The product between f(x) and g(x) is equal to -6x² + 37x - 35.
The product of two functions, (fg)(x), is defined as f(x)*g(x). Therefore, to find the product of the given functions, we simply need to multiply them together:
(fg)(x) = (6x -7)(5 - x)
Using the distributive property, we can simplify this expression:
(fg)(x) = (6x)(5) - (6x)(x) - (7)(5) + 7(x)
(fg)(x) = 30x - 6x² - 35 + 7x
Combining like terms, we get:
(fg)(x) = -6x² + (30x + 7x) - 35
(fg)(x) = -6x² + 37x - 35
So the product of the two functions is (fg)(x) = -6x² + 37x - 35.
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What is the measurement of the diameter of the circle? (Round to the nearest tenth if needed)
Answer:
The answer is 13.
Step-by-step explanation:
First, we must find the hypotenuse for right angle:
[tex]c^{2} = a^{2} + b^{2}\\ \\\\c^{2} = 12^{2} + 5^{2} \\c^{2} 144 + 25 \\c^{2} = 169 \\c^{2} =\sqrt{169}\\c^{2} = 13[/tex]
sandra contributed $400, jaclyn $600 and alecia $1000. they agreed that the profit would be divided among them based on how each person give as capital.how much percentage of the capitol did jacklyn contribute
The total percentage of capital contributed by Jacklyn is 30%
The total capital contributed by Sandra, Jaclyn, and Alecia is:
$400 + $600 + $1000 = $2000
To find the percentage of capital contributed by Jacklyn contributed,
Percentage contributed by Jaclyn = (Jaclyn's contribution / Total capital) x 100
= ($600 / $2000) x 100
= 30%
Therefore, Jacklyn contributed 30% of the capital.
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Part A
By the end of its fourth week, a movie had grossed $9.2 million. Assume the revenue y in millions of dollars is
proportional to the week x.
Movie Sales
Revenue (millions of dollars)
987654321
y
X
0 1 2 3 4 5 6 7 8 9
Week
Graph the equation on your own paper. Which of the following ordered pairs does your graph pass through? Select
all that apply.
A) (4,9.2)
B) (9.2,4)
C) (1,9.2)
D) (9.2,1)
E) (0,0)
F) (1,2.3)
The ordered pair which the graph pass through include the following:
A) (4, 9.2)
E) (0, 0)
F) (1, 2.3)
How to determine the constant of proportionality?In Mathematics, a proportional relationship is a type of relationship that generates equivalent ratios and it can be modeled by the following mathematical expression:
y = kx
Where:
x represent the number of week.y represent the revenue (in millions of dollars).k represent the constant of proportionality.In order to have a proportional relationship and equivalent ratios, the variables x and y must have the same constant of proportionality:
Constant of proportionality (k) = y/x
Constant of proportionality (k) = 9.2/4
Constant of proportionality (k) = 2.3
Therefore, the required equation is given by:
y = kx
y = 2.3x
In conclusion, we would use an online graphing calculator to determine the ordered pairs that the line passes through.
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Why do we use standard units when measuring an object? (Select all that apply) A. Standard units are used because they are relatively permanent. B. Standard units enable communication over time. C. Standard units enable communication over distance. D. Standard units are exact.
We use standard units when measuring an object because :
A. Standard units are used because they are relatively permanent.
B. Standard units enable communication over time.
C. Standard units enable communication over distance.
D. Standard units are exact.
We use standard units when measuring an object because they are relatively permanent and enable communication over time, distance, and exactness.
Standard units are used because they are relatively permanent. This means that they do not change over time and are consistent. Standard units enable communication over time. Since they are consistent, they allow for accurate communication of measurements even if they are taken at different times.
Standard units enable communication over distance. They allow for accurate communication of measurements even if the people communicating are in different locations. Standard units are exact. This means that they are precise and accurate, allowing for consistent measurements.
