The solutions for the given problems involving factorials and permutations are 8! = 40,320, 5P5 = 120, 8P4 = 1,680, 7P7 = 5,040 and the number of arrangement for 4 performers out of 9 is 3,024.
Explanation:The subject here is mathematics, specifically factorials and permutations. Factorials are represented by the symbol '!'. The factorial of a number n is the product of all positive integers less than or equal to n. For example, 8! = 8*7*6*5*4*3*2*1 = 40,320.
Permutations refer to the number of ways a set of objects can be arranged. 'nPn' always equals to the factorial of n, therefore 5P5 equals to 5! = 5*4*3*2*1 = 120 and 7P7 equals to 7! = 7*6*5*4*3*2*1 = 5,040.
'nPm' equals to n factorial divided by (n-m) factorial. So, 8P4 equals to 8!/(8-4)! = (8*7*6*5)/1 = 1,680.
For the last question, the number of ways 4 performers can stand in a row out of 9, is calculated as a permutation of '9P4' (because the order of performers matters). Thus, it will be 9!/(9-4)! = 9*8*7*6 = 3,024.
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55^2 = 50^2 + 35^2 − 2(50)(35)cos(A)
Answer:
A = 78.46
Step-by-step explanation:
55^2 = 50^2 + 35^2 − 2(50)(35)cos(A)
3025 = 2500 + 1225 - 3500 cos A
- 3500 Cos A = -3725 + 3025
-3500 cos A = -700
cos A = -700/-3500
= 0.2
A = cos^-1 0.2
A = 78.46
Answer:
A=78.46° (to 4sf)
Step-by-step explanation:
The only way to approach this problem is to solve the equation.
55²=50²+35²-2(50)(35)cos(A)
3025=2500+1225-3500cosA
3025=3725-3500cosA
Now collecting like terms together:
3025-3725=-3500cosA
-700=-3500cosA
CosA=0.2
A=78.46° (to 4sf)
Two plus two equals?
Two plus two is equal to four
Answer:
2+2=4
Step-by-step explanation:
1+1=2
1+1=2
^=4
HOPE THIS HELPEDS!!!!! XD
Can I have brain? please
Find the roots of the quadratic function by completing the square: x^2 + 4x - 1 = 0
Answer:
x = - 2 ± [tex]\sqrt{5}[/tex]
Step-by-step explanation:
Given
x² + 4x - 1 = 0 ( add 1 to both sides )
x² + 4x = 1
To complete the square
add (half the coefficient of the x- term )² to both sides
x² + 2(2)x + 2² = 1 + 2² ← complete the square on the left side
(x + 2)² = 5 ← take the square root of both sides
x + 2 = ± [tex]\sqrt{5}[/tex] ← subtract 2 from both sides
x = - 2 ± [tex]\sqrt{5}[/tex]
roots are x = - 2 - [tex]\sqrt{5}[/tex] or x = - 2 + [tex]\sqrt{5}[/tex]
Final answer:
To find the roots of the quadratic function x² + 4x - 1 = 0, the equation is rearranged, a perfect square is formed by adding (b/2)² to both sides, and the square root is taken to obtain the solutions x = √{5} - 2 and x = -√sqrt{5} - 2.
Explanation:
To find the roots of the quadratic function x² + 4x - 1 = 0 by completing the square, we need to follow a series of steps:
First, we'll arrange the equation so that the x-squared and x terms are on one side, leaving the constant on the other side. In this case, we add 1 to both sides to obtain x² + 4x = 1.
Next, we find the number that needs to be added to x^2 + 4x to make it a perfect square trinomial. This number is (b/2)², where b is the coefficient of x. Here, b is 4, so we need to add (4/2)² = 4 to both sides.
Our equation now reads x² + 4x + 4 = 5. Notice that the left-hand side is a perfect square, as it can be written as (x+2)².
Finally, we take the square root of both sides, giving us x + 2 = √{5} or x + 2 = -√{5}. Therefore, the solutions are x = √{5} - 2 and x = -√{5} - 2.
To check our solutions, we could substitute them back into the original equation and ensure that the left side equals zero.
What is the total amount of sap the trees produced that day
Answer:
wrong its 5gallons
Step-by-step explanation:
Tabitha earns $7 per hour working at the mail. Last week, she worked for 12 1/4 hours. Which best describes how much money she earned?
A- a little less than $65
B- a little less than $77
C- a little less than $84
D- a little less than $90
Answer:
D
Step-by-step explanation:
7 * 12 = 84
And 1/4 hours which is 7/4 which is $1.25 more.
