A pair of standard dice are rolled. Find the
probability of rolling a sum of 7 with these dice.

The triangles which are congruent are P and Q. So, the correct answer is A).

Triangles are congruent if they have the same shape and size. This means that all corresponding angles and sides are equal. In this case, we can use the side lengths to determine which triangles are congruent.

Triangle P and Q have the same side lengths, so they are congruent (A). Triangle Q and R do not have the same side lengths, so they are not congruent (not B). Triangle R and S do not have the same side lengths, so they are not congruent (not C).

Triangle S and P do not have the same side lengths, so they are not congruent (not D). Therefore, the answer is A. Triangles P and Q are congruent.

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--The given question is incomplete, the complete question is given

" 8.G.1.2 A mathematical puzzle uses four triangles with the dimensions show below. Which of the following triangles are congruent? A. P and Q B. Q and R C. R and S D. S and P "--

The solutions  of the system of equations are (0, -6) and (7, 8)

Here we have the system of equations:

y=x^2−5x−6

y=2x−6

To solve this, we need to solve the equation:

x^2 - 5x - 6 = 2x - 6

x^2 - 7x = 0

x*(x - 7) = 0

We can see that the two solutions are:

x = 0

x = 7

Evaluating the line in these values we get:

y = 2*0 - 6 = -6

y = 2*7 - 6 = 8

Then the solutions are (0, -6) and (7, 8)

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Answer: $3120

Step-by-step explanation:

3500-3240

=260

260*12=3120

Answer:

-------------------------

Given a rectangle with dimensions of:

Use area equation and find the value of x:

The second root is ignored as being negative.

The dimensions of the rectangle are:

The coordinates for the centroid of the triangle is (0, 2.67)

The centroid of a triangle is the place where the three medians of the triangle meet or intersect.

To find the centroid of the triangle, we use the formula

(x₁ + x₂ + x₃)/3 , (y₁ + y₂ + y₃)/3

We add all the x coordinates together and divide by 3, we also do the same for the y coordinates.

It becomes x = (-4 + 0 + 4)/3  

y = (8 + 2 + (-2))/3

Therefore, the centroid is (0, 2.67)

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It will take approximately 12.85 hours until only 1 gram of the element remains.

If after 2 hours, 36% of the initial amount remains, then the remaining amount is 0.36 times 100 grams, which is 36 grams. Let's define the amount of the element remaining after time t in hours as R(t). Since the element is decaying exponentially, we can model it with the equation R(t) = 100[tex](0.36)^t[/tex].

We want to find how many hours it will take until only 1 gram remains. We can set R(t) equal to 1 gram and solve for t:

1 = 100[tex](0.36)^t[/tex]

Taking the logarithm of both sides, we get:

ln(1) = ln(100[tex](0.36)^t[/tex])

0 = ln(100) + t*ln(0.36)

t = -ln(100)/ln(0.36)

Using a calculator, we find that t ≈ 12.85 hours.

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Answer:

h=.34

Step-by-step explanation:

1/2(bh)=A

1/2(5.5*h)=8.9375

5.5h=1.875

h=.34

The solution to each of the given polynomials are:

1)  x² + 3x - 4

2) x² - 4x + 4

3) x² - 7x + 8

4) x² - 13 - (15/(x + 3)

1) We want to divide the polynomial. Thus:

          x² + 3x - 4

x + 2| x³ + 5x² + 2x - 8

     -   x³ + 2x²

                 3x² + 2x

               - 3x² + 6x

                        - 4x - 8

                      -  -4x - 8

                                0

2) We want to divide the polynomial. Thus:

        x² - 4x + 4

x - 2| x³ - 6x² + 12x - 8

     -  x³ - 2x²

              - 4x² + 12x

              - -4x² + 8x

                         4x - 8

                      -  4x - 8

                                0

3) We want to divide the polynomial. Thus:

         x² - 7x + 8

x + 3| x³ - 4x² - 13x + 24

      -  x³ + 3x²

              - 7x² - 13x

              - -7x² - 21x

                         8x + 24

                      -  8x + 24

                                0

4) We want to divide the polynomial. Thus:

        x² - 13

x + 3| x³ + 3x² - 13x - 15

      -  x³ + 3x²

              - 0x² - 13x

              - -7x² - 13x

                       0 - 15

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The equation that models the balance y after t months is y = -75t + 1,500

To write an equation that models the balance y after t months, we need to use the given information to determine the rate at which the balance is decreasing. We can use the formula for a straight line, y = mx + b, where m is the slope and b is the y-intercept.