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the polynomial in factored form as a produ f(x)=12x^(5)+5x^(4)-39x^(3)+9x^(2)+19x-6
The polynomial in factored form as a product can be found by using the factor theorem and synthetic division. The factor theorem states that if f(a) = 0, then (x-a) is a factor of f(x). We can use synthetic division to find the factors of f(x)=12x^(5)+5x^(4)-39x^(3)+9x^(2)+19x-6.
First, we need to find a value of x that makes f(x) = 0. We can use the rational root theorem to find possible rational roots. The possible rational roots are ±1, ±2, ±3, ±6. We can use synthetic division to test these possible roots.
Using synthetic division with x = 1, we get a remainder of -1, so x = 1 is not a root. Using synthetic division with x = -1, we get a remainder of 0, so x = -1 is a root and (x+1) is a factor of f(x).
Now we can divide f(x) by (x+1) using synthetic division to get the quotient 12x^(4)-7x^(3)-32x^(2)+41x-6. We can repeat the process of finding possible rational roots and using synthetic division to find the factors of this quotient.
The possible rational roots of the quotient are ±1, ±2, ±3, ±6. Using synthetic division with x = 1, we get a remainder of 8, so x = 1 is not a root. Using synthetic division with x = -1, we get a remainder of 0, so x = -1 is a root and (x+1) is a factor of the quotient.
We can divide the quotient by (x+1) using synthetic division to get the new quotient 12x^(3)-19x^(2)-13x+6. We can repeat the process of finding possible rational roots and using synthetic division to find the factors of this new quotient.
The possible rational roots of the new quotient are ±1, ±2, ±3, ±6. Using synthetic division with x = 1, we get a remainder of -14, so x = 1 is not a root. Using synthetic division with x = -1, we get a remainder of 0, so x = -1 is a root and (x+1) is a factor of the new quotient.
We can divide the new quotient by (x+1) using synthetic division to get the new quotient 12x^(2)-31x+6. We can repeat the process of finding possible rational roots and using synthetic division to find the factors of this new quotient.
The possible rational roots of the new quotient are ±1, ±2, ±3, ±6. Using synthetic division with x = 1, we get a remainder of -13, so x = 1 is not a root. Using synthetic division with x = -1, we get a remainder of 49, so x = -1 is not a root. Using synthetic division with x = 2, we get a remainder of 0, so x = 2 is a root and (x-2) is a factor of the new quotient.
We can divide the new quotient by (x-2) using synthetic division to get the new quotient 12x-3. This quotient cannot be factored further, so the factors of f(x) are (x+1)(x+1)(x+1)(x-2)(12x-3).
Therefore, the polynomial in factored form as a product is f(x) = (x+1)(x+1)(x+1)(x-2)(12x-3).
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5. LetA=[142231]. Find a basis forRow(A)⊥using the dot product.
The vector x = [2, -3, 1] is a basis for Row(A)⊥. This means that any vector in Row(A)⊥ can be written as a multiple of x.
To find a basis for Row(A)⊥ using the dot product, we need to find a vector that is orthogonal to all the rows of A. This means that the dot product of the vector and each row of A should be equal to 0.
Let's say the vector we are looking for is x = [x1, x2, x3]. Then we need to solve the following system of equations:
x1 * 1 + x2 * 4 + x3 * 2 = 0
x1 * 2 + x2 * 2 + x3 * 1 = 0
x1 * 3 + x2 * 1 + x3 * 1 = 0
We can write this system of equations in matrix form as:
[1 4 2] [x1] = [0]
[2 2 1] [x2] = [0]
[3 1 1] [x3] = [0]
We can use Gaussian elimination to solve this system of equations. After performing the necessary row operations, we get:
[1 0 -2] [x1] = [0]
[0 1 3] [x2] = [0]
[0 0 0] [x3] = [0]
From the last equation, we can see that x3 can be any value. Let's choose x3 = 1. Then, from the second equation, we get x2 = -3, and from the first equation, we get x1 = 2.
So, the vector x = [2, -3, 1] is a basis for Row(A)⊥. This means that any vector in Row(A)⊥ can be written as a multiple of x.
