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Thanks Max,

I just spent time working through your solution by hand. I needed the practice.

I never would of got that first step without your guidence :)

We have

(3/5)j = 2/3(m)

j - m = (1/5)b

j = b + 35

Now this is just a three-variable system of equations. You should be able to use system of equations to solve this, and then figure out how much Mark has to give to Bill.

⚡⚡⚡

ShockFish

\(6 \over 19\) chance for the first marble

\(5 \over 19\) chance for the second marble

\(8 \over 19 \) chance for the third marble. 

Because they all "rely" on each other, the probability is the product of all 3 cases. 

Your question doesn't make sense as written! You must have a number such as 1,000, and the question should read: "On what number does the count of 1,000 land when counting up to 13 and down to 1" !!. Check your question carefully.

Integrating \(f'(x)\) with respect to \(x\) yields

\(f(x) = -\cos(x) + C\),

where \(C\) is a constant we can solve for given the initial condition. Because we are given \(f(0) = 10,\) we can say

\(10 = f(0) = -\cos(0) + C = -1 + C\).

This implies that \(C = 11.\) Hence,

\(f(x) = -\cos(x) + 11.\)

Let \(a_n\) denote the \(n\)th term of this arithmetic sequence. There are 27 terms occurring from \(a_5 = 9\) to \(a_{32} = 60\). Assuming a constant common difference between terms, then the terms increase by \(\frac{60-9}{27} = \frac{17}{9}\) with each \(n\) step. That is, \(a_{n+1} = a_n + \frac{17}{9}\).

There are 18 terms occurring from \(a_5 = 9\) to \(a_{23}.\) Then, there would be 18 additions of \(\frac{17}{9}\) from \(a_5\) to \(a_{23}\). That is,

\(a_{23} = a_5 + 18\left(\frac{17}{9}\right) = 9 + 34 = 43\).

Aww... Thank you!!!

0.25 * 21600 ==$5,400 - Max's share

0.30 * 21600==$6,480 - John's share.

21,600 - [5,400+6,480]==$9,720 -Bobby's and Ivan's share.

Let Bobby's share==B

Ivan's share ==1.7B

B  +  1.70B ==9,720

2.70B == 9,720

B ==9,720 / 2.70==$3,600 - Bobby's share

3,600 x 1.7 == $6,120 - Ivan's share

Excellent use of results from Number Theory and Algebra, Max!!!

Thank you! I couldn't have done that without you. 

I have revised your solution and I finally understand.

Already answered

Yes, that is correct!!!

Would the answer then be 4+sqrt5?

Thank you, but I am a bit confused.

Vertical throw up

Hello Guest!

g = 9.805m/s²;  t = 4sec; initial speed t; h = 0

\(h=v_0t-\frac{g}{2}t^2=0\\ v_0= \frac{1}{2}\ g\ t\)

\(\large v_0=\frac{g}{2}\cdot 4\ sec\)
Not a quadratic equation but valid.

  !

The perimeter of the trapezoid is the sum of the sides + the sum of the line cutting the square. 

The length of the sides (not including the line) is \(4a\). One of the sides is 2a (labeled a in the diagram), and the other two sides (b and c in the diagram) will always add to 2a. 

We can then use the Pythagorean Theorem for the length of the bisecting line, and we find that its length is \(a \sqrt 5 \). 

Thus, the perimeter of the shape is \(a(4 + \sqrt 5)\). Can you do the rest?

Diagram:

We have the system: 

\(\sqrt{xy} = 21\)

\({{x+y} \over 2} = 75\)

We can simplify both equations to: 

\(xy= 441\)                (1)

\(x+y=150\)           (2)

Now, we just have to solve this system. Can you do it?

D==Dennis,  J==James

5/6[0.25D) =what Dennis spent

1 /6[0.25D] - what Dennis has left.

1/3[120  +  0.10J] ==2/3[120  +  0.10J] - what James has left.

0.25D ==120  +  0.10J..................................(1)

2/3[120  +  0.10J]== 1 /6[0.25D] + 70........(2), solve for D, J

D==$560 - what Dennis had at the start

J ==$200 - what James had at the start.

a maximum height of 45 meters.
 

Hello SBAHarper!

The launch velocity is  \(v_0.\)

The launch angle to the plane is \(\alpha.\)

Let the running time be t.

\({\color{blue}g}=9.805m/sec^2\)

\(h=-\frac{g}{2}\cdot t^2+v_0\cdot sin\ \alpha \cdot t\\ \dfrac{dh}{dt}=-g\cdot t+v_0\cdot sin\ \alpha=0\\ t=\dfrac{v_0\cdot sin\ \alpha}{g}\)

\(h_{max}=45m\\ =-\frac{g}{2}(\dfrac{v_0\cdot sin\ \alpha}{g})^2+v_0\cdot sin\ \alpha \cdot \dfrac{v_0\cdot sin\ \alpha}{g}\\ =-\dfrac{v_0^2\cdot sin^2\alpha}{2g}+\dfrac{2v_0^2\cdot sin^2\alpha}{2g}\)

\(h_{max}(=45m)=\dfrac{3}{2}\cdot \dfrac{v_0^2\cdot sin^2\alpha}{g}\)

Choose the angle of attack and calculate the launch speed

or

choose the launch speed and calculate the angle of attack.

  !

The answer is 3 + sqrt(6).

Because the circle has a radius of 1, we know the rectangle's dimensions are \(4 \times 2 \). 

Note that the perimeter of the shaded region is the perimeter of the rectangle plus the perimeter of the 2 circles. 

The circumference/perimeter of the circle can be found using the formula \(2 \pi r\). 

Can you take it from here?

Thank you!!

L==Lionel,     D==Dennis

[D - 12] - what Dennis gives to Lionel

[L + 12] - what Lionel receives from Dennis

[L - 26] ==[D + 26].................(1)

[L + 12] / [D - 12] ==5/1..........(2), solve for D, L

D ==31 sweets - that Dennis started with.

L==83 sweets - that Lionel started with.