12. Stretch Your Thinking Write and solve a story problem about gathering three kinds of vegetables. Use 10-partners.

Answer:

5, 3

Step-by-step explanation:

x-7 is a factor of [tex]x^5+5x^4-2x^3+4x^2+8x+3976$[/tex], we can use the Factor theorem which states that if f(a) = 0, then x-a is a factor of f(x).the remainder when we divide [tex]x^5+5x^4-2x^3+4x^2+8x+3976$[/tex] by x-7 is 4054.

To check if x-7 is a factor of [tex]x^5+5x^4-2x^3+4x^2+8x+3976$,[/tex] we can use the factor theorem which states that if f(a) = 0, then x-a is a factor of f(x).

To check if x-7 is a factor, we evaluate the polynomial at x=7:

[tex]f(7) = 7^5+5\cdot7^4-2\cdot7^3+4\cdot7^2+8\cdot7+3976 = 384004$.[/tex]

Since [tex]$f(7)\neq 0$[/tex], we can conclude that x-7 is not a factor of [tex]x^5+5x^4-2x^3+4x^2+8x+3976$.[/tex]

To find the remainder when we divide [tex]$x^5+5x^4-2x^3+4x^2+8x+3976$[/tex]by x-7, we can use polynomial long division or synthetic division. Using polynomial long division, we get:

  [tex]x^4 + 12x^3 + 82x^2 + 578x + 4046\\x-7 | x^5 + 5x^4 - 2x^3 + 4x^2 + 8x + 3976\\- x^5 + 7x^4[/tex]

    ------------

      [tex]12x^4 - 2x^3 + 4x^2 + 8x\\ 12x^4 - 84x^3[/tex]

          -------------

                     [tex]82x^3 + 4x^2 + 8x\\ 82x^3 - 574x^2[/tex]

                      --------------

                                  [tex]578x^2 + 8x\\ 578x^2 - 4046[/tex]

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

                                              4054

Therefore, the remainder when we divide [tex]$x^5+5x^4-2x^3+4x^2+8x+3976$ by $x-7$ is $4054$.[/tex]

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Answer:

⟶ We can express the sum 1^2 + 2^2 + 3^2 + ... + 750^2 using summation notation as:

∑i^2, where i goes from 1 to 750.

⟶ Here, the symbol ∑ represents the summation, and i is the index variable, which takes on the values 1, 2, 3, ..., 750. The expression i^2 is the term being summed.

⟶ So the sum can be written as:

∑i^2 = 1^2 + 2^2 + 3^2 + ... + 750^2

⟶ Note that the upper limit of the index variable is 750, which corresponds to the last term in the sum.

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Answer:

[tex]\sum \limits^{750}_{n=1} {n^2}[/tex]

Step-by-step explanation:

General explanation of summation notation

Any sum of terms whose format is the same can be represented more concisely using a Summation symbol.

The Summation symbol is a capital greek letter Sigma:  [tex]\sum[/tex]

The summation will take a general format of each term, and increase a "dummy variable" by 1, summing together terms until the summation is complete.  To write the summation expression, find the common format of each term, and write it using a "dummy variable" to represent the number that is changing.

If the change of the number within each term increases by 1 each time, that makes the transition easier because the dummy variable increases by 1.  If the change of the number within each term does not change by 1 each time, using summation notation can be more challenging, as one needs to develop a formula to represent the amount that it is changing each time.

The common format of each term & the changing number

In our case, each term is some number squared, and the number that is squared changes from 1, to 2, to 3, ... and so on .. up to 750 (changing by 1 each time).  Since the number that is square changes by 1 each time, that will be our "dummy variable".

Using "n" for the "dummy variable", our terms look like [tex]n^2[/tex]

The start and end of the list

Lastly, to specify where the list starts and ends, we write where the list starts (lowest number) at the bottom with the "dummy variable", and where the list ends (largest number) at the top.

So, in the end, the summation would look like this:  [tex]\sum \limits^{750}_{n=1} {n^2}[/tex]

This represents adding a bunch of terms that look like [tex]n^2[/tex], starting at n=1, each time, increasing the value of n by 1, and stopping with the last term when n=750.