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jenna borrows 8,000 for college at a yearly simple interest rate of 6 she takes 15 years
The amount of interest earned on Jenna's loan of $8000 for 15 years is $7200
How to determine the amount of interestThe given variables and their values from the question are listed as follows:
Principal = 8000
Rate = 6%
Time = 15 years
The general formula to calculate simple interest is:
Simple interest = Principal * Rate * Time
By replacing the given values into the equation mentioned above, we obtain the subsequent expression
Simple interest = 8000 * 6% * 15
Evaluate the products in the expression
Simple interest = 7200
Hence, the amount of interest is $7200
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Complete question
Jenna borrows 8,000 for college at a yearly simple interest rate of 6 she takes 15 years
Determine the amount of interest
PLEASE HELP
Tanner is spray painting an arrow on the side of a building to point to the entrance of his store. The can of gold spray paint he wants to use covers up to 12 square feet. Does Tanner have enough spray paint for his arrow?
Using the area formula, it is obtained that Tanner has enough spray paint to cover the arrow with one can of spray paint.
What is area?
An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.
To determine if Tanner has enough spray paint for his arrow, we need to find the total area of the arrow and compare it to the coverage of one can of spray paint.
The arrow consists of a rectangle and a triangle.
The rectangle has a length of 2 feet and a width of 5 1/3 feet, so its area is -
Area of rectangle = length × width
= 2 ft × 5 1/3 ft
= 10 2/3 sq. ft.
The triangle has a base of 3 feet and a height of the difference between the width of the rectangle (5 1/3 feet) and the width of the arrow (6 feet):
Height of triangle = 6 ft - 5 1/3 ft = 2/3 ft
Area of triangle = 1/2 x base x height = 1/2 x 3 ft x 2/3 ft = 1 sq. ft.
The total area of the arrow is the sum of the area of the rectangle and the area of the triangle -
Total area of arrow = Area of rectangle + Area of triangle
Total area of arrow = 10 2/3 sq. ft. + 1 sq. ft.
Total area of arrow = 32/3 sq. ft. + 1 sq. ft.
Total area of arrow = 11 2/3 sq. ft.
Therefore, since one can of spray paint covers up to 12 square feet, Tanner has enough spray paint to cover the arrow with one can of spray paint.
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Whe the preperties of esponents to simplify the evprestion, Write your answer nith pestive espenents enly. ((4x^(2)z^(-4))/(20y^(-3)))^(-4)
The simplified expression is [tex](z^{(16))}(y^{(12))}/(100663296)(x^{(8))}[/tex].
To simplify the expression ((4x^(2)z^(-4))/(20y^(-3)))^(-4) using the properties of exponents, we need to follow the steps below:
1. First, we need to distribute the exponent of -4 to each term inside the parentheses. This will give us:
(4^(-4))(x^(2*-4))(z^(-4*-4))/(20^(-4))(y^(-3*-4))
2. Next, we need to simplify the exponents by multiplying them. This will give us:
[tex](4^(-4))(x^(-8))(z^(16))/(20^(-4))(y^(12))[/tex]
3. Now, we need to simplify the terms with negative exponents by moving them to the opposite side of the fraction. This will give us:
[tex](z^(16))(y^(12))/(4^(4))(x^(8))(20^(4))[/tex]
4. Finally, we need to simplify the terms with the same base by adding their exponents. This will give us:
[tex](z^(16))(y^(12))/(4^(4))(x^(8))(2^(8))(5^(8))[/tex]
5. We can further simplify the expression by simplifying the terms with the same base. This will give us:
(z^(16))(y^(12))/(16^(2))(x^(8))(2^(8))(5^(8))
6. Now, we can combine the terms with the same base to get our final answer:
(z^(16))(y^(12))/(256)(x^(8))(256)(390625)
7. Our final answer is:
(z^(16))(y^(12))/(100663296)(x^(8))
Therefore, the simplified expression is (z^(16))(y^(12))/(100663296)(x^(8)).