So she got 85.25 dollars in total, which is a little less than $90.
if the point (7,3) is on the graph of an equation, which statement must be true
i need help asap
HELP ME
Which of the following can Cindy use to purchase
an item if she wants to pay for her purchase at a
later time?
A debit card
B check
C money transfer
D credit card
Answer:
credit card
Step-by-step explanation:
a credit card you pay every month not when you use it in the store
Part 2
TOO PART 1 Please help me
Answer:
Part A) The volume of the ice cream scoop is [tex]36\pi\ in^{3}[/tex]
Part B) The melted ice cream won't fill the cup
Part C) The melted ice cream exceeds the volume of Anna's cup
Part D) The height of the smallest cylindrical cup is [tex]h=8\ in[/tex]
Step-by-step explanation:
Part A) Find the volume of the ice cream scoop
we know that
The volume of the sphere (ice cream scoop) is equal to
[tex]V=\frac{4}{3}\pi r^{3}[/tex]
we have
[tex]r=3\ in[/tex]
substitute
[tex]V=\frac{4}{3}\pi (3)^{3}=36\pi\ in^{3}[/tex]
Part B) Find the volume of Anna's cylindrical cup
we know that
The volume of a cylinder is equal to
[tex]V=\pi r^{2}h[/tex]
we have
[tex]r=3\ in[/tex]
[tex]h=6\ in[/tex]
substitute
[tex]V=\pi (3)^{2}(6)=54\pi\ in^{3}[/tex]
[tex]54\pi\ in^{3}> 36\pi\ in^{3}[/tex]
The volume of Anna's cup (cylinder) is greater than the volume of melted ice cream scoop
therefore
The melted ice cream won't fill the cup.
Part C) Will two scoop of melted ice cream fit in Anna's cup?
Multiply the volume of ice cream scoop by 2 and compare with the volume of Anna's cup
so
[tex](2)36\pi\ in^{3}=72\pi\ in^{3}[/tex]
[tex]72\pi\ in^{3}> 54\pi\ in^{3}[/tex]
The volume of Anna's cup (cylinder) is less than the volume of two melted ice cream scoop
therefore
The melted ice cream exceeds the volume of Anna's cup
Part D) Find the smallest cylindrical cup that will hold two scoops of melted ice cream
we know that
The volume of a cylinder is equal to
[tex]V=\pi r^{2}h[/tex]
we have
[tex]V=72\pi\ in^{3}[/tex] ------> volume of two scoops of melted ice cream
[tex]r=3\ in[/tex]
substitute in the formula and solve for h
[tex]72\pi=\pi (3)^{2}h[/tex]
simplify
[tex]72=(9)h[/tex]
[tex]h=8\ in[/tex]
An 8-car Ferris wheel is shaped like a regular octagon. Does the Ferris wheel have rotational symmetry?
Answer:
Yes, it does.
Step-by-step explanation:
Every regular shaped figure will have rotational symmetry, since they are built with identical segments all around.
To find the answer, ask yourself « If I rotate the shape, is there a time where I’ll find the exact same shape again with a different angle? »
So, a square has rotational symmetry, but not a rectangle.
A equilateral triangle has rotational symmetry, but not any other type of triangle.
The Ferris wheel shaped like a regular octagon would have rotational symmetry.
What is rotational symmetry?The rotational symmetry of a shape makes it possible for an object when rotated on its own axis, the shape of the object looks the same.
For a regular polygon with n sides, there would be n number of rotational symmetry,
The Ferris wheel shaped like a regular octagon would have rotational symmetry.
Find out more on rotational symmetry at: https://brainly.com/question/15178808
Lines A and B are represented by the equations given below: Line A: 6x + 6y = 24 Line B: x + y = 4 Which statement is true about the solution to the set of equations? (1 point) Select one: a. It is (24, 4). b. There are infinitely many solutions. c. It is (4, 24). d. There is no solution.
Answer:
I believe it is C
Step-by-step explanation:
since x+y=4 that could mean, 2+2=4 which means x and y has the same value, that is 2.
so if we make the equation
6(2)+6(2)= it WOULD equal to 24, and for that reason I think it is C.
(24,4)
There are two mixtures. One mixture contains 5% alchol and the other mixture contains 12% alchol. How much of each should be mixed to make 1000 gallons with 10% alchol
Answer:
to make 1000 gallons with 10% alcohol, we need 714.3 gallons of mixture that contains 5% alcohol and 285,7 gallons of mixture that contains 12% alcohol.