Extension of knowledge (NOT about THIS problem)

For a similar example where the changing number doesn't increase by 1 each time, look at a similar sequence of, where it's just the odd numbers squared up to 99:

1^2 + 3^2 + 5^2 + … + 99^2

While the terms do have the format of a number squared, since the number that is squared isn't just increasing by 1 each time, there would need to be a formula to describe how they are increasing.  One way to represent that would be:

[tex]\sum \limits^{50}_{n=1} {(2n-1)^2}[/tex]

Note that this does cause the final endpoint of the dummy variable to not necessarily be intuitive, but the "99 squared" is the 50th term (observe 2*50-1 is 99).

Answer:

I'd like to explain more but I have limited comment

chat me up

A

Answer:

Step-by-step explanation: -36

(a) The result of row 1 and row 2 interchange is [3     -2    3]

                                                                                 [2     3     1]

(b) The result of multiplying and adding the matrix is [-1       5     -2]

                                                                                         [3      -2      3]

The row operation of the matrix is calculated as follows;

The given matrix;

[2     3     1]

[3     -2    3]

To interchange row 1 and row 2, we determine it as follows;

R₁ ↔ R₂   = [3     -2    3]

                  [2     3     1]

The product of row 2 and -1 is calculated as;

-1 (R₂) = [-3      2     -3]

The addition of -1(R₂) + R₁ is calculated as;

-1(R₂) + R₁ = [-3      2     -3] + [2     3     1]

-1(R₂) + R₁ = [-1       5     -2]

new matrix =  [-1       5     -2]

                       [3      -2      3]

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The required total number of grams is 13.5 gram.

From figure we have,

The masses of 4 samples are the masses are 1.5 gram, 3.5 gram , 5 gram and 5 gram respectively.

To calculate the total number of grams of salt in the 3 samples with the most mass,

Add the masses of the three largest samples.

In this case,

The three largest samples are 3.5 grams, 5 grams, and 5 grams.

We have: 3.5 grams + 5 grams + 5 grams = 13.5 grams

Therefore, the total number of grams of salt in the 3 samples with the most mass is 13.5 grams.

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If the volume is 133 1/3 in³, the value of x is equal to 5 inches.

In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:

Volume of a rectangular prism = L × W × H

Where:

By substituting the given dimensions (parameters) into the formula for the volume of a rectangular prism, we have the following;

133 1/3 = 10 × x ×  2 2/3

400/3 = 10 × x × 8/3

400/3 = 80x/3

80x = 400

x = 400/80

x = 5 inches.

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Answer:

The number of discharge days for a patient admitted January 23 to February 15 is 31 - 23 + 15 = 23

Answer:

19.......

Step-by-step explanation:

correct

Answer:

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

Solve the equation for x:

The area of the composite figure is 81 yd². Option D

The composite figure is made up of a rectangle, a triangle and a trapezoid

The formula for the area of a rectangle is expressed as;

Area = length × width

Substitute the values

Area = 7(6)

Multiply the values

Area = 42 yd²

Area of the triangle = 1/2 base ×height

Area of the triangle = 1/2 × 6 ×5 = 30/2 = 15yd²

Area of the trapezoid = a+b/2 h

Substitute the values

Area = 7 + 9/2 (3) = 16/2(3) = 8(3) = 24yd²

Total area = 24 + 15 + 42 = 81 yd²

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The area of the composite figure is 81 cm²

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object.

Any figure that we can break and form more than one basic shape from is called a composite figure.

The figure consist of two trapezoid

Area of trapezoid 1 = 1/2( a+b) h

= 1/2 ( 12 + 7) 6

= 1/2 × 19 × 6

= 19 × 3

= 57cm²

area of trapezoid 2

= 1/2(a+b) h

= 1/2 ( 7+9) 3

= 1/2 × 16 × 3

= 8 × 3

= 24 cm²

Therefore the area of the composite figure

= 57 +24 = 81 cm²

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The reason that Rachel could give for her proof is given as follows:

<1 ≅ <5 by the Corresponding Angle Theorem.

When two parallel lines are cut by a transversal, corresponding angles are pairs of angles that are in the same position relative to the two parallel lines and the transversal.

Corresponding angles are always congruent, which means that they have the same measure.

A pair of corresponding angles in this problem is given as follows:

<1 and <5.

Hence they are congruent. <8 is opposite by the same vertex with <5, hence they are also congruent, and by the transitive property of congruence, we have that:

<8 ≅ <1, as <8 ≅ <5 and <5 ≅ <1.

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The probability that a salesperson hit the quota, given that the salesperson was promoted is 0.88. Therefore, option D is the correct answer

From the tree diagram, there are two ways to obtain the promotion:

0.41 of 0.68 (hit the quota).

