Step-by-step explanation:
There are 720 ways to arrange the letters in the word GRIFFIN.
To see why, we can use the formula for permutations of a set with n elements, which is n! (n factorial).
The word GRIFFIN has 7 letters, so we have:
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
However, since the letters in GRIFFIN are not all unique (there are 2 Fs and 2 Is), we need to divide by the number of ways the repeated letters can be arranged. This is the product of their factorials:
2! x 2! = 4
So the final answer is:
7! / (2! x 2!) = 5040 / 4 = 720
Examples of different arrangements of the letters in GRIFFIN are:
- FIGRINF
- IRGIFNF
- NGRIFFI
- IFRIGNF
- GFRINIF
- NRIGIFF
- FFIRIGN
And so on, for a total of 720 possible arrangements.
Does this answer your question.
The pages of a book are numbered 1, 2, 3, ... . In total, it takes 489 digits to number
all the pages of the book.
What is the number on the last numbered page?
Answer:989
Step-by-step explanation: 1,2,3 989
What square number divided by another square number = 9
The two numbers will be 9 and 3 and when squared and divided will give 9.
In mathematics, a number system is a set of numbers and the rules that govern how to manipulate them. The most commonly used number system is the decimal system, which is based on the use of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
On squaring the number 9 the result is,
9² = 81
On squaring the number 3 the result is,
3² = 9
The division of the two numbers is,
E = 81 / 9
E = 9
Therefore, the two numbers will be 9 and 3 and when squared and divided will give 9.
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f(x)=-4x+4
g(x)=2x^2-3x
Answer:
for x inter let y=0
0=4x+4
4x=-4
x=-1
suppose the actual number of sandwiches sold today was 41 describe the number of sandwiches sold as a percent increase or decrease
The number of sandwiches sold as a percent decrease can be described as 8.89% decrease.
Given that,
Number of sandwiches sold yesterday = 45
Number of sandwiches sold today = 41
It is clear that the number of sandwiches sold has been decreased.
So there will be a percent decrease.
Percent decrease = (original value - New value) / original value
= (45 - 41) / 45
= 0.0889
= 8.89%
Hence the percent decrease is 8.89%.
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The complete question is given below.
A local Deli sold 45 sandwiches yesterday.
Suppose the actual number of sandwiches sold today was 41. Describe the number of sandwiches sold as a percent increase or decrease, rounded to the nearest hundredth.
Mission Foods produces two flavors of tacos—chicken and fish—with the following characteristics.
Chicken Fish
Selling price per taco $ 3.90 $ 5.30
Variable cost per taco 1.95 2.65
Expected sales (tacos) 201,000 303,000
The total fixed costs for the company are $107,000.
Required:
a. What is the anticipated level of profits for the expected sales volumes?
b. Assuming that the product mix would be 42 percent chicken and 58 percent fish at the break-even point, compute the break-even volume using weighted-average contribution margin.
c. If the product sales mix were to change to four chicken tacos for each fish taco, what would be the new break-even volume?
a. To calculate the anticipated level of profits, we need to calculate the total revenue and total variable costs for each product, and then subtract the total variable costs and fixed costs from the total revenue.
Chicken:
Total revenue = $3.90 x 201,000 = $783,900
Total variable costs = $1.95 x 201,000 = $392,295
Contribution margin = $391,605
Fish:
Total revenue = $5.30 x 303,000 = $1,607,900
Total variable costs = $2.65 x 303,000 = $803,295
Contribution margin = $804,605
Total contribution margin = $391,605 + $804,605 = $1,196,210
Total fixed costs = $107,000
Anticipated profits = Total contribution margin - Total fixed costs = $1,196,210 - $107,000 = $1,089,210
b. Weighted-average contribution margin = (0.42 x $1.95) + (0.58 x $2.65) = $2.36
Break-even volume = Total fixed costs ÷ Weighted-average contribution margin = $107,000 ÷ $2.36 = 45,339 tacos
c. If the product sales mix changes to four chicken tacos for each fish taco, then the new product mix is 80% chicken and 20% fish.