We can start by finding the change in the balance over the three-month period from four to seven months after the purchase:

Change in balance = $1,200 - $975 = $225

The rate of change can be calculated by dividing the change in the balance by the number of months:

Rate of change = $225 / 3 months = $75 per month

We can use this rate of change as the slope of the equation:

y = -75t + b

To find the y-intercept b, we can use the fact that the balance was $1,200 four months after the purchase:

1,200 = -75(4) + b

b = 1,500

Therefore, the equation that models the balance y after t months is:

y = -75t + 1,500

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My own multi-step combination problem is given below:

To solve this problem, Amanda  need to use the formula for combinations and it is:

nCr = n! / (r! x (n-r)!)

where:

n = total number of items to select from

r is the number of items to select.

First, we have to calculate the number of ways to select 3 fruit from 5, hence it will be:

5C3

=  5! / (3! x (5-3)!)

= 10

Next, we have to calculate the number of ways to select 2 meatpie from 4 and it will be

4C2

= 4! / (2! x  (4-2)!)

= 6

Lastly,, we need to calculate the number of ways to select 2 desserts from 3 and it will be:

3C2

= 3! / (2! x  (3-2)!)

= 3

To have the total number of dinner menus, we have to multiply these three numbers together:

= 10 x 6 x 3

= 180

Therefore, one can say that Amanda have 180 different dinner menus that she can be create.

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Answer:

A!!!!!!!!!!!!!!!!!!!!!!!!!

The equation of the line passing through the points (6, 5) and (8, -2) is[tex]y = -\frac{7}{2} x + 26[/tex].

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

Determine the slope of the line.

Given the two points are (-7, -3) and (1, 2)

We can find the slope of the line by using the slope formula:

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values:

m = (-2 - 5) / (8 - 6)

m = -7/2

Using the point-slope form, plug in one of the given points and slope m = -7/2 to find the equation of the line.

Let's use the point (6, 5):

y - y₁ = m(x - x₁)

[tex]y - 5 = -\frac{7}{2}( x - 6 )\\\\y - 5 = -\frac{7}{2} x + 21\\\\y = -\frac{7}{2} x + 21 + 5\\\\y = -\frac{7}{2} x + 26[/tex]

Therefore, the equarion of the line is [tex]y = -\frac{7}{2} x + 26[/tex].

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Some of the common order of operation mistakes that students make when evaluating expressions are the use of basic mathematical operations.

You should know that an error or fault results from defective judgment, deficient knowledge, or carelessness.

A combination of numbers, letters, and relational operators containing at least one operation is deemed a mathematical expression. Additionally, it must also include variables and/or numbers, excluding an equal sign.

Some prevalent errors include using both a positive and a negative sign, using two negative signs, and misusing brackets.

By being unable to comprehend the problems accurately, learners often commit significant errors while assessing formulations, which underscores the importance of employing fundamental mathematical operations.

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18.2

Step-by-step explanation:

AB² =AC² + CB² -2(AC)(CB)CosA

X²=(17)² + (10)² - 2 (17)(10) Cos80

x² = 289 + 100- 340 cos 80

x= √289+100-340cos80

x= 18.2

Answer:

The answer is A.

Step-by-step explanation:

For this graph, you are using the linear equation formula: y = mx + b

The value of the constant term k in the exponential model of the bacteria population is k ≈ 0.0972

An exponential function is one in which the input variable is part of the exponent of the expression for the function.