0.14 of 0.32 (did not hit the quota).

So, the probability of getting a promotion is:

P(A) = 0.41 × 0.68 + 0.14 × 0.32 = 0.3236.

The probability of both getting a promotion and hitting the quota is:

P(A and B) = 0.41 × 0.68 = 0.2788.

So, the conditional probability is:

P(B|A) = P(A and B)/P(A) = 0.2788/0.3236 = 0.88 (approximately).

Therefore, option D is the correct answer.

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"Your question is incomplete, probably the complete question/missing part is:"

Salespeople at a certain company must meet a quota each month. On average, they hit the quota 68% of the time. If a salesperson hits the quota, the probability of being promoted in the next six months is 0.41. If a person doesn’t hit the quota, the probability of being promoted is 0.14.

Use the tree diagram to determine the probability that a salesperson hit the quota, given that the salesperson was promoted.

A. 0.28

B. 0.32

C. 0.41

D. 0.88

A transformation would accomplish this include the following: C. a rotation 180° counterclockwise about the origin.

In Mathematics and Geometry, the rotation of a point 180° about the origin in a clockwise or counterclockwise direction would produce a point that has these coordinates (-x, -y).

Furthermore, the mapping rule for the rotation of a geometric figure 180° counterclockwise about the origin is given by this mathematical expression:

(x, y)                                            →            (-x, -y)

Coordinates of point X (0, 4)  →  Coordinates of point Y = (0, -4)

Coordinates of point X (1, 3)  →  Coordinates of point Y = (-1, -3)

In conclusion, we can logically deduce that this transformation rule (x, y) → (-x, -y) would map Figure X onto Figure Y.

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The population of the town 12 years later will be 224,800 if the current population is 28,100 and the population doubles every 4 years.

Since the population doubles every 4 years, we can use the formula:

P = P0 x [tex]2^{t/4}[/tex]

where P0 is the initial population, t is the time elapsed in years, and P is the population after time t.

In this case, the initial population P0 is 28,100, and we want to know the population after 12 years, so t = 12. Substituting these values into the formula, we get:

P = 28,100  2^(12/4)

  = 28,100 x 2³

  = 28,100 * 8 = 224,800

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Prove in below steps:

Proved

The inverse of function f is [tex]f^{-1}(x)=\frac{4}{x}-3[/tex].

The domain of f is (-∞, -3) U (-3, ∞).

The range of f = (-∞, 0) U (0, ∞)

The domain of f⁻¹ = (-∞, 0) U (0, ∞).

The range of f⁻¹ = (-∞, -3) U (-3, ∞).

In this exercise, you are required to determine the inverse of the function f(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value or domain) and dependent value (y-value or range) as follows;

f(x) = y = 4/(3 + x)

x = 4/(3 + y)

3 + y = 4/x

f⁻¹(x) = 4/x - 3

[tex]f^{-1}(x)=\frac{4}{x}-3[/tex]

By critically observing the graph of this rational expression (function)   and its inverse shown in the image attached below, we can reasonably and logically deduce the following domain and range:

Domain of f = (-∞, -3) U (-3, ∞); {x|x ≠ -3}.

Range of f = (-∞, 0) U (0, ∞); {y|y ≠ 0}.

Domain of f⁻¹(x) = (-∞, 0) U (0, ∞); {y|y ≠ 0}.

Range of f⁻¹(x) = (-∞, -3) U (-3, ∞); {x|x ≠ -3}.

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The surface area of the pyramid is approximately 393,560 square feet.

To find the surface area of the pyramid, we need to find the area of each of the four triangular faces and the area of the square base.

The area of the square base is simply the area of a square with side length 500 feet:

Area of base =[tex](side length)^2 = (500 feet)^2[/tex]= 250,000 square feet.

To find the area of each triangular face, we need to know the length of one of the sides of the base and the slant height of the pyramid.

Since the base is a square, all sides have the same length of 500 feet.