Weighted-average contribution margin = (0.80 x $1.95) + (0.20 x $2.65) = $2.06
Break-even volume = Total fixed costs ÷ Weighted-average contribution margin = $107,000 ÷ $2.06 = 51,942 tacos
Answer Quick! A circular swimming pool has a diameter of the 26 feet. What is the area of the swimming pool? Use 3.14 to approximate for π. Enter your answer, as a decimal rounded to the nearest tenth, in the box.
Answer:
Area = π(13^2) = 169π = 530.9 ft^2
Since 3.14 is used for π,
Area = 3.14(13^2) = 530.7 ft^2
NEED HELP WILL GIVE BRAINLIEST AND 5 STARS I HAVE 20 MINUTES PLEASE HELP!
For this rational function, using proper limit notation, explain what happens to m(x) near x=-3
Answer: as x approaches -3, the rational function m(x) approaches infinity from one side and negative infinity from the other side, and it has a vertical asymptote at x = -3.
Step-by-step explanation: The mathematical expression denoted by m(x) is represented as 4(x-5)(x+8)/(x^2-2x-15) in this context.
When x approaches -3, the limit of the function m(x) is evaluated from both the left and right sides.
In accordance with established convention, it is feasible to express the aforementioned concept as follows:
As x tends towards negative three from the left, the value of m(x) can be expressed as the limit of the function [4(x-5)(x+8)/(x^2-2x-15)] evaluated at x approaching negative three from the left, symbolically represented as lim x → -3⁻.
In the context of contemporary society, the issues of environmental sustainability and conservation are of utmost importance. These concerns have arisen from the impact of human activities on the planet and the need to prevent further harm. Various strategies have been proposed and implemented to address this challenge, and stakeholders from different sectors have been involved in the efforts. Despite this, progress has been slow, and much more needs to be done to achieve the overarching goal of achieving a sustainable future for the earth and its inhabitants. Additionally, the challenges we face in this area are highly complex and multifaceted and require interdisciplinary approaches that can engage different perspectives and knowledge domains. The academic community, as a key player in this domain, has a crucial role to play in advancing knowledge, developing innovative solutions, and disseminating best practices to relevant stakeholders for maximum impact.
The limit as x approaches negative three from the right-hand side of the function m(x) is equivalent to the limit as x approaches negative three from the right-hand side of the function [4(x-5)(x+8)/(x^2-2x-15)].
The determination of the behavior of the function m(x) as it approaches -3 from the left-hand side (values of x less than -3) is denoted by the left-hand limit, lim x → -3⁻ m(x), while the right-hand limit, lim x → -3⁺ m(x), signifies the behavior of m(x) as it approaches -3 from the right-hand side (values of x greater than -3).
In order to assess these limits, the substitution of x = -3 in the m(x) expression may be pursued, however, such an endeavor would culminate in division by zero, a mathematically undefined operation.
The denominator of the expression for m(x) can be factored as stated below.
The given expression x^2 - 2x - 15 can be factored into the product of two binomials (x - 5) and (x + 3).
Consequently, it is possible to rephrase the equation for m(x) in the following manner:
The function m(x) can be expressed as 4(x - 5)(x + 8) / (x - 5)(x + 3) in the academic way of writing.
The expression may be simplified by factoring out the common factor of (x - 5) in both the numerator and denominator, resulting in:
The function m(x) can be expressed as 4(x + 8) / (x + 3), where x is the independent variable.
The limits as x approaches -3 from both the left and right sides can now be assessed.
In the context of mathematical analysis, it can be observed that the left-hand limit of the function m(x) as x approaches -3, denoted as lim x → -3⁻ m(x), can be established as the limit of the function [4(x + 8) / (x + 3)] as x approaches -3 from the left-hand side, denoted as lim x → -3⁻ [4(x + 8) / (x + 3)]. Upon applying mathematical calculations, it can be determined that this limit evaluates to -16/0, which in the realm of mathematics is regarded as undefined.
The mathematical statement "lim x → -3⁺ m(x) = lim x → -3⁺ [4(x + 8) / (x + 3)] = 20/6 = 10/3" may be expressed in a more formal and scholarly manner. Specifically, one may depict the aforementioned equation using mathematical symbols and terminology deemed appropriate within scholarly discourse. As such, one may denote that the limit, as x approaches negative three from the positive direction, of the function m(x) is equivalent to the limit, as x approaches negative three from the positive direction, of the quotient formed by four multiplied by the sum of x and eight, divided by the sum of x and three. Upon evaluating these limits, one arrives at the quotient 20/6, which reduces to the rational number 10/3.
