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Let's denote the side length of the square as s. Since PZXY is a square, we know that PZ = ZX = s, and QX = RY = 10 - s (since QX and RY are the remaining segments of length 10 after subtracting the side length of the square).

Using the Pythagorean theorem, we can set up two equations based on the right triangles PZQ and PXR:

s^2 + QX^2 = QP^2 (1) s^2 + RY^2 = RP^2 (2)

Substituting in the values we know:

s^2 + (10 - s)^2 = 100 (1) s^2 + (10 - s)^2 = 100 (2)

Expanding and simplifying:

2s^2 - 20s + 100 = 100 2s^2 - 20s = 0 2s(s - 10) = 0

This gives us two solutions: s = 0 (which doesn't make sense since the square has positive length) and s = 10.

Therefore, the side length of the square is 10.

To find the equation of the line that passes through the points (-1,7) and (3,-2), we first need to determine the slope of the line:

slope = (change in y) / (change in x) slope = (7 - (-2)) / (-1 - (-3)) slope = 9 / 2

Now that we know the slope of the line, we can use the point-slope form of the equation of a line to find the equation of the line:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is a point on the line.

Using the point (-1,7), we get:

y - 7 = (9/2)(x - (-1)) y - 7 = (9/2)(x + 1) y - 7 = (9/2)x + 9/2 y = (9/2)x + 16/2 y = (9/2)x + 8

Now we need to find y when x = -6, since the point (-6,y) is on the line:

y = (9/2)(-6) + 8 y = -27 + 8 y = -19

Therefore, when the point (-6,y) is on the line that passes through (-1,7) and (3,-2), y is equal to -19.

We want to find all integers m such that the quadratic equation x^2 - mx + 2m = 0 has integer solutions.

Let's use the quadratic formula to solve for x:

x = (m ± √(m^2 - 8m)) / 2

For x to be an integer, we need the discriminant m^2 - 8m to be a perfect square, say k^2. That is:

m^2 - 8m = k^2

Rearranging, we get:

(m - 4)^2 - 16 = k^2

Adding 16 to both sides and factoring, we get:

(m - 4 - k)(m - 4 + k) = 16

Since m is an integer, the factors on the left-hand side must also be integers. The only possible pairs of factors of 16 are (1, 16), (-1, -16), (2, 8), (-2, -8), and (4, 4). So we have five cases to consider:

Case 1: m - 4 - k = 1 and m - 4 + k = 16 Adding the two equations, we get:

2m - 8 = 17 m = 25/2

This is not an integer, so there are no solutions in this case.

Case 2: m - 4 - k = -1 and m - 4 + k = -16 Adding the two equations, we get:

2m - 8 = -17 m = -9/2

This is not an integer, so there are no solutions in this case.

Case 3: m - 4 - k = 2 and m - 4 + k = 8 Adding the two equations, we get:

2m - 8 = 10 m = 9

Substituting m = 9 into the original quadratic equation, we get:

x^2 - 9x + 18 = 0

Factoring, we get:

(x - 3)(x - 6) = 0

So the integer solutions are x = 3 and x = 6.

Case 4: m - 4 - k = -2 and m - 4 + k = -8 Adding the two equations, we get:

2m - 8 = -6 m = 1

Substituting m = 1 into the original quadratic equation, we get:

x^2 - x + 2 = 0

This quadratic equation has no integer solutions.

Case 5: m - 4 - k = 4 and m - 4 + k = 4 Adding the two equations, we get:

2m - 8 = 8 m = 8

Substituting m = 8 into the original quadratic equation, we get:

x^2 - 8x + 16 = 0

Factoring, we get:

(x - 4)^2 = 0

So the only integer solution is x = 4.

Therefore, the only integers m that satisfy the given condition are m = 8 and m = 9.

Sry, buy you are wrong :(

First, we can use the Pythagorean theorem to find the length of AC. Since AB is a diameter of the circle, we have:

AC^2 = 4^2 + 8^2 = 80

AC = 4*sqrt(5)

Now, let's draw the angle bisector of ∠ACB and label the point where it intersects the circle as M. We can use the angle bisector theorem to find the ratio of AM to MC:

AM/MC = AB/BC AM/MC = 8/4 AM/MC = 2

We also know that AM + MC = AC, so we can substitute in the values we found for AM/MC and AC:

2MC + MC = 4*sqrt(5)

3MC = 4*sqrt(5)

MC = (4/3)*sqrt(5)

Therefore, CM = (4/3)*sqrt(5).

The area of trapezoid ABCD is 16*sqrt(6).

7426 , 7428 , 7624 , 7628 , 7824 , 7826 , 9426 , 9428 , 9624 , 9628 , 9824 , 9826 , Total = 12 permutations.

Probability ==1 / 12 ==0.08333....

Here are the steps on how to express the equation y=x+7x−3​ in the form (x+a)(y+b)=c:

Multiply both sides of the equation by x+7.

(x+7)y=x−3

Distribute the x+7 on the left-hand side of the equation.

xy+7y=x−3

Factor out x from the left-hand side of the equation.

x(y+7)=x−3

Subtract x from both sides of the equation.

x(y+7)−x=x−3−x

Simplify the left-hand side of the equation.

xy+7=−3

Rewrite the equation in the form (x+a)(y+b)=c.