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Write an equation for a line parallel to y = 2 x + 1 and passing
through the point (1,5)
To find the equation for a line parallel to y = 2x + 1 and passing through the point (1,5), we need to remember that parallel lines have the same slope.
Since the slope of y = 2x + 1 is 2, the slope of the parallel line will also be 2.
Using the point-slope form of a linear equation, we can plug in the given slope and point to find the equation of the parallel line:
y - y1 = m(x - x1)
y - 5 = 2(x - 1)
Simplifying this equation gives us the equation for the parallel line:
y = 2x + 3
So, the equation for the line parallel to y = 2x + 1 and passing through the point (1,5) is y = 2x + 3.
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kierna is starting a lawn-mowing buisness in her neighborhood. she creats a graph to help her determine what to charge customers per lawn to maximize her profits. she uses c to represent the number of lawns she mows and y to represent her profit in dollars.
The profit is maximum when 40 lawns are mowed.
What is function?A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.
Given is that Kieran is starting a lawn-mowing buisness in her neighborhood. She creates a graph to help her determine what to charge customers per lawn to maximize her profits. She uses {c} to represent the number of lawns she mows and {y} to represent her profit in dollars.
The profit is maximum when 40 lawns are mowed as at this point the the peak of the parabola occurs.
Therefore, the profit is maximum when 40 lawns are mowed.
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What is the volume of a rectangular prism with a length of 4.7 feet, a width of 1.5 feet, and a height of 1.6 feet?
Responses
7.1 ft³
7.1 ft³
7.8 ft³
7.8 ft³
8.65 ft³
8.65 ft³
11.28 ft³
The volume of the rectangular prism is 11.28 feet³.
What is a rectangular prism?In terms of geometry, a rectangular prism is a polyhedron having two parallel, congruent bases. It also goes by the name cuboid. A rectangular prism is made up of six rectangles, each with twelve edges.
We are given that a rectangular prism has the following dimensions:
Length (l) = 4.7 feet
Width (w) = 1.5 feet
Height (h) = 1.6 feet
We know that
Volume (V) = [tex]l \times w \times h[/tex]
From this, we get
⇒V = [tex]4.7 \times 1.5 \times 1.6[/tex]
⇒V = 11.28 feet³
Hence, the volume of the rectangular prism is 11.28 feet³.
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Answer: Your answer is 11.28 ft³
Step-by-step explanation: 4.7 x 1.5 x 1.6 = 11.28 you have to do it in this order also I did the k12 quiz here's proof.
Hope it helped :D
You rolla fair 6-sided die what is p? ( rollgreater than 4)?
This is on kahn acedmy
The probability of rolling the dies and getting greater than 4 will be [tex]\dfrac{2}{6}[/tex].
Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Given that the 6-sided die is rolled, the probability of getting a number greater than 4 will be calculated as,
Sample space = {1, 2, 3, 4, 5, 6} = 6
Favourable outcomes = {5,6} = 2
[tex]\rm Probability = \dfrac{Favourable \ events}{Sample\ space}[/tex]
[tex]\rm Probability = \dfrac{2}{6}[/tex]
Therefore, the probability will be [tex]\dfrac{2}{6}[/tex].
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Classify each angle pair, then find the value of X
Please explain your work!!
Answer: The angles are classified as interior angles, where x=9
Step-by-step explanation:
The angle measure of a straight line is 180°, meaning that 7x-10 and 12x+19 must add up to 180.
It is clear from the visual that the former is an acute angle, and the latter is an obtuse angle. This means that when solving each, 7x-10 must be less than 90, and 12x+19 must be greater than 90, but less than 180.
If you set up the equation to solve for x; (7x-10) + (12x+19) = 180, you must combine like terms.
19x+9=180 is the new equation. Since x needs to be isolated, subtract 9 from each side, to get 19x=171. Dividing each side by 19 results in x=9.