Due to the inequality between the left-hand and right-hand limits, it is deduced that the limit of the function m(x) is not present at x = -3. Consequently, it can be stated that the function m(x) possesses a vertical asymptote at x = -3.
Which equation is satisfied by c = 7?
A 9-4+2(10) < c +8
B
18-4xc=6
C
22÷2-c<5
D
3c = 24
Equation (A) 9-4+2(10) < 8c sAtisfies when c = 7 respectively.
What are equations?Equation: Statement of equality between two expressions consisting of variables and/or numbers. equation: Formula used to express the equality of two expressions by joining them with the equals sign =.
In essence, equations are questions and attempts to systematically identify the solutions to these questions have been the driving forces behind the creation of mathematics.
So, when c = 7:
9-4+2(10) < 8c
Now, solve as follows: Use BODMAS
9-4+2(10) < 8c
9-4+2(10) < 8(7)
9-4+20 < 56
9+16 < 56
25 < 56
TRUE
(Since we got the equation that satisfies when c = 7, we don't need to solve further.)
Therefore, equation (A) 9-4+2(10) < 8c sAtisfies when c = 7 respectively.
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Correct question:
Which equation is satisfied by c = 7?
A 9-4+2(10) < 8c
B 18-4xc=6
C 22÷2-c<5
D 3c = 24
A company makes shirts, pants, belts, and socks. It makes 24,000 of each item. For each sample, predict the number of items likely to be defective.
6 defects in 500 shirts
Answer:
defective shirts = 288
Step-by-step explanation:
If there are 6 defects in a sample of 500 shirts, we can use this information to predict the number of defective shirts out of the total number of shirts produced.
First, we can calculate the defect rate for the sample:
defect rate = (defective items / total items) x 100%
defect rate = (6 / 500) x 100%
defect rate = 1.2%
Next, we can use this defect rate to estimate the number of defective shirts out of the total 24,000 shirts produced:
defective shirts = defect rate x total items produced
defective shirts = 1.2% x 24,000
defective shirts = 288
Therefore, we can predict that out of the 24,000 shirts produced, approximately 288 of them are likely to be defective based on the sample of 500 shirts with 6 defects.
Find the angles of △ABC given that a=11, b=20, and c=18.
Round each angle to the nearest tenth of a degree and use that rounded value to find the remaining angles.
The required angles are approximate:
A = 36.1 degrees
B = 55.9 degrees
C = 101.3 degrees
Using the law of cosines,
[tex]c^{2} = a^{2}+b^{2} -2ab*cos(C)[/tex]
Inserting the given values, we get,
[tex]18^{2} =11^{2} +20^{2}-(2*11*cos(C))\\\\ cos(C)= -17/440\\ C = 101.3 degrees[/tex]
Using the law of sines to find other angles
[tex]sin(A)/a = sin(C)/c\\\\sin(A)/11 = sin(101.3)/18\\\\A= 36.1 degrees[/tex]
[tex]sin(B)/b = sin(C)/c\\\\sin(B)/20 = sin(101.3)/18\\\\B= 55.9 degrees[/tex]
Therefore, the required angles are approximate:
A = 36.1 degrees
B = 55.9 degrees
C = 101.3 degrees
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Which graph represents the given function?
g(x) = -3(2)x - 1
Answer: Since you haven't given any photos. I graphed it on demos.
Step-by-step explanation:
If you would like to draw a graph, simply plug in random x-values. THis will give you y-values, which will allow you to plot coordinate points. Plot a few, at least 3-4, and draw a line though them.
why do you think he needed name's or word to tell" how many?
It is important to use word or names to tell how many in order to indicate quantity of the object in question.
How do we quantify in mathematics?There are two ways to quantify a propositional function:
universal quantification and existential quantification: existential quantification are written in the form of “∀xp(x)” and “∃xp(x)” respectively. To negate a quantified statement, change ∀ to ∃, and ∃ to ∀, and then negate the statement.In conclusion, quantification in logic is described as the attachment of signs of quantity to the predicate or subject of a proposition.