(x+0)(y+7)=−3

Therefore, the constants a, b, and c are 0,7,−3​.

The values of x for which f(x) is not defined are 5, -3.

Let's consider two points chosen at random on a circle of radius 1.

Without loss of generality, we can fix one point and consider the distribution of the other point with respect to the first one. Suppose we fix one point at the top of the circle. Then the second point can lie anywhere on the circle except for a segment of length 2 centered at the bottom of the circle.

To see this, imagine drawing a circle of radius 1 centered at the first point. The second point must lie somewhere on this circle. The distance between the two points is at most 1 if and only if the second point lies within a distance of 1 from the first point. This corresponds to a segment of length 2 centered at the bottom of the circle, as shown in the figure below.

The length of this segment is equal to the length of the diameter of the circle, which is 2. Therefore, the probability that the distance between the two points is at most 1 is equal to the ratio of the length of the segment of the circle that is allowed for the second point to the length of the circumference of the circle.

The length of the segment of the circle that is allowed for the second point is equal to the length of the circumference of the circle minus the length of the segment of length 2 centered at the bottom of the circle. The length of the circumference of the circle is 2π, and the length of the segment of length 2 is 2. Therefore, the length of the allowed segment is 2π - 2.

So the probability that the distance between the two points is at most 1 is:

(length of the allowed segment) / (length of the circumference of the circle) = (2π - 2) / (2π) = (π - 1)/π.

The answer is 134.

Why?

First of all, the question never specified that ACD was not 90 degrees, and without it there would still be sufficient info, but this just makes it easier to visualize things (You can prove that CP = PA without this assumption).  Angle CAP is 67 degrees, and let angle PBA be x degrees. We can also construct it so that CD is parallel to AB, and CA makes a 90 degrees angle to both lines. Thus, angle PAB is 90 - 67 = 23 degrees. Since E is collinear with CD, then CE is parallel to AB, and DE is parallel to AB, so shape ABED makes a parallelogram, thus angle ADE is 180 - 23 = 157 degrees. Angle ABE is also 157 degrees, so angle CBE is 157 - angle PBA = 157 - x degrees. Angle CDP is 180 - 157 = 23 degrees, and angle PCD is 90 - 67 = 23 degrees, thus CP = PD, and angle CPD is 180 - 23 - 23 = 134 degrees. By corresponding angles, angle CBE = angle CPD = 157 - x = 134 degrees, and because we are looking for angle CBE, angle CBE = 134 degrees (you dont even really need the x, but its more visuals).

He has retired for good.

1 - Pi x 2^2 x 8 ==100.531 cubic inches - volume of the first cup

2000/3 * Pi ==2094.395 cubic inches - volumes of water in the sink.

2094.395 / 100.531 ==20.833 ==21 - scoops to empty the water in the sink.

Note: Guest #2 forgot to square the radius of the cup.

Let x be the amount of money your brother currently has.

If you give your brother $5, then you will both have x+5 dollars.

If your brother gives you $12, then you will have x+12 dollars and he will have x−12 dollars.

We are given that x+5=x−12 and x+12=2(x−12). Solving the first equation, we get x=7. Substituting x=7 into the second equation, we get 7+12=2(7−12), which is true.

Therefore, your brother currently has $7​.

(1, 1, 2, 2, 4, 4, 4)=7!/3!2!2! =210 permutations 

(1, 2, 2, 2, 4, 4, 4)=7!/3!3! =140 p 

(1, 2, 2, 3, 4, 4, 4)=7!/3!2! =420 p 

(1, 2, 2, 4, 4, 4, 4)=7!/4!2! =105 p

(1, 2, 2, 4, 4, 4, 5)=7!/3!2! =420 p  

(1, 2, 2, 4, 4, 4, 6)=7!/3!2! =420 p

Total =210 + 140 + 420 + 105 + 420 + 420 =1,715 possible sequences of rolls

1 cm = 10 mm

1,000  x  10 = 10,000 mm

Volume of a cylinder:
V = π a^2 h    [a = radius] and [h=height]

1 - Pi x 4/2 x 8 =50.265 in^3 - volume of the first cup

2000/3 * Pi=2,094.395 in^3 - volume of water in the sink

2,094.395 / 50.265 =41.667 =42 scoops to empty the sink using first cup.

2 - Follow exactly the same procedure for the 2nd cup. 

5  +  7 ==12 - combined ratio of the 2 compounds

708 / 12 ==59 - constant ratio

59  x  5 ==295 mL of compound A will be needed

59  x  7==413 mL of compound B will be needed.

The answer is 265 milliliters of compound A are needed.

(a) 14 cups

(b) 38 cups

(10^5 * 10^77) / (10^15 ) / (10^15)^4

To start, we can set 10^5 * 10^77 to 10^82, as multiplying exponets is the same as adding them.

next, we can set (10^15)^4 to 10^60, as its the same thing as multiplying 10^15 to itself four times, which as we have stated is the same thing as adding. (15+15+15+15=60)

finally, dividing with exponents is the same as subtracting them, so we take 10^82, and subtract 10^15, and 10^60, (82-15-60), so our final exponent is 10^7.

10^7 is the same thing as 10,000,000, or 10 with 7 zeros behind it.

Our final answer is 7.

there are solutions, just not integers