For each part, write the equation that represents the line in slope-intercept form (y = mx + b) and standard form (ax + by = c where a, b, and c are integers and a is not negative).

The best estimate for the marginal cost in dollars, to produce the 57th unit of the product is C'(57) - C'(56). This is also known as the difference in cost between producing the 57th unit and the 56th unit, and is represented by the expression C'(57) - C'(56).

The marginal cost of producing a particular unit of a product is the cost of producing that unit minus the cost of producing the previous unit. So to find the marginal cost of producing the 57th unit of the product, you can subtract the cost of producing the 56th unit from the cost of producing the 57th unit.

This means that the best estimate for the marginal cost of producing the 57th unit of the product is given by:

M(57) = C(57) - C(56)

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Answer:

A)  Mr. Simpson's class; it has a larger median value 14.5 pencils.

Step-by-step explanation:

A box plot is a visual display of the five-number summary:

From inspection of the box plots (attached), the measures of central tendency (median) and dispersion (range and IQR) are:

Mr Johnson's class:

Mr Simpson's class:

In a box plot, the median is a measure of central tendency and tells us the location of the middle value in the dataset. It divides the data into two equal halves, with 50% of the values falling below the median and 50% above it.

The median number of pencils lost in Mr Simpson's class is greater than the median number of pencils lost in Mr Johnson's class. Therefore, Mr. Simpson's class has a larger median value.

The spread of data in a dataset can be measured using both the range and the interquartile range (IQR).

As Mr Simpson's class has a greater IQR and range than Mr Johnson's class, the data in Mr Simpson's class is more spread out than in Mr Johnson's class.

In summary, as Mr Simpson's class has a larger median 14.5 and a wider spread of data, then Mr Simpson's class lost the most pencils overall.

Answer:

3x

Step-by-step explanation:

39*x / 13

39/13 * x

3*x

3x

Answer:

3x

Step-by-step explanation:

We are given the expression:

39*x /13

We want to simplify this expression. It can be simplified because both the numerator (top number) and denominator (bottom number) can be evenly divided by 13.

(39*x /13) / (13/13)

(39x/13) / 1

3x / 1

When the denominator is 1, we can simply eliminate the denominator and leave the numerator as our answer.

3x

The expression 39*x/13 can be simplified to 3x

Answer:

The age of the pottery bowl is 12,378.7 years

Step-by-step explanation:

The amount of C-14 after t yeas is given by the following equation:

\(N(t) = N(0)e^{-kt}\)

In which N(0) is the initial amount and k is the decay rate.

In this question, we have that:

\(k = 0.0001\)

So

\(N(t) = N(0)e^{-0.0001t}\)

Age of the pottery bowl:

29% of its original amount of C-14. So we have to find t for which N(t) = 0.29N(0). So

\(N(t) = N(0)e^{-0.0001t}\)

\(0.29N(0) = N(0)e^{-0.0001t}\)

\(e^{-0.0001t} = 0.29\)

\(\ln{e^{-0.0001t}} = \ln{0.29}\)

\(-0.0001t = \ln{0.29}\)

\(t = -\frac{\ln{0.29}}{0.0001}\)

\(t = 12378.7\)

The age of the pottery bowl is 12,378.7 years

P(A and B) is equal to 6.

To find P(A and B), we can use the formula:

P(A and B) = P(A) + P(B) - P(A U B)

Given the information:

P(A U B) = 32 (the probability of either event A or event B occurring)

Universal set contains 52 elements (total number of possible outcomes)

P(A intersection B) = 6 (the probability of both event A and event B occurring)

P(A) = 12 (the probability of event A occurring)

P(B) = 26 (the probability of event B occurring)

We can substitute the known values into the formula:

P(A and B) = P(A) + P(B) - P(A U B)

P(A and B) = 12 + 26 - 32

Simplifying the expression:

P(A and B) = 38 - 32

P(A and B) = 6

Therefore, P(A and B) is equal to 6.

The result indicates that the probability of both event A and event B occurring simultaneously is 6 out of the total number of possible outcomes in the universal set. It means that out of the 52 elements in the universal set, 6 of them satisfy the conditions of both A and B.