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Carmen the trainer has two solo workout plans: Plan A and Plan B. Each client does either one or the other (not both). On Friday there were 12 clients who did plan A and 2 who did Plan B. On Saturday there were 3 clients who did Plan A and 5 who did Plan B. Carmen trained her Friday clients for a total of 9 hours and her Saturday clients a total of 9 hours. How long does each of the workout plans last?
Plan A lasts for 30 minutes and Plan B lasts for 90 minutes.
What is equation?An equation is a mathematical statement that asserts that two expressions are equal to each other. It typically consists of two sides separated by an equals sign (=). The expressions on either side of the equals sign can contain variables, constants, mathematical operations, and functions.
Let's call the duration of Plan A "x" and the duration of Plan B "y". We want to find the values of x and y.
From the given information, we can set up two equations:
[tex]12x + 2y = 9[/tex] (equation 1, for Friday)
[tex]3x + 5y = 9[/tex](equation 2, for Saturday)
We can solve this system of equations using elimination. Let's multiply equation 1 by 5 and equation 2 by 2 to get rid of the y terms:
[tex]60x + 10y = 45[/tex] (equation 1, multiplied by 5)
[tex]6x + 10y = 18[/tex] (equation 2, multiplied by 2)
Now we can subtract equation 2 from equation 1 to eliminate the y terms:
54x = 27
Dividing both sides by 54, we get:
x = 0.5
So Plan A lasts for 0.5 hours or 30 minutes.
To find y, we can substitute x=0.5 into one of the original equations (let's use equation 1):
[tex]12(0.5) + 2y = 9[/tex]
[tex]6 + 2y = 9[/tex]
[tex]2y = 3[/tex]
[tex]y = 1.5[/tex]
So Plan B lasts for 1.5 hours or 90 minutes.
Therefore, Plan A lasts for 30 minutes and Plan B lasts for 90 minutes.
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Plan A lasts for 0.5 hours which is 30 minutes and Plan B lasts for 1.5 hours which is 90 minutes.
What is Equation?An equation is a mathematical statement that shows the equality between two expressions. It consists of two sides, the left-hand side, and the right-hand side, separated by an equal sign.
To solve the following problem form the two linear equations according to the given conditions in the Question by representing the number of hours with two different variables.
Here we have
Carmen the trainer has two solo workout plans: Plan A and Plan B.
On Friday there were 12 clients who did Plan A and 2 who did Plan B.
Let the duration of Plan A is "x hours" and the duration of Plan B be "y hours".
Hence, 12x + 2y = 9
=> 12x + 2y = 9 ---- (1)
On Saturday there were 3 clients who did Plan A and 5 who did Plan B.
=> 3x + 5y = 9 ---- (2)
From (1) and (2)
=> 12x + 2y = 3x + 5y
=> 12x - 3x = 5y - 2y
=> 9x = 3y
=> 3x = y ---- (3)
From (2)
=> y + 5y = 9
=> 6y = 9
=> y = 9/6 = 1.5 hours
From (3)
=> 3x = 1.5
=> x = 1.5/3 = 0.5 hrs
Therefore,
Plan A lasts for 0.5 hours which is 30 minutes and Plan B lasts for 1.5 hours which is 90 minutes.
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What is the velocity of light in meters per second in a material with an index of 2
The velocity of light in a material with an index of 2 is approximately 149,896,229 meters per second.
The velocity of light in a material can be calculated using the formula:
v = c/n
where v is the velocity of light in the material, c is the velocity of light in a vacuum (which is approximately 299,792,458 meters per second), and n is the refractive index of the material.
In this case, if the refractive index of the material is 2, we can plug that into the formula:
v = c/n = 299,792,458 / 2 = 149,896,229 meters per second
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Question 1 of 25
How does the graph of g(x)= x + 4 compare with the graph of the parent
function, f(x) = x?
A. The graph of g(x) is compressed vertically by a factor of 4.
B. The graph of g(x) is shifted 4 units up.
C. The graph of g(x) is shifted 4 units to the right.
D. The graph of g(x) is compressed horizontally by a factor of 4.
← PREVIOUS
SUBMIT
The graph of g(x) is shifted 4 units up from the parent function
How does the graph of g(x) compare with the parent function f(x)?The equation of the parent function f(x) is given as
f(x) = x
The transformed function is given as
g(x) = x + 4
Substitute the known values in the above equation, so, we have the following representation
g(x) = f(x) + 4 or g(x) = f(x + 4)
This means that the function is shifted to the left by 4 units or it is shifted up by 4 units
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3. Julia has 7 pennies.