Step-by-step explanation:
According to the statement, there are two mixtures. One mixture contains 5% alcohol and the other contains 12% alcohol.
We want to make 1000 gallons with 10% alcohol, so we have the following system of equations:
A + B = 1000 gallons. [1]
0.05A + 0.12B = 0.1(1000) ⇒ 0.05A + 0.12B = 100 [2]
(Where 'A' represents the mixture that contains 5% alcohol and 'B' the one that contains 12% alcohol).
Solving the system of equations:
A + B = 1000 ⇒ A = 1000 - B [3]
[3] → [2]
0.05(1000 - B) + 0.12B = 100 ⇒ 50 - 0.05B + 0.12B = 100
⇒ 0.07B = 50 ⇒ B = 714.28 ≈ B=714.3 [4]
[4] → [1]
A + B = 1000 ⇒ A + 714.3 = 1000 ⇒ A = 285,7
Therefore, to make 1000 gallons with 10% alcohol, we need 714.3 gallons of mixture 'A' and 285,7 gallons of mixture 'B'
What is the measure of 1
76
92
118
88
Answer:
[tex]m<1=92\°[/tex]
Step-by-step explanation:
we know that
The sum of the interior angles in a quadrilateral must be equal to 360 degrees
so
[tex]62\°+92\°+114\°+m<1\°=360\°[/tex]
solve for the measure of angle 1
[tex]268\°+m<1\°=360\°[/tex]
[tex]m<1=360\°-268\°=92\°[/tex]
Answer:
The answer is
B. 92 degrees.
Step-by-step explanation:
a delivery company charges an extra fee for a package that weighs more than 100 pounds write an inequality that shows the weight w, in pounds, of a package that sonia can ship without having to pay the extra fee
Answer:
[tex]w\leq 100\ pounds[/tex]
Step-by-step explanation:
Let
w------> the weight in pounds
we know that
A delivery company charges an extra fee for a package that weighs more than 100 pounds
so
For
[tex]w> 100\ pounds[/tex] -----> the company charges an extra fee
therefore
For
[tex]w\leq 100\ pounds[/tex] ----> the company not charge an extra fee
The inequality w ≤ 100 expresses the weight limit for Sonia's package. Jesse should use kilograms to weigh the box in the mailroom.
The inequality is: w ≤ 100
Since Sonia can ship packages without paying the extra fee for weights up to 100 pounds. Therefore, any package weighing 100 pounds or less would not incur the additional charge.
Jesse should use kilograms to weigh the box of books in the mailroom because it's a more appropriate metric unit considering the size and weight of the objects in the box.
may somebody help me please
Answer:
The answer would be D because both ordered pair that was substituted in the equation didn't equal to each other.
Explanation:
All you do is substitute the x and y numbers. (x,y)
A:
Substitute the x and y number
7(2) - 5 = 4(4) - 6
Multiply the number in the parentheses by the outside number on both sides.
14 - 5 = 16 - 6
Subtract both sides to get your solution.
9 = 10
Doesn't equal so that means it's not the solution.
B:
Substitute
7(3) - 5 = 4(6) - 6
Multiply the number in the parentheses by the outside number on both sides.
21 - 5 = 24 - 6
Subtract both sides to get your solution.
16 = 18
It also doesn't equal so that means that it's not the solution either.
C:
Both ordered pair didn't equal to each other so this is not the answer.
Hope this helps!
Interest in the trainings offered at the pet shelter decreased from summer to winter. This change can be represented as -83. What is the absolute value of -83?
It would be 83.
Reason it 83 units away from 0.
I hope this helps :)
Answer:
83
Step-by-step explanation:
What is the answer to this question?
Answer: [tex]x\geq6[/tex]
Step-by-step explanation:
g°h indicates that you must plug the function h(x) into the function g(x) as you can see below:
[tex]g\°h=\sqrt{(2x-8)-4}[/tex]
Now you must simplify by adding like terms, as following:
[tex]g\°h=\sqrt{2x-12}[/tex]
By definition you have that:
[tex]2x-12\geq0[/tex]
Theen you must solve for x:
[tex]2x\geq12\\x\geq6[/tex]
Therefore, the domain is:
{[tex]x[/tex] ∈R:[tex]x\geq6[/tex]}
Then the answer is [tex]x\geq6[/tex]
Answer:
Restriction on the domain is x ≥ 6.