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The length of line x in the figure of the cube given is 19.

Calculate the length of the base of the cube, which is the diagonal of the lower sides :

base length = √10² + 6²

base length = √136

The length of x is the diagonal of the cube

x = √(√136)² + 15²

x = √136 + 225

x = √361

x = 19

Therefore, the length of line x in the figure given is 19.

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The Pythagorean theorem states that in a right triangle, the square of the hypotenuse = sum of the squares of the legs.

We have,

The Pythagorean theorem that relates to the sides of a triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (known as the legs). Mathematically, it can be expressed as a² + b² = c², where "a" and "b" are the lengths of the legs, and "c" is the length of the hypotenuse.

Now,

In a right triangle, the legs are the two sides that form the right angle, and the hypotenuse is the side that is opposite to the right angle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse = sum of the squares of the legs. So, the relationship between the legs and the hypotenuse can be described by this theorem. In other words, if we know the length of the two legs of a right triangle, we can use the Pythagorean theorem to find the length of the hypotenuse, and vice versa. The hypotenuse is always the longest side of the right triangle, and it is also the side that connects the two legs.

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complete question:

Applying the Pythagorean Theorem In this activity, you will explain your understanding of mathematical relationships and use the Pythagorean theorem to solve real-world problems. Question 1 In your own words, explain the relationship between the legs and the hypotenuse of a right triangle.

By writing and solving a system of equations, we will see that the club sold 32 oranges and 68 bars.

Here the solution will be the number of oranges and protein bars sold.

Let's define:

x = number of oranges.

y = number of bars.

We know that they sell 100 items and win a total of $304, then we can write the system of equations:

x + y = 100

x*$1 + y*$4 = $304

Isolating x on the first equation we get:

x = 100 - y

Replacing that in the other equation we get:

(100 - y)*$1 + y*$4 = $304

$100 - y*$1 + y*$4 = $304

y*$3 = $304 - $100 = $204

y = $204/$3 = 68

y = 68

And the value of x is:

x = 100 - y = 100 - 68

x = 32

That is the solution, x = 32 and y = 68.

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If the bucket is allowed to fall, then its linear acceleration is 8.687 m/s^2.

From the given question,

We have to solve, if the bucket is allowed to fall, then its linear acceleration

Mass of Pulley(M)= 0.692 kg

Radius of pulley(r)= 0.131 m

Mass of bucket(m)= 2.70 kg

Net torque on pulley,

\(\Sigma\) r = rT = Iα

T = Iα/r

T= tension on string(connecting pulley and bucket)

I= moment of inertia of pulley

α= angular acceleration of pulley

linear acceleration of bucket(a),

a = rα

α = a/r

Pulley is disc shaped so

I = 1/2 Mr^2

Now, T= Iα/r = (1/2 Mr^2) (a/r)/r

T = 1/2 Ma

Now considering the motion of bucket

ma = mg-T

ma = mg-1/2 Ma

ma+1/2 Ma = mg

a = m/(m+1/2 M) g

Now putting the value

a = (2.70)×9.8/ (2.70+1/2 (0.692))

After solving

a = 8.687 m/s^2

If the bucket is allowed to fall, then its linear acceleration is 8.687 m/s^2.

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The right question is;

A 2.70 kg bucket is attached to a disk-shaped pulley of radius 0.131 mm and mass 0.692 kg . If the bucket is allowed to fall, What is its linear acceleration.

The length of MQ is 5 units.

To find MQ we have a line segment with two parts:

One represented by 4x - 1 and the other by 12x - 17.

The entire line segment, MQ, is represented by 5. To find the value of MQ, we need to set up an equation using the given terms:

(4x - 1) + (12x - 17) = 5

Now, let's solve the equation step by:

Combine like terms: 16x - 18 = 5 2.

Add 18 to both sides of the equation:

16x = 23 3.

Divide both sides by 16:

x = 23/16

Now that we have the value of x, we can find the length of MQ, which is equal to 5.

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Answer:

The smaller number is 4√2 and the larger number is (4√2 + 1).