How many more pennies does she
need in order to have the same value
as I dime?
more pennies
Answer: 3
Step-by-step explanation:
Dime worth: 10
Julia's pennys: 7
Subtract: 10 - 7: 3
Answer: 3
Mya claims (m23 + m24) = m21, as shown in the triangle below.
A
1/2
Which equations explain why Mya's claim must be true?
(m21 + m22) = 90° and (m23+ m24) = 90°
(m21 + m22) = 180° and (m23+ m24) = 180°
(m21 + m22) = 90° and (m23 + m24+ m22) = 90°
(m21+ m22) = 180° and (m23+ m24+ m22) = 180°
B
3
C
5
4
Answer:
D) (m∠1+ m∠2) = 180° and (m∠3+ m∠4+ m∠2) = 180°---------------------
The given equation describes the value of the exterior angle of a triangle:
The exterior angle is the sum of two remote interior angles.The proof comes from the two properties:
1) Linear pair is 180°.
This is shown as (m∠1+ m∠2) = 180°.
2) Sum of interior angles of a triangle is 180°.
This is shown as (m∠3+ m∠4+ m∠2) = 180°.
The two equations above help us to prove Mya's claim.
The matching answer choice is D.
The equation that agrees with Maya's claim is (D) (m∠1 + m∠2) = 180° and (m∠3 + m∠4 + m∠2) = 180° as the sum of all the 3 angles in the triangle is 180° and ∠1 and ∠2 is a straight angle.
What is a triangle?A polygon with three edges and three vertices is called a triangle. It is one of the fundamental geometric shapes.
The specified object is a triangle having vertices A, B, and C. triangle with equal sides the typical triangle.
While the exterior angles of a triangle are equal to the sum of the two interior angles that are not adjacent to it, the interior angles of a triangle always add up to 180°.
By deducting the angle of the target vertex from 180°, one can also determine a triangle's exterior angle.
So, Maya claims:
(m∠3 + m∠4) = m∠1
Now, this is true by the statement:
(m∠1 + m∠2) = 180° and (m∠3 + m∠4 + m∠2) = 180°
As the sum of all the 3 angles in the triangle is 180° and ∠1 and ∠2 is a straight angle.
Therefore, the equation that agrees with Maya's claim is (D) (m∠1 + m∠2) = 180° and (m∠3 + m∠4 + m∠2) = 180° as the sum of all the 3 angles in the triangle is 180° and ∠1 and ∠2 is a straight angle.
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The sum of three numbers is 13. the sum of twice the first number, 4 times the second number, and 5 times the third number is 40. the difference between 5 times the first number and the second number is 5. find the three numbers.
If sum of three numbers is 13. the sum of twice the first number, 4 times the second number, and 5 times the third number is 40. the difference between 5 times the first number and the second number is 5 then three numbers are 15/4, -9/2 and 55/4.
Let x, y and z are the three numbers
The sum of three numbers is 13
x+y+z=13...(1)
The sum of twice the first number, 4 times the second number, and 5 times the third number is 40.
2x+4y+5z=40...(2)
The difference between 5 times the first number and the second number is 5.
5x-y=5...(3)
x=5+y/5...(4)
By solving the four equations we get
x=15/4, y=-9/2 and z=55/4
Hence, the three number are 15/4, -9/2 and 55/4.
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what is the slope of the line?
Answer:
-1
Step-by-step explanation:
the slope of the line is given by the equation m=(y2-y1)/(x2-x1)
here the line passes with the coordinate (-2,0),(0,-2)
m=(-2-0)/(0-(-2))
= -2/2
= -1
i need to find the surface area of the figure below
Answer: 548 mi^3
Step-by-step explanation:
Find the areas of all the sides by doing length times width and add these areas together.
120+120+84+84+70+70=548
devonta throws a rock straight down off the edge of a cliff that overlooks the ocean. The distance (d) the rock falls after t seconds can be represented by the equation d = 16t^2 + 24t. if the ocean's surface is 16.4 feet below the cliff, to the nearest tenth, how many seconds will it take for the rock to hit the ocean's surface?