Using the Pythagorean theorem, we can find the height of each triangular face:

height = √(slant heigh[tex]t^2[/tex] - base lengt[tex]h^2[/tex]/4)

height = √([tex]60^2 - (500/2)^2[/tex])

height = √(3600 - 62500/4)

height = √(3600 - 15625)

height = √20375

height ≈ 142.76 feet

Now we can find the area of one of the triangular faces:

Area of triangular face = (1/2) x (base length) x (height)

Area of triangular face = (1/2) x 500 feet x 142.76 feet

Area of triangular face ≈ 35,690 square feet

Since there are four triangular faces, the total surface area of the pyramid is:

Total surface area = 4 x (Area of triangular face) + (Area of base)

Total surface area = 4 x 35,690 square feet + 250,000 square feet

Total surface area ≈ 393,560 square feet

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Calculating the total number of deaths for all countries requires collecting data from various sources and summing up the individual death counts. Here is a step-by-step guide on how to approach this calculation:

Identify reliable sources: Look for reputable sources that provide updated and accurate information on global death counts.

Gather country-specific data: Access the data provided by these sources and collect the individual death counts for each country.

Validate and verify data: Cross-reference the data from multiple sources to ensure consistency and accuracy.

Sum up the death counts: Add up the individual death counts for each country to obtain the total number of deaths.

Regularly update the data: As the situation evolves, keep track of new data releases and updates.

Calculating the total number of deaths for all countries requires collecting data from various sources and summing up the individual death counts. Here is a step-by-step guide on how to approach this calculation:

Identify reliable sources: Look for reputable sources that provide updated and accurate information on global death counts. Examples include official government health agencies, international health organizations (e.g., WHO), and reputable databases (e.g., Worldometer, Johns Hopkins University).

Gather country-specific data: Access the data provided by these sources and collect the individual death counts for each country. Ensure the data is recent and covers a comprehensive list of countries.

Validate and verify data: Cross-reference the data from multiple sources to ensure consistency and accuracy. Different sources may have slight variations, so it's important to verify the data and use the most reliable figures.

Sum up the death counts: Add up the individual death counts for each country to obtain the total number of deaths. Use a spreadsheet or calculator to perform the calculations accurately.

Regularly update the data: As the situation evolves, keep track of new data releases and updates. Update the total number of deaths accordingly to maintain accuracy.
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The question probable may be:

How to calculate the total number of deaths for all the countries?

​

The two numbers are 15 and 15. The product of the two numbers is 225.

Let x and y be the two numbers we are searching for. We know that x + y = 30, so we can settle for y to get y = 30-x. The result of x and y is then, at that point:

[tex]P(x, y) = x(30 - x) = 30x - x^2[/tex]

To find the greatest item, we can take the subsidiary of P(x, y) concerning x and set it equivalent to 0:

dP(x, y)/dx = 30 - 2x = 0

Settling for x, we get x = 15. Subbing x = 15 into y = 30-x, we get y = 15. Hence, the two numbers that have an amount of 30 and produce the biggest conceivable item are 15 and 15.

The result of the two numbers is:

P(15, 15) = 15 * 15 = 225

So the result of the two numbers is 225.

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Answer:

0.34

Step-by-step explanation:

4 or below go down low

5 or above fly like a dove

so the 33 part is lower than 50 so round down

The solution of the equation, | x + 10 / 2x + 1 | = 20 is x = 10 / 39 or x = -30 / 41.

The absolute equation can be solved as follows:

| x + 10 / 2x + 1 | = 20

Therefore,

x + 10 / 2x + 1 = 20  or  x + 10 / 2x + 1 = - 20

cross multiply

x + 10 / 2x + 1 = 20

x + 10 = 20(2x + 1)

x + 10 = 40x + 20

x - 40x  = 20 - 10

-39x = -10

x = 10 / 39

x + 10 / 2x + 1 = - 20

cross multiply

x + 10 = -20(2x + 1)

x + 10 = -40x - 20

x + 40x = -20 - 10

41x = -30

divide both sides of the equation by 41

x = - 30 / 41

Therefore, x = 10 / 39 or x = -30 / 41

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The diameter of the second pile of phosphate is found to be approximately 62.6 feet.

The volume of a right circular cone is given by the formula,

V = (1/3)πr²h where volume is V, radius of the base is r, height is h, and π is approximately 3.14159. For the first pile of phosphate,

h = 15.3 feet

d = 48.6 feet

r = d/2 = 24.3 feet

Using the formula for the volume of a cone,

V = (1/3)πr²h

V = (1/3)π(24.3)²(15.3)

V = 24,249.8 cubic feet

For the second pile of phosphate,

h = 15.3 - 3.6 = 11.7 feet

r = ? (to be determined)

Using the same formula for the volume of a cone,

V = (1/3)πr²h

V = (1/3)πr²(11.7)

Since the volume of both piles is the same, we can set the two volume formulas equal to each other,

(1/3)π(24.3)²(15.3) = (1/3)πr²(11.7)

Simplifying,

(24.3)²(15.3) = r²(11.7)

r² = (24.3)²(15.3)/11.7

r = 31.3 feet (rounded to the nearest tenth)

Therefore, the diameter of the second pile of phosphate is d = 2r = 62.6 feet (rounded to the nearest tenth).