Step-by-step explanation:

Let the two numbers be x and y, where y is the larger of the two numbers.

Since y is the larger number, it is one more than the smaller number. So:

\(y=x+1\)

When negative two times the smaller is added to the square of the larger, the result is 33. In other words:

\(-2x+y^2=33\)

Substitute:

\(-2x+(x+1)^2=33\)

Solve for x. Square:

\(-2x+(x^2+2x+1)=33\)

Simplify:

\(x^2+1=33\)

Subtract one from both sides:

\(x^2=32\)

And take the square root of both sides:

\(x=\pm\sqrt{32}=\pm 4\sqrt{2}\)

Since y is positive, we can ignore the negative case. So, the smaller number is:

\(x=4\sqrt{2}\approx5.66\)

And the larger number is:

\(y = 4\sqrt{2} + 1 \approx6.66\)

Explanation:

We have 8 choices for the first slot, 7 for the second, and so on. We count our way down until we have 1 choice left for the eighth slot.

We'll get 8*7*6*5*4*3*2*1 = 40,320 different permutations.

You could use the nPr formula

\(_nP_r = \frac{n!}{(n-r)!}\)

with n = 8 and r = 8 to get the same result. The exclamation marks indicate factorials.

The 8 violinists can be seated in 40,320 ways.

A permutation is an arrangement of items in a specific direction or sequence. One should consider both the selection and the arrangement while dealing with permutation. In permutations, ordering is crucially important. The permutation is seen as an ordered combination, in other words.

Given In the orchestra, there are eight violinists in the front row,

the sitting arrangement of violinists is done with the help of permutation,

ⁿPₓ = n!/(n - x)!

here n = 8 and x = 8

(n - x)! = (8 - 8)!

(n - x)! = = 0! = 1

⁸P₈ = 8!/1

⁸P₈ = 8 x 7 x 6 x 5 x 4 x 3 x 2 x1

⁸P₈ = 40,320

Hence they can be seated in 40,320 ways.

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Answer:

m = -2

Step-by-step explanation:

-m^3 + 17m + 44 = -(m + 2)(m^2 - 2m - 22)

m + 2 = 0 or m^2 - 2m - 22 = 0

m = -2 or m = 1 ± √23

Answer:

Step-by-step explanation:

The value of x and y are: x = 5 2/3 and y = 1 1/3

Perimeter is the distance around the edge of a shape. Learn how to find the perimeter by adding up the side lengths of various shapes.

The perimeter of a rectangle = 2(l+w)

P = 2(l+w)

P= 2( x+2y +x)

28 = 2(2x+2y)

28 = 4x +4y

divide all by 4

x+y = 7

the perimeter of a triangle = a+b+c

24 = 3x + 3y + 4y

24 = 3x + 7y

relating the two perimeters

x+y = 7 equation 1

3x + 7y = 24 equation 2

x = 7 -y

3( 7 -y) +7y = 24

21 -3y +7y = 24

21 +4y = 24

4y = 3

y = 4/3 = 1 1/3

x+ y = 7

x = 7-4/3

x = (21-4)/3

x = 17/3 = 5 2/3

therefore the value of x = 5 2/3 and y = 1 1/3

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The value of a in the relative frequency table for the survey results a = 27%

From the given Venn diagram, the total number of players =  11 + 3 + 7 + 5 = 26

The number of players preferred magazines books which are not coupon books = 7

The relative frequency for the players who preferred magazines books which are not coupon books (a)=

magazine books but not coupon books/ total player = 7 / 26 = 0.269

In percent,

7/26 x 100 = 26.9 = 27%

Hence, a = 27%

Therefore, the value of a in the relative frequency table for the survey results a = 27%

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Answer:

9 and -i

Step-by-step explanation:

12-3 + 5i-4i

9 + -i

9-i

The y-intercept is -14

Parameters:

Slope, m = 2

Point = (5, -4)

x = 5, y = -4

The equation of a line is

y = mx + c

Substitute the given parameters

-4 = 2(5) + c

-4 = 10 + c

c = -4 - 10

= -14

The y-intercept is -14

The ratios which are equivalent are as follows:

1) 16:4

4) 480:120

6) 4:1

Step-by-step solution:

1) 16:4

\(=\frac{16}{4}\)

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

4:1

2) 2:1

\(=\frac{2}{1}\)

2:1

3) 32:2

\(=\frac{32}{2}\)

\(=\frac{16}{1}\)

=16:1

4) 480:120

\(=\frac{480}{120}\)

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

=4:1

5) 4:16

\(=\frac{4}{16}\)

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

=1:4

6) 4:1

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

=4:1

Therefore, the correct solution to this question is:

1) 16:4

4) 480:120

6) 4:1

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Answer:

16:4 and 480:120 and 4:1 are equivalent.

Step-by-step explanation:

The depth of the crater in yards is 387.2 yards

Since the crater is 13,937.63 in deep, we convert this to yards.

Since 36 in = 1 yard, we use this conversion factor to convert to yards

13,937.63 in × 1 yard/36 in = 387.16 yards

≅ 387.2 yards to the nearest tenth

So, the depth of the crater in yards is 387.2 yards

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Answer:

1000 tiles

Step-by-step explanation:

Determine total floor space
800 x 10 = 8000 squared cm total floor space.
Divide floor space by size of tile
8000 / 8 = 1000 tiles now required to cover the floor.

passive income

dividends, real estate, interest income

active income

salary, business income,  

Answer:

5y(x^2 - 3)

Step-by-step explanation:

Hope this helps :)

(-∞, -4] U [1, ∞)

The given function is:

Since f(x) ≥ 0, we are going to find the range of values of x for which f(x) ≥ 0

By carefully observing the graph shown:

f(x) ≥ 0 for values of x less than or equal to -4, and also for values of x greater than or equal to 1.

This can be written mathematically as:

x ≤ -4 or x ≥ 1

This can be written in interval notation as:

(-∞, -4] U [1, ∞)

The number of adult tickets sold and senior tickets sold are 88 and 52 respectively.

Given that:-

Total number of tickets sold = 140

Adult ticket price = $ 13

Senior ticket price = $ 10

Total receipts of adult and senior tickets = $1,664

We have to find the number of adult tickets sold and senior tickets sold.

Let the number of adult tickets be x and,

The number of senior tickets sold be y.

Hence, we can write,

x + y = 140 ...(1)

13x + 10y = 1664  ... (2)

Here we have a set of linear equations.

We can solve it by using elimination method.

Multiplying (1) by 10, we get,

10x + 10y = 1400  ...(3)

(2) - (3)

3x + 0y = 264

3x = 264

x = 264/3

x = 88

Putting x = 88 in (1), we get,

88 + y = 140

y = 140 - 88

y = 52

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Answer:

1)  The functions are not inverses.

\(\textsf{2)} \quad g^{-1}(n)=&\dfrac{3}{8}n-\dfrac{7}{8}\)

\(\textsf{3)} \quad g^{-1}(x)&=\sqrt[3]{\dfrac{1}{2}-\dfrac{1}{2}x}\)

Step-by-step explanation:

The inverse composition rule states that if two functions are inverses of each other, then their compositions result in the identity function.

Given functions:

\(g(x) = 4 + \dfrac{7}{2}x \qquad \qquad f(x) = 5 - \dfrac{4}{5}x\)

Find g(f(x)) and f(g(x)):

\(\begin{aligned} g(f(x))&=4+\dfrac{7}{2}f(x)\\\\&=4+\dfrac{7}{2}\left(5 - \dfrac{4}{5}x\right)\\\\&=4+\dfrac{35}{2}-\dfrac{14}{5}x\\\\&=\dfrac{43}{2}-\dfrac{14}{5}x\\\\\end{aligned}\)                 \(\begin{aligned} f(g(x))&=5 - \dfrac{4}{5}g(x)\\\\&=5 - \dfrac{4}{5}\left(4 + \dfrac{7}{2}x \right)\\\\&=5-\dfrac{16}{5}-\dfrac{14}{5}x\\\\&=\dfrac{9}{5}-\dfrac{14}{5}x\end{aligned}\)

As g(f(x)) or f(g(x)) is not equal to x, then f and g cannot be inverses.