Please help me with step by step explanation, thank you
d
2/3 is the same as 4/6 and 25/100 is the same as 1/4. -1.3 is transferred to the beginning of the equation
Irons is trying to lay out the bases for a game of kickball such that the infield is a square as shown. She would like bases to be 25 feet apart. She first placed home base and then places first and third base 25 feet from home base
How far to the nearest tenth of a foot should first base and third base be from each other justify.
The answer of the given question based on the Irons is trying to lay out the bases for a game of kickball such that the infield is a square the answer is first base and third base should be about 35.4 feet apart.
What is Pythagorean theorem?The Pythagorean theorem is fundamental concept in geometry that relates to three sides of right triangle. It states that in right triangle, the square of length of hypotenuse (the side opposite the right angle) is equal to sum of the squares of lengths of other two sides.
We can use the Pythagorean theorem to find this distance.
Let's call the distance between home base and first base "a". We know that the distance between home base and third base is also "a", because the infield is a square. We also know that the distance between home base and second base (which is the hypotenuse of the right triangle formed by home base, first base, and second base) is 25 feet.
Using the Pythagorean theorem, we can solve for "a":
a² + 25² = a² + a²
2a² = 25²
a² = (25²)/2
a = sqrt((25²)/2) ≈ 17.7 feet
So the distance between first base and third base should be approximately 35.4 feet (2a). Rounding to the nearest tenth of a foot, this is 35.4 feet. Therefore, first base and third base should be about 35.4 feet apart.
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iven the polynomial 4x^(2)y^(4) - 9x^(2)y^(6), rewrite as a product of polynomials
The polynomial 4x^(2)y^(4) - 9x^(2)y^(6) can be rewritten as a product of polynomials as:
x^(2)y^(4)(2y + 3)(2y - 3)
The polynomial 4x^(2)y^(4) - 9x^(2)y^(6) can be rewritten as a product of polynomials by factoring out the common factor of x^(2)y^(4). This leaves us with:
x^(2)y^(4)(4 - 9y^(2))
Now, we can factor the polynomial inside the parentheses as a difference of squares:
x^(2)y^(4)(2y + 3)(2y - 3)
Therefore, the polynomial 4x^(2)y^(4) - 9x^(2)y^(6) can be rewritten as a product of polynomials as:
x^(2)y^(4)(2y + 3)(2y - 3)
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Ian placed 263 stamps on 9 pages of his collection book. He wanted to place the same number
of stamps on each page with as few as possible left over. How many stamps are on each page?
How many were left to put in an envelope?
Enter the correct value to complete each sentence.
Blank stamps were placed on each page.
Blank stamps were put in an envelope.
Answer: To find the number of stamps on each page, we can divide the total number of stamps by the number of pages:
263 stamps ÷ 9 pages = 29.22 stamps per page (rounded to two decimal places)
Since we want to place the same number of stamps on each page with as few left over as possible, we can round down to the nearest whole number:
29 stamps per page
To find out how many stamps were left to put in an envelope, we can multiply the number of pages by the number of stamps per page, and then subtract from the total number of stamps:
263 stamps - (29 stamps per page x 9 pages) = 8 stamps left over to put in an envelope
Therefore, the completed sentences are:
29 stamps were placed on each page.
8 stamps were put in an envelope.
Step-by-step explanation:
A large container of breath mints has a mass of 50 g. A small container has a mass of 32 g. What is the percent decrease from the mass of the large container to the mass of the small container?
Answer:
The percent decrease is 36%.
Step-by-step explanation:
The rate of change of f(x) = 2(2) is __ the rate of change of the function in the graph:
A. equal to
B. less than
C. greater than
The rate of change of f(x) = 2(2) is equal to the rate of change of the function in the graph.
What is a function?A function is a relation between two sets of values in mathematics. It is a mathematical process that takes an input and produces an output. It is represented as an equation which describes the relationship between the input and the output. A function can also be used to represent a mathematical rule or procedure that takes a set of inputs and produces a set of outputs.
