What is the length of the side labeled x cm?

10.8
13.7
9.9
20.8

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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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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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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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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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 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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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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The order of the equations from least number of solutions to greatest number of solutions is: C, B , and A.

In algebra, an expression is made up of several numbers, variables, and mathematical operations including addition, subtraction, multiplication, and division. One expression is, for instance, 3x + 2y - 5.

In algebra, an equation is a claim that two expressions are equal to one another. Often, it asks us to determine the value of a variable that would make the equation true.

Equation A:

4x + 12 = 2(2x + 3) + 6

4x + 12 = 4x + 12

This equation has infinitely many solutions, as 0 = 0 is always true.

Equation B:

5(x - 1) = 3(x + 7)

5x - 5 = 3x + 21

2x = 26

x = 13

This equation has one solution, which is x = 13.

Equation C:

2x - 4 = 2(x + 18)

2x - 4 = 2x + 36

-4 = 36

This equation has no solutions.

Hence, the order of the equations from least number of solutions to greatest number of solutions is: C, B , and A.

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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

There are 4 + 5 + 7 + 8 = 24 buttons in total.

After the cat knocks 2 buttons onto the floor, there are 22 buttons remaining on the table. The probability that the first button knocked onto the floor is blue or red can be calculated as follows:

P(blue or red) = P(blue) + P(red)

P(blue or red) = 4/24 + 5/24

P(blue or red) = 9/24

P(blue or red) = 3/8

Therefore, the probability that the button on the floor was blue or red is 3/8 or 0.375.Answer:

Step-by-step explanation:

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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Answer: $3120

Step-by-step explanation:

3500-3240

=260

260*12=3120

Answer:

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

Answer:

The answer is A.

Step-by-step explanation:

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

Answer:

120

Step-by-step explanation:

25% is equal to 1/4. If you divide 30 by 1/4 you get 120. To check this, if you multiply 120 by 1/4 you get 30.

Answer: 120

Step-by-step explanation:

In order to find the missing value, divide the provided value by the percentage as follows:

30 / 0.25 = 120

This can be double-checked by plugging the number into the statement afterwards: 25% of 120 is 30, which is true

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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The probability that both events A and B will occur is 1/3 or approximately 0.3333.

To find the probability of both events A and B occurring, we need to calculate the individual probabilities of each event and then multiply them together.

Event A: The first die is a 4 or less.

In a standard six-sided die, there are four outcomes (1, 2, 3, 4) that satisfy this condition. Out of the six possible outcomes, four are favorable, so the probability of event A is 4/6 or 2/3.

Event B: The second die is even.

In a standard six-sided die, there are three even outcomes (2, 4, 6) out of the six possible outcomes.

Therefore, the probability of event B is 3/6 or 1/2.

To find the probability of both events occurring, we multiply the probabilities of each event:

[tex]P(A and B) = P(A) \times P(B) = (2/3) \times (1/2) = 1/3[/tex]

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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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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:

Answer:

30 and 10

Step-by-step explanation:

Main concepts:

Concept 1.  Geometric Sequence

Concept 2. Common ratio

Concept 3. Finding the next terms

Concept 1.  Geometric Sequence

Geometric sequences are a specific type of sequence with a pattern where a specific number is multiplied to each term to get the next term.

For instance, a completely different geometric sequence 1,2,4,8,16,... is geometric because you multiply any term by 2 to get the next term.

Concept 2. Common ratio

In a geometric sequence, this number that you can multiply by to get the next term (in the "1,2,4,8,16,..." example, the number 2) is called the common ratio because taking the ratio of two sequential terms of the sequence (dividing two terms in a row) always gives the same answer:

This process of finding ratios will yield a common ratio only if the sequence is a Geometric Sequence.

Concept 3. Finding the next terms

Since we're told that the sequence we're given in the question is a Geometric sequence, it must have a common ratio (otherwise, it wouldn't be a geometric sequence).

Find the common ratio by dividing sequential terms:

270/810 = 1/3

90/270 = 1/3

So, for the sequence in the question, the common ratio is 1/3.  Each term is multiplied by 1/3 to get the next term in the sequence.  (This is equivalent to dividing each term by 3 to get the next term in the sequence, if you prefer division and/or to avoid fractions)

So, to get the next two terms in the sequence, multiply by the common ratio:

90 * 1/3 = 30

30 * 1/3 = 10

So, the next two terms of the sequence are 30 and 10

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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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 "--

Answer:

Step-by-step explanation:

No

Answer:

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

The missing term is 4n.

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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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 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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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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Answer:

h=.34

Step-by-step explanation:

1/2(bh)=A

1/2(5.5*h)=8.9375

5.5h=1.875

h=.34