The function for the population of the bacteria indicates;

[tex]P(t) = \frac{22,000 }{1 + 1.2\cdot e^{(-k\cdot t)}}[/tex]

The population of the bacteria after t = 1 hour is P(1) = 10,532

Therefore;

[tex]P(1) = \frac{22,000 }{1 + 1.2\cdot e^{(-k\times 1)}} = 10,532[/tex]

[tex]1 + 1.2\cdot e^{(-k\times 1)} = \frac{22,000}{10,532}[/tex]

[tex]1.2\cdot e^{(-k\times 1)} = \frac{22,000}{10,532} - 1[/tex]

[tex]e^{(-k\times 1)} = \frac{\frac{22,000}{10,532} - 1}{1.2}[/tex]

Taking the natural logarithm of both sides, we get;

[tex]\ln (e^{(-k\times 1)}) = \ln (\frac{\frac{22,000}{10,532} - 1}{1.2})[/tex]

[tex]\ln (e^{(-k)}) = \ln (\frac{\frac{22,000}{10,532} - 1}{1.2})[/tex]

[tex]\ln (e^{(-k)}) = -k = \ln (\frac{\frac{22,000}{10,532} - 1}{1.2})[/tex] ≈ -0.0972 (rounded to 4 decimal places)

-k ≈ -0.0972

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Using some logarithm properties we will get the expanded expression:

log(y^10) = 10*ln(y)/ln(10)

So there are two logarithm properties that we can use here, these are:

logₙ(x) = ln(x)/ln(n)

and:

logₙ(x^a) = a*logⁿ(x)

Now let's go to our expression, it is:

log(y^10)

Using the second property we can remove the exponent, then we will get:

log(y^10) = 10*log(y)

Now we can use the first rule, remember that log(x) has a base of 10, then we can write:

10*log(y) = 10*ln(y)/ln(10)

That is the expession expanded.

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The length of the curve y = 5 - 4x, -2 ≤ x ≤ 2, is 4√(17).

To use the arc length formula, we need to find the derivative of the curve y = 5 - 4x:

y' = -4

Then, we can use the arc length formula:

L = ∫[a,b] √(1 + (y')^2) dx

where a and b are the limits of integration. In this case, a = -2 and b = 2.

L = ∫[-2,2] √(1 + (-4)^2) dx

= ∫[-2,2] √(17) dx

= √(17) * [x]_(-2)^(2)

= 4√(17)

So the length of the curve y = 5 - 4x, -2 ≤ x ≤ 2, is 4√(17).

To check this answer, we can note that the curve is a line segment with endpoints (-2, 13) and (0.75, 1), and we can calculate its length using the distance formula:

L = √((0.75 - (-2))^2 + (1 - 13)^2)

L = √(18.25 + 144)

L = √(162.25)

L = 4√(17)

This matches our previous answer, so we can be confident that the length of the curve is 4√(17).

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The vector u+v can be represented as < √85, -41.19° >, which is obtained by adding vector u marked on the y-axis from 0 to 6 and vector v marked on the x-axis from 0 to -7 on a coordinate plane.

To add vectors on the coordinate plane. The vector u is marked on the y-axis from 0 to 6, and the vector v is marked on the x-axis from 0 to -7.

To add them, we start at the tail of u and move horizontally to the left by 7 units (since v has a negative x-component) and then vertically up by 6 units (since u has a positive y-component) to reach the head of the resulting vector. This gives us the vector u+v.

So, the vector u+v can be represented as

Magnitude √(6^2 + 7^2) = √85

Direction atan(6/(-7)) = -41.19 degrees (measured counterclockwise from the positive x-axis)

Notation < √85, -41.19° >

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The Taylor polynomial t3(x) for f(x) = cos(x) centered at a = pi/2 is t3(x) = -(x-pi/2) + (x-pi/2)³/3!.

A polynomial is  described as an expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

The Taylor polynomial of degree n for a function f(x) centered at a is given by the formula:

We get the derivatives :

pn(x)=f(c)+f′(c)(x−c)+f′′(c)2!....

f(x) = cos(x)

f'(x) = -sin(x)

f''(x) = -cos(x)

f'''(x) = sin(x)

We the  evaluate these derivatives at x = pi/2:

f(pi/2) = cos(pi/2) = 0

f'(pi/2) = -sin(pi/2) = -1

f''(pi/2) = -cos(pi/2) = 0

f'''(pi/2) = sin(pi/2) = 1

We substitute  these values into the formula for the Taylor polynomial, and simplify to get Taylor polynomial T3(x) for the function:

the Taylor polynomial t3(x) for f(x) = cos(x) centered at a = pi/2 is t3(x) = -(x-pi/2) + (x-pi/2)³/3!.

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Answer:

4mn + 3m + 8n + 6 = (m + 2)(4n + 3)

The missing term is 4n.