The rate of change, or slope, is the measure of how quickly one variable changes when the other changes. In the graph, the rate of change is a constant, meaning that for every one unit the x-value increases the y-value increases by two. This is the same as the rate of change of f(x) = 2(2), as for every two units the x-value increases the y-value increases by four. Both the graph and the equation of the function have a constant rate of change which is equal to two, so the rate of change of f(x) = 2(2) is equal to the rate of change of the function in the graph.
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Does $10,000 invested at 6% interest double its value in half the time as $10,000 invested at 3% interest? Show your work.
The answer is $21,989.34 and Yes, $10,000 invested at 6% interest will double its value in half the time as $10,000 invested at 3% interest.
Now, For the 6% investment:
$10,000 invested at 6% interest will double in 12 years:
$10,000 × (1.06)^12 = $21,989.34
For the 3% investment:
$10,000 invested at 3% interest will double in 24 years:
$10,000 × (1.03)^24 = $21,989.34
Therefore, it will take half the time (12 years) for the 6% investment to double its value compared to the 3% investment (24 years).
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The cell phone plan from Company C costs $10 per month, plus $15 per gigabyte for data used. The plan from Company D costs $80 per month, with unlimited data. Rule C gives the monthly cost, in dollars, of using g gigabytes of data on Company C's plan. Rule D gives the monthly cost, in dollars, of using g gigabytes of data on Company D's plan.
Which is less, C(4) or D(4)? What does this mean for the two phone plans?
Answer:
C(4) is less
Step-by-step explanation:
Let C(x) represent Rule C and let D(x) represent Rule D.
C(x) = 10x + 15
D(x) = 80x
C(4) = 10(4) + 15 = 40 + 15 = 55
D(4) = 80(4) = 320
Thus, C(4) is less.
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Use the defined variables for the verbal model to write an equation in slope-intercept form that relates the variables. $2.00 pound Pounds of peaches + An equation is y = $1.50 pound Pounds of apples Let a represent the number of pound of peaches. Let y represent the number of pounds of apples. - $15
plss help
Answer:
15=2.00x+1.50y
Step-by-step explanation:
Please help!
Simplify arctan1/3 + arctan2/3
(round to the nearest degree).
a. 38°
b. 72°
c. 52°
The simplified expression is 45°, which is closest to option a. 38°.
What is lines and angles?Lines are one-dimensional geometrical objects that extend infinitely in both directions. Angles are formed when two lines intersect, and they measure the amount of rotation between the lines.
We can use the formula:
arctan(x) + arctan(y) = arctan[(x + y) / (1 - xy)]
to simplify the expression.
Setting x = 1/3 and y = 2/3, we have:
arctan(1/3) + arctan(2/3) = arctan[(1/3) + (2/3) / (1 - (1/3)(2/3))]
= arctan(1)
= 45°
Therefore, the simplified expression is 45°, which is closest to option
a. 38°.
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8 is 14% of what number? Round your answer to the nearest hundredth if necessary.
Answer:
I think the answer is 57.14
Answer:
57.14
Step-by-step explanation:
57.14 is the result of rounding 57.14 to the nearest 0.01
3/4d-11=1/4d-6 1/2 please help!!!
The solution for d in the one-variable equation as required to be determined is; d = 9.
What is the value of d in the given equation?As evident in the task content; the given equation is;
3/4d - 11 = 1/4d - 6½
¾d - 11 = ¼d - 13/2
Therefore, we have that;
¾d - ¼d = 11 - 13/2
½d = 9/2
d = 9
Ultimately, the value of d which holds true for the given equation is; 9.
