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The area of the square is 4. The points that satisfy x+y<3 form a triangle with vertices (0, 0), (1, 1), and (0, 2). The area of this triangle is 1/211 = 1/2. Therefore, the probability that x+y<3 is 1/2 / 4 = 1/8.

Alternatively, we can solve this problem by using the fact that the probability of a point falling in a certain region is equal to the ratio of the area of that region to the area of the entire square. The region that satisfies x+y<3 is a triangle with vertices (0, 0), (1, 1), and (0, 2). The area of this triangle is 1/211 = 1/2. The area of the entire square is 4. Therefore, the probability that x+y<3 is 1/2 / 4 = 1/8.

To solve the puzzle, we can approach it systematically by filling in the squares one by one. Here's one possible solution:

 1 | 2 | 3 | 4
--------------
 3 | 4 | 1 | 2
--------------
 2 | 1 | 4 | 3
--------------
 4 | 3 | 2 | 1
 

In this solution, each row, column, and outlined square contains the numbers 1, 2, 3, and 4 exactly once.

Note that there might be other valid solutions as well. The important thing is that each row, column, and outlined square contains the numbers 1, 2, 3, and 4 without repetition.

We can use the given information to set up a system of equations to solve for Grogg's and his dad's current ages. Let g be Grogg's current age and d be his dad's current age. We know that 7 years ago, Grogg's dad was 8 times as old as Grogg, so we have the equation d−7=8(g−7). We also know that 4 years ago, Grogg's dad was 5 times as old as Grogg, so we have the equation d−4=5(g−4). We can solve this system of equations by first multiplying the second equation by 8 and then subtracting the first equation from the second equation. This gives us the equation 7d−28=35g−160. We can then divide both sides of this equation by 7 to get d−4=5g−23. Finally, we can add 4 to both sides of this equation to get d=5g−19.

Now that we know the relationship between Grogg's and his dad's ages, we can plug in the given information to solve for Grogg's current age. We know that 7 years ago, Grogg's dad was 8 times as old as Grogg, so we have the equation d−7=8(g−7). Plugging in d=5g−19, we get 5g−26=8(g−7). We can then solve for g to get g=13.

Now that we know Grogg's current age is 13, we can plug it into the equation d=5g−19 to solve for his dad's current age. This gives us d=5(13)−19=46. Therefore, Grogg's dad is currently 46 years old.

Grogg's age 7 years ago = x Grogg's dad's age 7 years ago = 8x Grogg's age 4 years ago = x+3 Grogg's dad's age 4 years ago = 5(x+3)

8x+7 = 5(x+3) + 4 3x = 8 x = 8/3

Grogg's dad's age now = 8x+7 = 8(8/3) + 7 = 31

So the answer is 31

 d−4=5(g−4)................(1)

d==5(g - 4) + 4

d−7=8(g−7)................(2)

5(g - 4) + 4 - 7==8(g - 7)

5g - 20 + 4 - 7==8g - 56

56 - 20 + 4 - 7==8g - 5g

33 == 3g

g==33/3==11 - sub this into (2) above

d - 7==8(11 - 7)

d - 7 ==32

d ==32 + 7 ==39 years - his dad's current age.

Let Grogg's age be g and his dad's age be d. We can set up a system of equations to represent the information given in the problem:

Seven years ago, Grogg's dad was 8 times as old as Grogg: d−7=8(g−7)

Four years ago, his dad was 5 times as old as Grogg: d−4=5(g−4)

We can solve this system of equations by substitution. First, we can solve the first equation for g:

g=8d−7​+7

We can then substitute this value of g into the second equation:

d−4=5(8d−7​+4)

We can then solve this equation for d:

d=30

Therefore, Grogg's dad is currently 30 years old.

d - 7 = 8(g - 7)..........................(1)    

d - 4 = 5(g - 4)..........................(2), solve for d, g

Use substitution to get:

g ==11 years old

d==39 years old

There are 5 odd digits: 1, 3, 5, 7, 9

There are 4 even digits: 2, 4, 6, 8

The arrangements must begin with an odd number and end in an odd number:

Therefore, there are: 5!  x  4! ==120  x  24 ==2,880 such even arrangements possible.

Let g = Grogg's age and d = Grogg's dad's age.

