GYanggg

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Hi Svtturtles!

We can solve this problem using a system of equations. 

Let p = the number of pineapple bags and a = the number of apricot bags. 

9p + 6a = 39. 

p + a = 5.  

Solving gives p = 3 and a = 2. 

Therefore, Sharon unpacked 3 pineapple bags and 2 apricot bags. 

I hope this helped,

Gavin. 

Hi Svtturtles!

We can solve this problem using a system of equations. 

Let p = the number of pineapple bags and a = the number of apricot bags. 

9p + 6a = 39. 

p + a = 5.  

Solving gives p = 3 and a = 2. 

Therefore, Sharon unpacked 3 pineapple bags and 2 apricot bags. 

I hope this helped,

Gavin. 

Hi guest!

We can set up some equations solving for the radius. The formula to find the lateral surface area of any right circular cylinder is 2πrh and the formula to find the volume of any right circular cylinder is πr²h. Therefore, we can write 2πrh = 3 and πr²h = 3. Since we are only solving for r, we can set the equations equal and write 2πrh = πr²h. This simplifies to r = 2.  

I hope this helped,

Gavin. 

Hi Guest!

Let x be the number of quarters Gisele has. Thus Gisele must have x+16 nickels. 

To calculate cents, we can write 25x + 5(x+16) = 590

Simplifying, 25x + 5x + 80 = 590 or 30x = 510

Solving for x, we get x = 17

Gisele has 17 quarters. 

Hello guest!

There are a total of 18 students. Of the 18 students, there are \(\binom{18}{3}=816\) ways to choose three students. 

Of the 6 boys, there are \(\binom{6}{3}=20\) ways to choose the boys. 

Therefore, the total probability is 20/816 = 5/204

I hope this helped, 

Gavin. 

Hello Guest!

If we set the number of three point baskets he made as x, we can write the following equation for the total amount of points he scored. 

22 = 2 * (4x) + 3 * x 

22 = 8x + 3x

22 = 11x

2 = x

x = 2

Since x = 2, we know he scored 2 three pointers. 

I hope this helped, 

Gavin. 

Hi wiseowl!

Connect the centers like in the following diagram: https://imgur.com/WoRzy2n

This forms a right triangle with equal legs of 2 units. 

The hypotenuse is equal to the diameter of the smaller circle plus 2. 

We can right the equation: \(2^2+ 2^2=(2+2x)^2\\ 8=4+8x+4x^2\\ x^2+2x-1=0\) 

Solving for x, we get x = √2 - 1

I hope this helped,

Gavin

Hello wiseowl! 

The diameter must pass through the midpoint of the circle. Furthermore, the midpoint of the diameter is the center of the circle. 

Therefore, from the diameter, we can find the radius length and center, finding the equation in the end. 

The midpoint of (-1,-3) and (5,1) is \((\frac{-1+5}{2},\frac{-3+1}{2})\) or \((2,-1)\).

The diameter length is \(\sqrt{(5-(-1))^2+(1-(-3))^2}=2\sqrt{13}\)

Then the radius must be \(\sqrt{13}\)

The equation of the circle must be \((x-2)^2+(y+1)^2=13\)

I hope this helped, 

Gavin

Hi Rollingblade, 

I think I finally worked it out. 

Since the cookies and candies are identical, only the number of cookies affect each distribution. Therefore, the ways to distribute the cookies are \(a + b + c + d = 6\), where  \(0\le a,b,c,d\le 3\).

This is equal to 44, so the total ways to distribute is 44. 

For the chairs, there are 15 remaining chairs after 5 have been chosen. So \(a + b + c + d + e + f = 15\), where only a and f can be zero. The number of solutions are 4368. 

I hope this helped, 

Gavin

There are a total of 5 (1/2)'s. 5 * 1/2 = 5/2

1/4 + 1/8 + 1/16 + 1/32 = 8/32 + 4/32 + 2/32 + 1/32 = 15/32

5/2 = 80/32. 80/32 + 15/32 = 95/32

I hope this helped, 

Gavin. 

If the first three terms form an arithmetic sequence, the arithmetic sequence goes: 6, -9, x

Therefore, 6 - (-9) = -9 - x, x = -24

Since the last three terms is a geometric sequence, the geometric sequence goes: -9, -24, y

Therefore, -24/(-9)=y/(-24) 

After you solve for y, you get your answer.

Gavin, 

\(\frac{1}{x}+\frac{1}{x+2}=\frac9{40}\\ \frac{x+2+x}{x(x+2)}=\frac{9}{40}\\ \frac{2x+2}{x^2+2x}=\frac{9}{40}\\ 9x^2+18x=80x+80\\ 9x^2-62x-80=0\\ (9x+10)(x−8)=0\\ x_1=\frac{-10}{9}, x_2=8\)

only 8 is an integer. 

The binomial theorem is as follows:

\((a+b)^2=\sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^k \)

The formula never helped me much either, but here is the explanation. Looking at any expansion, there are coefficients and exponents. For example, the expansion for \((a+b)^3\) is as follows:

\((a+b)^3=a^3+3a^2b+3ab^2+b^3\)

Just looking at the coefficients, you get 1, 3, 3, and 1. This is the fourth row of the Pascal triangle. 

For the exponents, you get 3, 2, 1, 0 for a; and 0, 1, 2, 3 for b. From this, we can deduce what \((2x-3y)^5\) will be. 

The expansion for just \((x-y)^5\):

The exponents for x will be 5, 4, 3, 2, 1, 0 and the exponents for y will be 0, 1, 2, 3, 4, 5. 

The coefficients will be the sixth row of the Pascal triangle, which is 1, 5, 10, 10, 5, 1

Therefore, the expansion will be \((x-y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5\)

However, we are not done yet. We need to add the positive/negative signs, alternating. 

\((x-y)^5=x^5-5x^4y+10x^3y^2-10x^2y^3+5xy^4-y^5\)

Using this equation, you can plug in the values to find the given expression.

I hope this helped,

Gavin.