tan=\frac {opposite}{adjacent}= \frac {5}{1}\), and by the Pythagorean Identity, we can find that the sides are 5,1, and \(\sqrt 26\). Therefore, cos x = \(\frac{adjacent}{hypotenuse}\)= \(\frac{1}{\sqrt 26}\) = \(\frac{\sqrt 26}{26}\).

The shortest distance is the edge of a cube plus a diagonal of the base square. (Draw a diagram, and after a bit of trial and error & intuition, you should see why). So the answer is \(1+\sqrt2\). 

How am I plagarizing?

I drew the diagram myself using asymptote.

The answer is, 6 dogs who have not learned any tricks.

This IS the answer fo the given question.

This is indeed.

I have personally checked.

Here is the proof:

62 have blue: 7+23+16+16 = 30 + 32 = 62

55 have green, 16+16+15 + 8 = 32 + 23 = 55

63 white: 23+15+16 + 9 = 38 + 25 = 63

Any further prrof needed?

I’m sure the AoPS answer is correct. However, this is not the answer for the posted question. If you are going to plagiarize, you should at least make sure you are copying the correct answer.

You still have about 4 hours to post the correct one. You’ll still get 20 points, although I may deduct 50 points for plagiarizing. You need to clearly indicate the source when you post solutions or diagrams that are not your original work. 

I’ll post a (non-plagiarized) Venn diagram (including all the supplementary data), after the time expires.

Tick-Tock ……….

Ok I got it! Thanks

The sum of the coefficients is 3^{16}.

\(\frac{AD*BC}2=\frac{BE*AC}2\) because in a triangle, all of the altitudes * the corresponding bases divided by two equal the area. They are in the same triangle, so the areas must be equal. 

Now, substitute the values and solve for BE. 

You should be able to finish the arithmetic. 

Oh got it

sin(4x)cos(8x)-cos(4x)sin(8x)=-.4      for smallest positive solution

Hello carlos!

\(sin(4x)cos(8x)-cos(4x)sin(8x)=-0.4 \)

\(sin(\alpha-\beta)=sin(\alpha)cos(\beta)-cos(\alpha)sin(\beta)\ \color{black}\small{(Bartsch\ Leipzig\ 1980)}\)

    \(\alpha=4x\\ \beta =8x\)

\(sin(4x-8x)=sin(4x)cos(8x)-cos(4x)sin(8x)=-0.4\)

   \(sin(-4x)=-0.4\)

  \(-4x=arcsin(-0.4)\\ x= \frac{arcsin(-0.4)}{-4}\)

  \(x=0.102879211517\)

  !

an + b = 7n + 1.

Let's start off by finding the equation of the line perpendicular to \(y=\frac{1}{2}x+5\). To do this, we note that the slope of two perpendicular lines is the negative reciprocal of each other. So we have \(y=-2x+b\). Now we plug in the point (6,2) to find that b=14. Our perpendicular line is \(y=-2x+14\). 

Next, we find the intersection point of the two lines by solving for the two equations. We get the point \((\frac{18}{5}, \frac{34}{5})\). 

Now, we can use the Mid-Point Formula to find our desired point, since \((\frac{18}{5}, \frac{34}{5})\) must be the midpoint of our desired point (a, b) and (6,2).

\(\frac{a+6}{2}=\frac{18}{5}\) and \(\frac{b+2}{2}=\frac{34}{5}\). 

Solve for these two equations and you have your point (a, b)!

2. We start off by rotating the point (-3, 2) 90 degrees clockwise around the origin. Now the coordinates are (2,3). Reflecting this point over the line y=x means flipping the x and y values. Therefore, the coordinates of this new point C is (3,2).

1.

a) First, from the circle equation, we can see that the center of the circle is (2, -3). Now we use the distance from point to line formula to figure out the distance from the center of the circle to the line. 

Formula: the distance between (x₁, y₁) and a line of the form ax + by + c = 0 is

\(\frac{|ax_{1}+by_{1}+c|}{\sqrt(a^2+b^2)}\)

Now, plugging the values in, \(\frac{|-8-15+20|}{\sqrt(16+25)}\), and solving for this equals \(\frac{3}{\sqrt(41)}\). Rationalizing the denomiator, we get that the answer is \(\frac{3\sqrt(41)}{41}\).

b) The greatest distance is the distance we just found above plus the circle's radius. (To see why, draw a diagram- it's just a line from the end of the circle to the point)

However, your equation implies that the circle's radius is the square root of -4, which is not a real number. Did you type it wrong?

The data is indeed correct.

Unless if AoPS is incorrect?

I rendered the image myself, but the numbers come straight from both mine and AoPS' calculations.

If you'll be so kind as to point out the errors?

Total and no-tricks added

Answer: 12/17 of the fruit is rotten

Let a=# of apples and p=# of pears.

(2/3)*a=(3/4)*p

so

a=(3/2)*(3/4)*p

p=(8/9)*a

The total amount of rotten fruit is: 2*(2/3)*a=(4/3)*a

The total fruit in box is: a+p=a+(8/9)*a=(9/9)*a+(8/9)*a=(17/9)*a

so the fraction of the fruit rotten is: (4/3)/(17/9)

simplifying it, we get 12/17

🐼

Hugo,

The data on this Venn diagram are incorrect.  The diagram is also missing the total number of dogs and the number of dogs that didn’t win any ribbons. 

This is sloppy work! You still have time to fix the mistakes.

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Also is EDT GMT-4, Eastern standard?

 GMT -4 [AST] is Atlantic Standard Time. The designator is optional for Standard Time

GMT -4 EDT is Eastern Daylight Time

GMT -5 [EST] is Eastern Standard Time

GMT -5 CDT is Central Daylight Time