Answer:

Step-by-step explanation:

No

Tyler will need a amount of  79,200 grams (or 79.2 kilograms) of glitter to cover the prism completely.

To calculate the amount of glitter needed to cover the prism completely, we first need to find the total surface area of the prism.

The prism consists of five faces: two rectangular faces with dimensions 18 m x 10 m, two triangular faces with base 18 m and height 10 m, and one rectangular face with dimensions 18 m x 25 m.

The total surface area of the prism can be calculated by adding the areas of all the faces together:

Total Surface Area = 2(18 m x 10 m) + 2(0.5 x 18 m x 10 m) + 18 m x 25 m

Total Surface Area = 360 m² + 180 m² + 450 m²

Total Surface Area = 990 m²

Given that 80 g of glitter is needed to cover each m², we can now calculate the total amount of glitter needed:

Total Glitter = Total Surface Area x Glitter per m²

Total Glitter = 990 m² x 80 g/m²

Total Glitter = 79,200 g

Therefore, Tyler will need 79,200 grams (or 79.2 kilograms) of glitter to cover the prism completely.

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The equation of the line parallel to the graph of 2x - 3y = 7 and containing the point (-3,-3) is y = (2/3)x - 1.

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

Given the equation of the graph: 2x - 3y = 7.

First, we need to rearrange the given equation into slope-intercept form (y = mx + b) by solving for y:

2x - 3y = 7

-3y = -2x + 7

y = (2/3)x - 7/3

The slope of the given line is 2/3.

Hence, any line parallel to it will also have a slope of 2/3.

Using the point-slope form, we determine the equation of the line that passes through (-3,-3) and has a slope of 2/3:

y - y1 = m(x - x1)

y - (-3) = (2/3)(x - (-3))

y + 3 = (2/3)x + 2

y = (2/3)x - 1

Therefore, the equation of the line is y = (2/3)x - 1.

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The proportion relation obtained from the similar triangles indicates that the x/3 = 10/x

Similar triangles are triangles that have the same shape but their sizes may vary.

The proportion equation from the image based on the ratio of the corresponding sides of the similar right triangles can be found as follows;

The side x is the hypotenuse side of the smallest right triangle

The ratio x/3, is equivalent to the ratio of the hypotenuse side to the shortest side of the right triangle

The corresponding ratio in the medium triangle is ((x² - 3²) + 7²)/(x² - 3²)

((x² - 3²) + 7²)/(x² - 3²) = (x² + 40)/(x² - 3²)

The corresponding ratio in the large similar right triangle is (7 + 4)/x = 10/x

Therefore, the corresponding ratio is; x/3 = 10/x

The proportion is; x/3 = 10/x

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To plot the above linear regression on a graph, follow the procedures outlined below.

You may use the following steps to plot the supplied data on a graph:

1) Create an x- and y-axis coordinate plane.

2) Label the x-axis with the numbers 1 through 13 (to represent the 13 data points).

3) Label the y-axis with a suitable scale that encompasses the data's greatest and lowest values (e.g., 0 to 2200 with 200 increments).

4) For each point, try to find  the associated x -value on the x-axis and the corresponding y-value on the y- axis, and then place a point at the intersection of the two values.

For example, if the first point is (1, 1000), you would identify the x-axis point labeled "1" and the y-axis point labeled "1000" and add a point at it  intersection.

5) based  on the pattern in the data, connect the points of the coordinares using  a line or a curve. Because the data appears to increase gradually with some fluctuations in this case, a line connecting the points would be appropriate.

After plotting the data on the graph, you may study the pattern and draw inferences about the data's trend over time. Please see the attached sample graph.

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A. Using the formula for area of rectangle, the length of the rectangular space is: 2x - 3

B. The expression to prove this is: (2x - 3) * 4x = 8x² - 12x

To find the area of a rectangle, multiply the width and the length together. This means: Area = length * width.

Part A: We are given that area of the rectangular space is expressed as 8x² - 12x, if the width is 4x, then:

Length of the rectangular space = (8x² - 12x) / 4x = 4x(2x - 3)/4x

Length = 2x - 3

Part B: To prove this, we have:

length * width = 8x² - 12x

Substitute:

(2x - 3) * 4x = 8x² - 12x

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Check the picture below.