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a + b = 21.

There are 28 triples (a,b,c).

The possible values of t are 5 and 8.

Let the number of cupcakes in the box==C 

Andrew gets==2/5C + 2/5 ==[2C + 2] / 5

C - [2C + 2] / 5 ==[3C - 2] / 5 - cupcakes left in the box.

Bernard receives==2/5 x [3C - 2]/5 + 3/5==[6C + 11] / 25.

[3C - 2]/5 - [6C + 11]/25==[9C - 21]/25 - cupcakes remain in box.

[9C - 21] / 25 ==6, solve for C

C ==19 cupcakes - original number in the box.

Andrew gets 8 cupcakes. Bernard gets 5 cupcakes. Cathy get the last 6 cupcakes.

There are 1400 arrangements.

The area of triangle ACE is 280.

From x + 3y = -5, y = -x/3 - 5, so the slope of the line is -1/3. The slope of the new line is also -1/3, so y = -x/3 + B.  Pugging in x = 2 and y = -7, we get -7 = -2/3 + B, so B = -19/3.

Then the line is y = -x/3 - 19/3.  Then 3y = -x - 19, so 3y + x = -19.  We want the right-hand side to be 3, so we mutiply both sides by -3/19: -9/19*y - 3/19*x = 3.  Therefore, B - A = -19/3 - 3 = -28/3.

The answer is 900000

So the first step would be to plot the information given:

Then you want to figure out the equation of the line that passes through your point - Which in this case is \(x+3y=-1\)

Because the line needs to be parellel you dont change the A or B - That is the(1)x and 3y.

Therefore A(3) - B(1) = 2

I hope that all made sense  

The same question is here, you can check it out if you want!

https://web2.0calc.com/questions/too-many-combinations

Hint:

the integral is lnx =   the integral of (1*lnx)  

do that first.

You need to use integration by parts twice.

I have not done it, I don't have time at present, but that is how I would proceed.

What is the largest possible value of b?

Hello Guest!

\(\{R_g,R_f\}\subset\{-5\le 3\ and -2\le1\}\\ a\le R_{g+f}\le b\\ \color{blue}b_{max}=(3+1)=4\)

  !

Please present your questions properly.

Note that 2 lines form the diameter that is perpendicular to the slope of the line. 

The slope of the line is \({7 \over 7} = 1 \), meaning the slope of the line is \(\color{brown}\boxed{-1}\)

thanks

Note that we have a right triangle, because \(5^2 + 12^2 = 13^2 \)

This means that 2 of the altitudes are 5 and 12 because they are perpendicular to another side. 

The third altitude can be found with this formula: \(\text{hypotenuse} \times \text{altitude} \div 2 = \text{Area}\). 

We know the area is 30 and the hypotenuse is 13, meaning the altitude is \({60 \over 13}\). 

Can you find the sum from here?

The first point in Quadrant 1 will be \((0,18)\). 

The last point will be \((18,0)\)

Note that there will be a point for every x value, between 0 and 18. 

Now, can you tell me how many lattice points that line passes through?

Note that 2 points will have a y-coordinate of 18, and 2 will have a y-coordinate of 8. 

If the y-coordinate is 18, the points are \((1 \pm 13, 18)\), which are \((-12, 18) \) and \((14,18)\)

If the y-coordinate is 8, we have a right triangle, where the hypotenuse is 13 units and 1 leg is 10, meaning the other leg is \(\sqrt{69}\).

This means the 2 points here are: \((1 \pm \sqrt{69}, 8)\), which are \((1 + \sqrt {69}, 8)\) and \((1 - \sqrt {69}, 8)\). 

Adding everything up, we find the sum is \(\color{brown}\boxed{56}\)

Simplify to: \(5b^2 + 5b + 4 = (3b + 4)(b + 5)\)

Simplify the right-hand side to: \(5b^2 + 5b + 4 = 3b^2 + 19+ 20\)

Move everything to the left-hand side to make a quadratic: \(2b^2 - 14b - 16 = 0\)

Factor as: \(2(b+1)(b-8) = 0\)

The only solutions are -1 and 8, but b must be positive, meaning \(b = \color{brown}\boxed{8} \)

a) \(10! \) ways to arrange the boys and \(8!\) ways to order the girls. There are also 2 ways to order the groups (10 boys, 8 girls; 8 girls, 10 boys). 

This makes for \(8! \times 10! \times 2 = \color{brown}\boxed{292,626,432,000}\) ways.

b) \(8! \) ways to order the girls. Now, treat the 8 girls as 1 person. Now, there are \(11!\) ways to order everything else. 

This makes for \(8! \times 11! = \color{brown}\boxed{1,609,445,376,000} \) ways.

Here: https://web2.0calc.com/questions/number-theory_30

The sum of the x and y coordinates is 52.

The answer is 18.

x^4 + y^4 = 20

Mrs. Read can make 40 pairs of mittens.