7 years ago, Grogg's dad was 8 times as old as Grogg:

d - 7 = 8(g - 7)

4 years ago, his dad was 5 times as old as Grogg:

d - 4 = 5(g - 4)

Solving for g and d, we get:

g = 11 d = 44

Therefore, Grogg's dad is currently 44 years old.

To find the greatest possible length of each of the smaller pieces of rope, we need to find the greatest common factor (GCF) of 300, 321, and 396. The GCF is the largest number that divides evenly into all three numbers.

To find the GCF, we can factor each number into its prime factorization. The prime factorization of 300 is 2 * 2 * 3 * 5 * 5, the prime factorization of 321 is 3 * 107, and the prime factorization of 396 is 2 * 2 * 3 * 3 * 11.

The common factors of 300, 321, and 396 are 3 and 2. The GCF is the product of the common factors, which is 6.

Therefore, the greatest possible length of each of the smaller pieces of rope is 6 cm.

That's what I thought at first but I don't think you're correct.

Hello! The answer is 54 and here is my solution:

Since the ratio of yellow, green, blue, and white marbles is 1:2:3:4 –

yellow = x

green = 2x

blue = 3x

white = 4x

So x + 2x + 3x + 4x = 180 marbles. Then we simplfy to 10x = 180, therfore x = 18. So there are 18 yellow marbles, now we have to find how many blue marbles there are which is 3*18 = 54.

part a is wrong, the answer is 200 not 167

I think this is wrong because point P is (-2, 15), not (-2, 2). Thank you for your answer though.

Here's how you can solve this problem without using calculus:

Since the line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P, the line and the parabola share a common point. This means that the line's equation can be written in the form of a parabola's equation.

The equation of a parabola in vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

The line y = –11x – 7 can be written in the form of a parabola's equation by completing the square. Completing the square gives us y = –11(x + 1)^2 + 2.

The vertex of the parabola y = –11(x + 1)^2 + 2 is (-1, 2).

Since the line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P, and the x-coordinate of P is –2, we know that the point P is (-2, 2).

Since the point P is (-2, 2), we know that a + b = 2.

Therefore, the value of a + b is 2.

are we supposed to know the area of the circle or not

first of all, what even is a triangle AB

just use the angle bisector theorem from B to solve for BC

then use it again from C to solve for BZ and the fact that BZ+ZA = BA

1. The formula for area of a triangle is \(A=\frac{1}{2}bh\). Plugging in the side lengths it's \(\frac{1}{2}(6)(8)=24\) square units.

2. The sum of the angles in a triangle is 180 degrees. The sum of the other two angles given is \(135+30=165\). So to find the unknown angle you subtract 180-165, which is 15. Therefore angle \(BAC=15º\).

b=6.5179 - this is the approx. value of b

6.5179 / 7==0.931128571428 this is the first term of an infinite GS

7 ==0.931128571428 / [1  -  r], solve for r

r ==0.866981632653061 - this is the common ratio of the infinite GS

Sum=0.931128571428 / [1  -  0.866981632653061] ==7

The equation will have either

1, no real roots, meaning that the line doesn't intersect with the parabola,

2, two real roots, meaning that the line intersects the parabola at two (different) points, or

3, two equal roots, meaning that the line contacts the parabola at a single point. In that case the line will be a tangent to the parabola.

Subtracting 26 from both sides of the congruence gives 13(3x - 2) = 0 (mod 8). Dividing both sides by 13 gives 3x - 2 = 0 (mod 8). Adding 2 to both sides gives 3x = 2 (mod 8). Dividing both sides by 3 gives x = 2/3 (mod 8).

The positive integer solutions to this congruence are 2/3, 10/3, 18/3, and 26/3. The sum of these solutions is 2 + 10 + 18 + 26 = 56.

Therefore, the answer is 56.

Find the ordered triple (p,q,r) that satisfies the following system:  

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

We'll identify the three equations as equations 1, 2, and 3.  

p - 2q = 3           (eq 1)  

q - 2r = -2 + q     (eq 2)  

p + r = 9 + p        (eq 3)  

per (eq 3)                                            p + r  =  9 + p  

subtract p from both sides                         r  =  9  

per (eq 2)                                           q – 2r  =  –2 + q  

subtract q from both sides                      –2r  =  –2  

divide both sides by –2                               r  =  1  

If the system is to be valid, then  

r cannot be equal to both 9 and 1,  

so there must be something wrong  

with the problem as stated.                  cannot be solved  

_.