The rock will hit the ocean's surface in approximately 1 second.
We are given the equation d = [tex]16t^2 + 24t[/tex], where d is the distance the rock falls in feet and t is the time in seconds.
We are also told that the ocean's surface is 16.4 feet below the cliff. This means that when the rock hits the ocean's surface, the distance it falls will be equal to 16.4 feet. So we can set d equal to 16.4 and solve for t:
[tex]16.4 = 16t^2 + 24t[/tex]
Moving all the terms to one side, we get:
[tex]16t^2 + 24t - 16.4 = 0[/tex]
Dividing both sides by 8, we get:
[tex]2t^2 + 3t - 2.05 = 0[/tex]
We can solve this quadratic equation using the quadratic formula:
t[tex]= (-b \pm \sqrt(b^2 - 4ac)) / 2a[/tex]
Here, a = 2, b = 3, and c = -2.05. Substituting these values, we get:
[tex]t = (-3 \pm \sqrt(3^2 - 4(2)(-2.05))) / (2(2))[/tex]
Simplifying:
[tex]t = (-3 \pm\sqrt(33.8)) / 4[/tex]
To the nearest tenth, the positive value of t is:
t ≈ 1.0 second
Therefore, it will take approximately 1 second for the rock to hit the ocean's surface.
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create one problem of ordering numbers in real life
One problem of ordering numbers in real life is ranking the defenses of NFL teams according to least points allowed.
How to order numbers?Numbers can be ordered in both ascending or descending formats, as follows:
Ascending order: Arrange the numbers from lowest to highest. For example, the ascending order of 3, 1, 6, 2, 8 is 1, 2, 3, 6, 8.Descending order: Arrange the numbers from highest to lowest. For example, the descending order of 3, 1, 6, 2, 8 is 8, 6, 3, 2, 1.In ascending order, one example is ranking the defenses of NFL teams according to least points allowed, as the fewer points the defense allow, the better they are.
In descending order, one example is ranking authors according to book sales, as the more books they sell, the more high-selling they are.
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Four members of a marching band notice that they've formed the vertices of a parallelogram, if the field they were on had coordinates like the coordinate plane. Cindy is at (-2, 2), Arturo is at (3, 1), and Rowan is at (-1, 4).
Part 1: What are all possible coordinate locations for Hailey, the fourth band member?
Part 2: How do you know that you have found all possible coordinates? Justify your answer using words, mathematics, and/or patterns as needed.
What shape best describes a horizontal cross-section of a triangular pyramid?
OA. triangle
OB. square
OC. rectangle
OD. parallelogram
Without using a calculator, match each expression to the correct point.
f
de
←++ ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
-5
-3
Point Expression
3
2
d
e
-4
f
-√8
-π
-2
-1
+
0
The number line of each expression to the correct point is added as an attachment
Matching each expression to the correct point.From the question, we have the following parameters that can be used in our computation:
d = -3/2
e = -√8
f = -π
When the above expressions are evaluated, we have
d = -1.5
e = -2.83
f = -3.14
This means that the locations of the points are
d = -1.5; halfway between -1 and -2e = -2.83; between -2 and -3f = -3.14; between -3 and -4Using the above as a guide, we have the following:
We plot the points d, e and f between the numbers highlighted above
See attachment for the new number line
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please simplify answer i need the correct answer
The volume of the sphere is 32 cubic feet.
To calculate the volume of sphere, apply the formula given below;
The formula for the volume of a sphere can be described as:
V = (4/3)πr³
where r is the representation of the radius of the sphere.
Given that the radius of the sphere is 2 feet and π = 3, we can substitute these values into the formula to get:
V = (4/3)π(2³)
= (4/3)π(8)
As per the given information π = 3,
V = (4/3)(3)(8)
V = 32 cubic feet.
Therefore, the required volume is 32 cubic feet.
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HELP ME PLSSSSSSS
A square pyramid has a base measuring 10 inches on each side. The height of the pyramid is 5 inches.
A similar square pyramid has a base measuring 2.5 inches on each side.
How do the surface areas of these pyramids compare?
Drag a value to the box to correctly complete the statement.
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The surface area of the larger pyramid is Response area times the surface area of the smaller pyramid.