\(\hrulefill\)

To find the inverse of a function, swap the dependent and independent variables, and solve for the new dependent variable.

Calculate the inverse of g(n):

\(\begin{aligned}y &= \dfrac{8}{3}n + \dfrac{7}{3}\\\\n &= \dfrac{8}{3}y + \dfrac{7}{3}\\\\3n &= 8y + 7\\\\3n-7 &= 8y\\\\y&=\dfrac{3}{8}n-\dfrac{7}{8}\\\\g^{-1}(n)&=\dfrac{3}{8}n-\dfrac{7}{8}\end{aligned}\)

Calculate the inverse of g(x):

\(\begin{aligned}y &= 1-2x^3\\\\x &= 1-2y^3\\\\x -1&=-2y^3\\\\2y^3&=1-x\\\\y^3&=\dfrac{1}{2}-\dfrac{1}{2}x\\\\y&=\sqrt[3]{\dfrac{1}{2}-\dfrac{1}{2}x}\\\\g^{-1}(x)&=\sqrt[3]{\dfrac{1}{2}-\dfrac{1}{2}x}\\\\\end{aligned}\)

Answer:

1.

If the composition of two functions is the identity function, then the two functions are inverses. In other words, if f(g(x)) = x and g(f(x)) = x, then f and g are inverses.

For\(\bold{g(x) = 4 + \frac{7}{2}x\: and \:f(x) = 5 -\frac{4}{5}x}\), we have:

\(f(g(x)) = 5 - \frac{4}{5}(4 + \frac{7}{2}x)\\ =5 - \frac{4}{5}(\frac{8+7x}{2})\\=5 - \frac{2}{5}(8+7x)\\=\frac{25-16-14x}{5}\\=\frac{9-14x}{5}\)

\(g(f(x)) = 4 + (\frac{7}{5})(5 - \frac{4}{5}x) \\=4 + (\frac{7}{5})(\frac{25-4x}{5})\\=4+ \frac{175-28x}{25}\\=\frac{100+175-28x}{25}\\=\frac{175-28x}{25}\)

As you can see, f(g(x)) does not equal x, and g(f(x)) does not equal x. Therefore, g(x) and f(x) are not inverses.

Sure, here are the inverses of the functions you provided:

2. g(n) = (8/3)n + 7/3

we can swap the roles of x and y and solve for y to find the inverse of g(n). In other words, we can write the equation as y = (8/3)n + 7/3 and solve for n.

y = (8/3)n + 7/3

n =3/8*( y-7/3)

Therefore, the inverse of g(n) is:

\(g^{-1}(n) = \frac{3}{8}(n - \frac{7}{3})=\frac{3}{8}*\frac{3n-7}{3}=\boxed{\frac{3n-7}{8}}\)

3. g(x) = 1 - 2x^3

We can use the method of substitution to find the inverse of g(x). We can substitute y for g(x) and solve for x.

\(y = 1 - 2x^3\\2x^3 = 1 - y\\x = \sqrt[3]{\frac{1 - y}{2}}\)

Therefore, the inverse of g(x) is:

\(g^{-1}(x) =\boxed{ \sqrt[3]{\frac{1 - x}{2}}}\)

Answer:

21

Step-by-step explanation:

A ratio of 3:4 means that for every 3 boys there are 4 girls and vice versa.

12 girls / 4 girls = 3

3 boys * 3 = 9 boys

There are 9 boys in the class.

9 boys + 12 girls = 21 total students

Answer:

269/99

Step-by-step explanation:

The measure of the minor arc m∡rv = 111°

Recall that according to the principle of angle of intersecting secants, (a-b)/2 = c

Where a = ∡RV
b = ∡SU = 37°:

and c = ∠T (the angle between intersecting secants.

Thus, if (a-b)/2 = c, and b = c = 37 degrees, then we can substitute these values into the equation to get:

(a - 37) / 2 = 37

Multiplying both sides by 2 gives:

a - 37 = 74

Adding 37 to both sides gives:

a = 111

Therefore, a is equal to  m∡rv = 111°.

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