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A 12-sided solid has equal-sized faces numbered 1 to 12.
a. Find P(number greater than 8).
b. Find P(number less than 6).
c. Is the solid fair? Explain.
a. P(number greater than 8) = %
(Type an integer or decimal rounded to the nearest tenth as needed.)
h
a)P(number greater than 8) = 4/12 = 1/3 ≈ 0.3
b)P(number less than 6) = 5/12 ≈ 0.4
c)If each face has an equal chance of rolling, the solid is fair. The solid is fair since the faces are numbered sequentially from 1 to 12 and each face has an equal chance of being rolled.
what is decimal?One of the number types in algebra that has a whole integer and a fractional portion separated by a decimal point is a decimal. The decimal point is the dot that appears between the parts of a whole number and a fraction. An example of a decimal number is 34.5.
from the question:
a) A solid has 12 equal-sized faces with numbers ranging from 1 to 12. The chance of getting a number larger than 8 is calculated by dividing the total number of faces by the number of faces with numbers greater than 8. Given that there are 4 faces (12 - 8) with numbers greater than 8, the likelihood of drawing one is:
P(number more than 8) = 4/12 = 1/3 = 0.35
b) Similarly, the chance of receiving a number less than 6 is calculated by dividing the total number of faces by the number of faces that have numbers less than 6. Given that there are 6 - 1 = 5 faces with numbers lower than 6, the likelihood of drawing one is as follows:
P(less than six) = 5/12= 0.4
c) If each face has an equal chance of rolling, the solid is fair. The solid is fair since the faces are numbered sequentially from 1 to 12 and each face has an equal chance of being rolled.
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Nork Facior out the GCF from the polynomial a^(5)b^(7)-a^(3)b^(2)+a^(2)b^(6)-a^(2)b^(2)
The GCF of the polynomial a^(5)b^(7)-a^(3)b^(2)+a^(2)b^(6)-a^(2)b^(2) is a^(2)b^(2), and the factored form of the polynomial is a^(2)b^(2)(a^(3)b^(5)-a+b^(4)-1).
The GCF, or greatest common factor, is the largest factor that all terms in a polynomial have in common. In this case, we need to find the GCF of the polynomial a^(5)b^(7)-a^(3)b^(2)+a^(2)b^(6)-a^(2)b^(2).
First, we need to look at the exponents of each term to determine the GCF. The smallest exponent for a is 2, and the smallest exponent for b is 2. Therefore, the GCF for this polynomial is a^(2)b^(2).
Next, we need to factor out the GCF from each term in the polynomial. This is done by dividing each term by the GCF and then multiplying the GCF by the resulting polynomial.
So, the factored form of the polynomial is:
a^(2)b^(2)(a^(3)b^(5)-a+b^(4)-1)
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For any invertible matrix BEM nxn(K), define a function TB: Mnxn(K) → M nxn(K) by
TB (A) = BAB-¹,
where A EM nxn(K). Prove that TB is an isomorphism.
TB is both one-to-one and onto, it is an isomorphism.
To prove that TB is an isomorphism, we need to show that it is both one-to-one and onto.
One-to-one: Assume TB(A) = TB(C) for some A, C EM nxn(K). Then, BAB-¹ = CBC-¹. Multiplying both sides by B-¹ on the left and B on the right gives B-¹BAB = B-¹CBC. Since B is invertible, B-¹B = I and we have A = C. Therefore, TB is one-to-one.
Onto: Let D EM nxn(K). Then, we can define A = B-¹DB. Since B is invertible, A EM nxn(K) and TB(A) = BAB-¹ = B(B-¹DB)B-¹ = D. Therefore, TB is onto.
Since TB is both one-to-one and onto, it is an isomorphism.
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sin
^-1 (sin A) ≠ A
A. implies that A is not in the domain
b. requires that A = 0
c. is not possible because arcsin reverses sin
d. happens when A is not in [-pi/2, pi/2]
The correct answer is d. happens when A is not in [-pi/2, pi/2]. The inverse sine function, sin^-1, or arcsin, is the function that reverses the sine function.
It is defined for values in the range [-1, 1] and has a range of [-pi/2, pi/2]. This means that if A is not in the range [-pi/2, pi/2], then sin^-1 (sin A) will not equal A.
For example, if A = pi, then sin A = 0, but sin^-1 (0) = 0, not pi. This is because pi is not in the range [-pi/2, pi/2], so the inverse sine function cannot return it as an answer.
Therefore, The correct answer is d. happens when A is not in [-pi/2, pi/2].
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