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Finding the Area of Triangle XYZ

The area of a right triangle is (base * height)/2.

In this case, base = XY = 12 and height = YZ = 10.

So, area of triangle XYZ = (12 * 10)/2 = 60.

Relating the Area of XYD to the Height from D to XY

Let h be the height of triangle XYD from point D to side XY.

The area of triangle XYD = (base * height)/2 = (12 * h)/2 = 6h.

Finding the Maximum Height for an Area of 24

We want the area of XYD to be at most 24.

So, 6h <= 24.

Solving for h, we get h <= 4.

Calculating the Probability

The probability of choosing a point D with a height less than or equal to 4 is equal to the ratio of the area where the height is less than or equal to 4 to the total area of triangle XYZ.

If we draw a line parallel to XY at a distance of 4 units from XY, it will divide the triangle into two similar triangles. The smaller triangle will have a height of 4 and its area will be (1/4) * (area of triangle XYZ) = (1/4) * 60 = 15.

So, the probability of choosing a point D with a height less than or equal to 4 (and thus an area of XYD at most 24) is 15/60 = 1/4.

Therefore, the probability that the area of triangle XYD is at most 24 is 1/4.

Understanding the Problem

We have four random points on the unit interval [0, 1]. We form two intervals, I and J, from these points. We want to find the probability that these intervals overlap.

Approach

To calculate the probability, we'll consider the complementary event: the probability that the intervals do not overlap. If we can find this probability, we can simply subtract it from 1 to get the desired probability.

Calculating the Probability of Non-Overlapping Intervals

For the intervals to not overlap, one interval must completely lie to the left of the other. There are two possibilities for this:

I is entirely to the left of J:

x1 < x2 < x3 < x4

Probability of this arrangement is 1/4! (since there are 4! equally likely orderings of the four points).

J is entirely to the left of I:

x3 < x4 < x1 < x2

Probability of this arrangement is also 1/4!

The total probability of non-overlapping intervals is the sum of these two probabilities:

P(non-overlapping) = 1/4! + 1/4! = 2/4! = 1/12

Calculating the Probability of Overlapping Intervals

Finally, the probability of overlapping intervals is:

P(overlapping) = 1 - P(non-overlapping) = 1 - 1/12 = 11/12

Therefore, the probability that intervals I and J overlap is 11/12.

x² – mx + 24 = 10   

x² – mx + 14 = 0   

the coefficient of x is the sum of the factors of 14   

the integer factors of +14 are (+2, +7) and (–2, –7)   

                                         and (+1, +14) and (–1, –14)   

So m could equal +9 or –9 or +15 or –15   

Therefore, m could have 4 possible values    

_.

p - 2q = 3               (eq 1)   

q - 2r = -2 + q         (eq 2)   

p + r = 9 + p            (eq 3)   

By eq 2  r = 1

By eq 3  r = 9   

Both can't be true so there is no solution.   

_.

The prime numbers between 10 and 15, inclusive, are 11 and 13.   

So the largest possible number of students in Jon's class is 13.   

Any number other than a prime number can be divided.   

_.

The first prime numbers after 100 are   

101, 103, 107, 109, and 113   

So the fifth smallest after 100 is 113.   

_.

We can wrte down 2 equations for this then use substitution to solve it.

\(6a=8c\)

\(\frac{3}{4}a+280=c+\frac{1}{4}a\)

\(\frac{3}{4}a=c\)

\(c+280=c+\frac{1}{4}a\)

\(\frac{1}{4}a=280\)

\(a=1120\)

\(\frac{3}{4}(1120)=c\)

\(c=840\)

Let's double check:

\(6(1120)=8(840)\)

\(6720=6720\)

\(\frac{3}{4}(1120)+280=840+\frac{1}{4}(1120)\)

\(840+280=840+280\)

\(1120=1120\)

So both equations are true, therefore, Andrew has 1120 stamps and Caden has 840 stamps.

If you consider the cases where the order doesn't matter (so 1,2,3,4,5,6 is the same as 6,4,5,3,1,2), there are 4 possibilities. Find those, then use basic combinatorics to find the number of possibilities.

Let's call m the number of cards Mike has and a  the number of cards Aaron has. We can write the following equations:

\(m = a+56\)

\(a = \frac{3}{5}m+72\)

Let's use elimination to solve this

\(5a = 3m + 360\)

\(-3a=-3m+168\)

\(2a=528\)

\(a=264\)

\(m = (264)+56\)

\(m = 320\)

Let's double-check our answers:

\((320)= (264)+56\)

\(320=320\)

\((264) = \frac{3}{5}(320)+72\)

\(264=192+72\)

\(264=264\)

So both numbers are correct. Therefore, Mike has 264 cards and Aaron has 320 cards.

Understanding the Problem

Franklin starts at (0, 0).

He can move up, down, left, or right one unit at each step.

We need to find the total number of unique points he can reach after 12 steps.

Solution Approach

This problem can be solved using a combinatorial approach.

Number of steps in each direction: Since Franklin can move in four directions, we can think of distributing 12 steps among these four directions.

Combinations: We need to find the number of ways to distribute 12 identical objects (steps) into 4 distinct boxes (directions). This is a classic stars and bars problem.

Solving the Problem

Using the stars and bars formula, the number of ways to distribute n identical objects into k distinct boxes is:

C(n + k - 1, k - 1)

In our case, n = 12 (steps) and k = 4 (directions).

So, the number of ways to distribute the steps is:

C(12 + 4 - 1, 4 - 1) = C(15, 3) = 455

Therefore, Franklin the fly can end up at 455 different points after 12 steps.

I graphed it, and made sure for all integers x from 11 - 20 (10 doesn't work) that there is no integer B such that 1/10 - 1/x = 1/B

First, we want to get rid of the nasty fractions. 

Let's do this by multiplying both sides by the LCM of the denominators. 

By doing this, we have

\(10\left(x+4\right)=5\left(x-3\right)\left(x+5\right)+2\left(x+7\right)\left(x+5\right)\)

Distributing every number in and moving all terms to one side, we have

\(7x^{2}+24x-45=0\)

Using the quadratic equation and finding what x is, we have

\(x=\frac{3\sqrt{51}-12}{7}\\ x=\frac{-3\sqrt{51}-12}{7}\)

Thanks! :)

First, let's simplify all like terms and bring all terms to one side. 

We get the equation

\(2x^{2}+2x+7=0\)

Now, using the quadratic equation, we finally get two answers. We find that we have

\(x=-\frac{1}{2}+\sqrt{13}\cdot (\frac{1}{2}i)\\ x=-\frac{1}{2}-\sqrt{13}\cdot (\frac{1}{2}i)\)

Now we know what A and B are. We find that

\(a=-1/2; b = \frac{\sqrt{13}}{2}\)

Multiplying them toeghet, we finally come up with the answer. 

We finally have 

\(ab=-\frac{\sqrt{13}}{4}\)

Thus, our final answer is -sqrt(13)/4

Thanks! :)

First, let's focus on the expression we need A and B to be equal to. 

In order to find what we need, we need to be able to compare coefficients. let's convert it to standard form. 

We have

\((x - 2)(x + 2)=x^2-4\)

However, the constans don't match. Thus, we need to divide what we have by -4 to get a constant of +1. 

We have

\(-\frac{1}{4}x^2+1\)

Now. we cam compare coefficients. Since A is the coefficient of x^2, we have

\(A=\frac{-1}{4}\)

Since B is the coefficient of x, we have \(B=0\)

Thus, we finally have \(A+B=-\frac{1}{4}+0 = -\frac{1}{4}\)

So -1/4 is our final answer. 

Thanks! :)