All Answers 358087 Answers

Think of the dice being different colours

There are 6*6 = 36 possible roll

How many ways can you roll a 3 ?

How many ways can you roll a 12?

Add those two together and the answer is that number over 36

x = 40*25/5 = 200.

I don't want to give away the answer, but it sounds like you're onto something :)

Note that the only way to have a real number is to have both x and y have the same parity (look at the powers of i under guest's answer)

If both x and y are even, there are 50 odd numbers to choose from. Note that if we choose any pair of 2 distinct even numbers, there will be 1 case for x and y. So, in this case, there are \({50 \choose 2} = 1225\) cases. 

However, if x and y are odd, we need one to have a remainder of 1 when divided by 4 and the other to have a remainder of 3. There are 25 numbers for x and y, which makes for \(25^2 = 625\) cases.

So, there are a total of \(1225 + 625 = \color{brown}\boxed{1850}\) cases.

Divide it up by casework: 

If the twins get 0 pieces - There are 10 pieces of candy and 3 children for 10 stars and 2 bars which makes \({12 \choose 2} = 66\) cases

If the twins get 1 piece - There are 8 pieces of candy and 3 children for 8 stars and 2 bars which makes \({10 \choose 2} = 45\) cases

If the twins get 2 pieces - There are 6 pieces of candy and 3 children for 6 stars and 2 bars which makes \({8 \choose 2} = 28\) cases

If the twins get 3 pieces - There are 4 pieces of candy and 3 children for 4 stars and 2 bars which makes \({6 \choose 2} = 15\) cases

If the twins get 4 pieces - There are 2 pieces of candy and 3 children for 2 stars and 2 bars which makes \({4 \choose 2} = 6\) cases

If the twins get 5 pieces - There are 0 pieces of candy and 3 children for 0 stars and 2 bars which makes \({2 \choose 2} = 1\) case

So, there are \(66 + 45 + 28 + 15 + 6 + 1 = \color{brown}\boxed{161}\) cases.

A diagram would help!

The f(n) = 1/(n+1) is the simplified form of the function.

(2n)/n - (2n)/(n + 1) = f(n) (2n)/n

Simplifying:

(2n)/n - (2n)/(n + 1) = 2f(n)

(2n)/2n - (2n)/2(n + 1) = 2f(n)/2

1 - n/(n+1) = f(n)

(n+1 - n)/(n+1) = f(n)

1/(n+1) = f(n)

I do not know

The key here is to find the volume of a cone with height 6 inches and radius 1 inch. Using \(V=\dfrac{\pi r^2h}{3}\), we have \(V=\dfrac{6\pi}{3}=2\pi\approx6.28\). For the purposes of this problem, approximating it to \(6.28\) is close enough.

Dividing, we get \({370\over6.28}=58.9\). Unfortunately, the 0.9 does not count; it asks for full cups. 0.9 is not full. Therefore, the answer is \(58.\)

(c)

The sum of n natural numbers is \(n(n+1)\over 2\), so we have \(\dfrac{99(100)}{2}=4950\)

(b)

He didn't write down a 0. Using \(\text{Probability}=\dfrac{\text{number of successful outcomes}}{\text{total possible outcomes}}\), the probability is 0.

(a)

Let's look at a simplified version: 1, 2, 3, 4. How many numbers are in that?

Obviously 4. That's the last number in the sequence. 

Therefore, using the same logic, Sam wrote down 99 digits.

Using the factor theorem (This states that if \(r\) is a solution to \(f(x)=0\), then \((x-r)\) is a factor of \(f(x)\)), we have \(f(x)=(x+1)(x-2)(x-3)=x^3-4x^2+x+6\).

This question is slightly confusing; are you asking for the area of a circle with radius 48? Are you asking for the area of a square inscribed in a circle of radius 48? Or are you asking for the area of a square in which a circle of radius 48 is inscribed?

Each of those questions is answered below. 

Area of a circle with radius 48:

\(A=\pi r^2\implies A=\pi(48)^2=(3.14)(2304)=7234.56\;\rm ft^2\)

Area of a square inscribed in a circle of radius 48:

Here's a diagram.

We see that the side length of the square is \(48\sqrt2\), so the area of the square is 4608.

Area of a circle inscribed in a square:

Here's a diagram:

We see that the radius is 48, so the side length of the square is 96. Therefore, the area of the square is 96²=9216.

I think the question you were asking was the second one, whose answer was 4608

Ah ok. So then to prove the equality, I use part of the Cauchy-Schwarz to show it would be a degenerate triangle. And then the inequality part of Cauchy-Schwarz to prove it would apply for any positive values of a, b, and c because it would make a triangle?

Cauchy-Schwarz is closely related to cosine law.  Keep looking at this approach, and I think you'll get it.

Hey thank you. Any way to apply the law of cosines? I noticed that the terms of the inequality kinda looks like the law of cosines if you write it like this: a^2-2ab+b^2=cos C. That's difficult because there is a 2ab, and I don't know how to get rid of it. I think maybe involving trying to find the area of a triangle with 1/2 * ab sin angle C. I think I figured out the equality part because it would be a degenerate triangle. But how do I prove the inequality? I know that part is related to the triangle inequality theorem.

Hey Thank you for the response! Yea I was thinking along the lines of that to help cancel the square root, but I just am not sure how to go from there. I labeled a triangle with c= sqrt(a^2 - ab +b^2) b= sqrt(a^2 - ac +c^2) a= sqrt(b^2 + bc +c^2) Then, I tried putting values into the law of cosines and got: c^2= (b^2 + bc +c^2) + (a^2 - ac +c^2) -2(sqrt(b^2 + bc +c^2)*sqrt(a^2 - ac +c^2)) I don't know how this proves anything. I used the triangle inequality theorem to substitute the terms as a, b, and c in triangle ABC with the lower case letters representing the opposite angles.

If you need to show your work, first convert 4 1/2 into an improper fraction, which would be 9/2, then conver 1 4/5 into an improper fraction too, which would be 9/5 then find a common denominator, in this case it would be 10 making it 45/10 * 18/10 which equals 810/100 which is 8 1/10 or 8.1

Hope that helps!
PD

The area of a triangle is given by:

Area = 1/2 x base x height

We have,

A triangle ABC.

Angle bisectors are AD, BE, and CF, which intersect at I.

DI = 3

DI = AI = 3

Height of the triangle = AD = AI + DI = 3 + 3 = 6

BD = 4 = BC

Base of the triangle = BC = BD + DC = 4 + 4 = 8

The area of a triangle:

= 1/2 x base x height

= 1/2 x 8 x 6

= 1 x 4 x 6

= 24

Thus,

The area of the triangle is 24.

Since I'm in 9th grade math, all I know is that c^2=a^2+b^2 is the pythagoreon theorom.
Hope that helps in some way!