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Complete question - As phosphate is mined, it moves along a conveyor belt, falling off of the end of the belt into the shape of a right circular cone, as shown.

Height of the cone is 15.3 feet

Diameter of the base of cone is 48.6 feet

A shorter conveyor belt also has phosphate falling off of the end into the shape of a right circular cone. The height of the second pile of phosphate is 3.6 feet shorter than the height of the first. The volume of both piles is the same. To the nearest tenth of a foot, what is the diameter of the second pile of phosphate?

The two triangles are similar because angle C = C' and angle A = A'. The scale factor is 4/3

Similar triangles are triangles that have the same shape, but their sizes may vary. The corresponding angles of similar triangles are congruent.

Also, the ratio of corresponding sides of similar triangles are equal.

Therefore to show that that the two triangles are equal;

Tan A = 4/3

Tan A = 1.333

A = 53.1°

Tan A' = 5.3/4

Tan A' = 1.33

A' = 53.1°

Therefore the two triangles are similar.

The scale factor = 4/3

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Answer:

A≈201.06cm²

Step-by-step explanation:

A=4πr2

d=2r

Solving forA

A=πd2=π·82≈201.06193cm²

After considering the given data we conclude that the value of the sub question are
A) 16
B) 50
C) 5
D) 7
E) 7
F) 15
G) 2
H) 6
I) 15
J) -15
And the marking the true and false
a) true
b) false
c)true
d) true
e) false

The number properties rule is a set of rules that help you solve mathematical equations. The rule is also known as the order of operations. The acronym PEMDAS is used to remember the order of operations. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right) ¹².

Let's solve the equations using the number properties rule:

A) 5 + 16 = 16 + __

We can solve this equation by subtracting 16 from both sides of the equation.

5 + 16 - 16 = 16 + __ - 16

5 = 16

The equation is not true.

B) 450 + ___ = 50 + 450

We can solve this equation by subtracting 450 from both sides of the equation.

450 + ___ - 450 = 50 + 450 - 450

___ = 50

The answer is 50.

C) 2 + ( 3 + 5 ) = ( 2 + 3 ) + _____

We can solve this equation by simplifying the parentheses first.

2 + (3 + 5) = (2 + 3) + _____

2 + 8 = 5 + _____

10 = 5 + _____

We can solve this equation by subtracting 5 from both sides of the equation.

10 - 5 = 5 - 5

5 = _____

The answer is 5

D) (7 + 4) + 6 = ___+ (4+6)

We can solve this equation by simplifying the parentheses first.

(7 + 4) + 6 = ___+ (4+6)

11 + 6 = ___+ 10

17 = ___+ 10

We can solve this equation by subtracting 10 from both sides of the equation.

17 - 10 = ___+ 10 - 10

7 = ___

The answer is 7

E) 7 + 0 = ________

The answer is 7

F) 15 – 0 = _______

The answer is 15

G) (12 – 8 ) – 2 = _______

We can solve this equation by simplifying the parentheses first.

(12 – 8 ) – 2 = _______

4 - 2 = _______

2 = _______

The answer is 2

H) 12 – (8 – 2 ) = _________

We can solve this equation by simplifying the parentheses first.

12 – (8 – 2) = _________

12 - 6= _________

6= _________

The answer is 6.

I)20 – 5= _____

20 - 5= _____

15= _____

The answer is 15.

J)5 –20= _______

5 - 20= _______

-15= _______

The answer is -15.

Mark as true or false.

a)16+4=4+16 ______________

True

b)15–5=5–15 ______________

False

c)(3+4)+5=3+(4+5) ______________

True

d)(12–4)–2=12–(4–2) ______________

True

e)(10-5)+5=10–(5+5) ______________

False
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Answer:

Step-by-step explanation:

[tex]8x^5 + 3x^4 - 23x\\= 8x^5+3x^4+0x^3+0x^2-23x+0[/tex]