Answer:
The surface area of the larger pyramid is 16 times area times the surface area of the smaller pyramid.
Step-by-step explanation:
10^2= 100
2.5^2= 6.25
100/6.25=16
Please someone help me with this. My teacher hasn't taught us to do an equation of this difficulty.
The area of the shaded region is 0.6204
What is the area of the shaded region?To find the area of the shaded region in the figure given the curves r = sin3θ and r = 0.5 + 0.5sinθ, we first need to find the points of intersection.
So, equating both expressions, we have that sin3θ = 0.5 + 0.5sinθ,
sin(2θ + θ) = 0.5 + 0.5sinθ
Using the trigonometric identity sin(A + B) = sinAcosB + cosAsinB, we have that
sin(2θ + θ) = sin2θcosθ + cos2θsinθ
= sin2θcosθ + cos2θsinθ
= sin2θcosθ + cos2θsinθ
using trigonometric identity cos2A = 1 - sin²A and sin2A = 2sinAcosA, we have that
sin(2θ + θ) = sin2θcosθ + cos2θsinθ
= 2sinθcosθcosθ + (1 - sin²θ)sinθ
= 2sinθcos²θ + sinθ - sin³θ
Using the trigonometric identity
cos²A = 1 - 2sin²A, we have that
2sinθcos²θ + sinθ - sin³θ = 2sinθ(1 - 2sin²θ) + sinθ - sin³θ
= 2sinθ - 2sin³θ + sinθ - sin³θ
= 2sinθ + sinθ - 2sin³θ - sin³θ
= 3sinθ - 3sin³θ
So, sin(2θ + θ) = 0.5 + 0.5sinθ
3sinθ - 3sin³θ = 0.5 + 0.5sinθ
3sinθ - 3sin³θ - 0.5 - 0.5sinθ = 0
- 3sin³θ + 3sinθ - 0.5sinθ - 0.5 = 0
- 3sin³θ + 2.5sinθ - 0.5 = 0
3sin³θ - 2.5sinθ + 0.5 = 0
6sin³θ - 5sinθ + 1 = 0
Let sinθ = y
So, 6y³ - 5y + 1 = 0
Checking if y - 1 is a factor
6(-1)³ - 5(-1) + 1 = 0
-6 + 5 + 1 = -1 + 1 = 0
So, y - 1 is a factor
So, dividing 6y³ - 5y + 1 by y - 1, we have
6y³ - 5y + 1/(y - 1) = 6y² + 6y + 1
So, factorizing 6y² + 6y + 1, we have
Using the quadratic formula, we find y
So, [tex]y = \frac{-b +/-\sqrt{b^{2} - 4ac} }{2a}[/tex] where a = 6, b = 6 and c = 1
So, we have that
[tex]y = \frac{-6 +/-\sqrt{6^{2} - 4\times6\times1} }{2\times6}\\y = \frac{-6 +/-\sqrt{36 - 24} }{12}\\y = \frac{-6 +/-\sqrt{8} }{12}\\y = \frac{-6 - 2.83 }{12} or y = \frac{-6 + 2.83 }{12}\\y = \frac{-8.83 }{12} or y = \frac{-3.17 }{12}\\y = -0.7358 or y = -0.2642[/tex]
So, we have that the solution of
6y³ - 5y + 1 = 0 is
y = 1, y = -0.7358 and y = -0.2642
Since y = sinθ, we have that
sinθ = 1, sinθ = -0.7358 and sinθ = -0.2642
Taking inverse sine, we have that
θ = sin⁻¹(1), θ = sin⁻¹(-0.7358) and θ = sin⁻¹(0.2642)
θ = π/2, θ = -0.83 and θ = -0.27
So, the area of the shaded region
A = 1/2∫r²dθ
= 1/2∫(0.5 + 0.5sinθ)²dθ
= 1/2[∫0.25dθ + 0.5sinθdθ + 0.25sin²θdθ]
= 1/2[0.25∫dθ + 0.5∫sinθdθ + 0.25∫sin²θdθ]
Since sin²θ = (1 - cos2θ)/2, we have that
= 1/2[0.25∫dθ + 0.5∫sinθdθ + 0.25∫(1 - cos2θ)/2dθ]
= 1/2[0.25∫dθ + 0.5∫sinθdθ + 0.25/2∫dθ - ∫cos2θdθ]
So, integrating from θ = -0.83 to θ = -0.27 to θ = π/2, we have that
= 1/2[0.25∫dθ + 0.5∫sinθdθ + 0.25/2∫dθ - ∫cos2θdθ]
= 1/2[0.25[θ] + 0.5[-cosθ] + 0.25/2[θ] - sin2θ/2]
= 1/2[0.75[θ]/2 + 0.5[-cosθ] - sin2θ/2]
= [0.75[θ] + 0.25[-cosθ] - sin2θ/4]
= [0.75{[-0.27 - (-0.83)] + [π/2 - (-0.27)]} - 0.25{[cos(-0.27) - cos(-0.83)] + [cos π/2 - cos(-0.27)]} - (sin2(-0.27)/4 - sin2(-0.83)/4 + sin2(π/2)/4 - sin2(-0.27)/4]
= [0.75{[-0.27 + 0.83)] + [1.57 + 0.27]} - 0.25{[0.9638 - 0.6749] + [0 - 0.9634]} - (sin(-0.54)/4 - sin(-1.66)/4 + sinπ/4 - sin(-0.54)/4]
= [0.75{[0.56)] + [1.84]} - 0.25{[0.2889] + [ - 0.9634]} + 0.5141/4 + 0.9960/4 + 0/4 + 0.5141)/4]
= [0.75{2.4} - 0.72225 - 0.9634 + 0.5141/2 + 0.9960/4 + 0]
= [0.75{2.4} - 0.72225 - 0.9634 + 0.25705 + 0.2490 + 0]
= [1.8 - 0.72225 - 0.9634 + 0.25705 + 0.2490]
= 0.6204
So, the area of the shaded region is 0.6204
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What is the value of p in this proportion? 3/10 / p=4/5 / 1/4 Enter your answer as a simplified fraction in the box.
The correct value of the p will be 32/3.
To solve the proportion, we can cross-multiply:
(3/10) / p = (4/5) / (1/4)
(3/10) / p = (4/5) * 4/1
(3/10) / p = 16/5
Now we can solve for p by multiplying both sides by p and then dividing by 16/5:
(3/10) = (16/5) * (1/p)
(3/10) * (5/16) = 1/p
(3/32) = 1/p
Multiplying both sides by p gives:
p * (3/32) = 1
Dividing both sides by 3/32 gives:
p = 32/3
Therefore, the value of p in the proportion is 32/3.
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A ladder is leaning against a brick chimney as shown in the picture below. The ladder is 25 feet long and is placed 15 fee up on the chimney. In order for the ladder to be stable, its base must be 15 feet or less away from the base of the chimney. Is the current position of the ladder stable?
A. No, the distance of the ladder from the base of the chimney is 20 feet.
B. No, the distance of the ladder from the base of the chimney is 40 feet.
C. Yes, the height of the ladder and the distance from the base of the chimney are both 15 feet.
D. Yes, the distance of the ladder from the base of the chimney is 10 feet.
[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{25}\\ a=\stackrel{adjacent}{b}\\ o=\stackrel{opposite}{15} \end{cases} \\\\\\ b=\sqrt{ 25^2 - 15^2}\implies b=\sqrt{ 625 - 225 } \implies b=\sqrt{ 400 }\stackrel{ \textit{the ladder is not stable} }{\implies \boxed{b=20}}[/tex]
Answer:
Letter A.
Step-by-step explanation:
The Pythagorean theorem formula is [tex]a^{2}+b^{2}=c^{2}[/tex]. The length of the hypotenuse, the longest side, is your [tex]c[/tex] value. The lengths of the legs If you substitute 25 for [tex]c[/tex] and 15 for [tex]a[/tex], you would get [tex]15^2 +b^2=25^2[/tex]. [tex]25^2[/tex] equals [tex]625[/tex] and [tex]15^2[/tex] equals [tex]225[/tex], so you would get [tex]225 +b^2=625[/tex]. [tex]625-225=400[/tex], so [tex]b^2=400[/tex]. If you take the square root of both sides, you would get [tex]b=20ft[/tex]. Since the base of the ladder has to be fifteen feet or less away from the base of the chimney and [tex]15ft\ngeq20ft[/tex], the ladder is therefore unstable.