Step-by-step explanation:
We have given two functions.
g(x) = √x-4 and h(x) = 2x-8
We have to find the restrictions on the domain of (g o f).
(g o h)(x) = g(h(x))
(g o h)(x) = g(2x-8)
(g o h)(x) = √2x-8-4
(g o h)(x) = √2x-12
Hence, 2x-12 ≥ 0
2x ≥ 12
x ≥ 6
Hence, restriction on the domain is x ≥ 6.
For what value of a do the lines 5x–2y=3 and x+y=a intersect at a point on the y-axis?
So what they are trying to say is that they both have the same y-intercept. Easy!
Make y the subject:
5x - 2y = 3
2y = 5x - 3
y = 2.5x - 1.5
The y-intercept is -1.5.
Make y the subject:
x + y = a
y = -x + a
Therefore, a must be -1.5 !
The system of linear equations has a solution on the y-axis only when a = -3/2
When the system has one solution?
We have the system of linear equations:
5x - 2y = 3
x + y = a
Now let's write both in slope-intercept form:
y = (-3/2) + (5/2)x
y = a - x
If we want the lines to intersect at the y-axis, then we will have x = 0, replacing that we get:
y = -3/2
y = a
So, for the system to be consistent, we must have a = -3/2.
If you want to learn more about systems of equations, you can read:
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Alyssa and Elijah are classmates who received the same homework assignment. Alyssa spent 45 minutes on the assignment, and Elijah spent 2 hours.what is the ratio in simplest form of the amount of time on the assignment to the amount of time Elijah spent?
Answer:
3 : 8
Step-by-step explanation:
Two hours = 120 minutes
Alyssa:Elijah
45 : 120
3 : 8
34) Apply the distributive property to simplify the expression. 5(3x − 7) A) 15x + 35 B) 15x − 35 C) −15x + 35 D) −15x − 35
Answer:
Answer:
B.
Explanation:
If you want to multiply a parenthesis by a number, you simply distribute the number to all the terms in the parenthesis.
Step-by-step explanation:
Answer:
B. [tex]15x-35[/tex]
Step-by-step explanation:
We are asked to simplify the expression [tex]5(3x-7)[/tex] using distributive property.
Distributive property states that we can multiply a quantity to sum by multiplying each addend separately and then add the products as:
[tex]a(b+c)=a\cdot b+a\cdot c[/tex]
Using distributive property, we will multiply 5 by [tex]3x[/tex] and [tex]-7[/tex] as:
[tex]5\cdot 3x-5\cdot 7[/tex]
[tex]15x-35[/tex]
Therefore, the simplified form of the given expression would be [tex]15x-35[/tex] and option B is the correct choice.
Triangle ABC undergoes a series of transformations to result in triangle DEF .
Is triangle DEF congruent to triangle ABC ?
Select Congruent or Not congruent for each description.
question 1 is congruent. 2 is not congruent and 3 is congruent
Answer: 1) Congruent
2) Not congruent
3) Congruent
Step-by-step explanation:
We know that a rigid transformations preserves side-lengths and angle measures of a figure in such a way that the figure doesn't shrink or get enlarger. It creates congruent figures.
The three main rigid transformations are:a) reflections, b) rotations, c) translations
On the other hand a dilation changes the size of the image when the scale factor is not equal to 1. It does not produces congruent images.
PLEASE HELP WILL MARK BRAINIEST! I really need help and dont get this problem please help (explaining will also be greatly appreciated and right answers)
Based on the graph below, what are the solutions to the equation f(x) = g(x)?
x = −2.4, 1.4, 3.9
x = −2.4, 0.25, 1.4
x = −2, 2, 3
x = −2, 1, 3
The possible values of x are -2.4,1.4,3.9
Step-by-step explanation:
F(x)=3x - 1
G(x)=x³-3x²-4x
when we will plot we are given two functions f(x) and g(x) as:
the graph of these two functions we could clearly see that the point of intersections are the possible solutions of the equation .
F(x)=G(x)
the points of the intersection of this equation is x= -2.4,1.4,3.9
Hence the solutions are x= -2.4,1.4,3.9.
Using the completing-the-square method, rewrite f(x) = x2 - 6x + 2 in vertex form.
Answer:
[tex]\large\boxed{f(x)+(x-3)^2-7}[/tex]
Step-by-step explanation:
[tex]\text{The vertex form of equation}\ y=ax^2+bx+c:\\\\y=a(x-h)^2+k\\\\\text{We have the equation:}\\\\f(x)=x^2-6x+2=x^2-2(x)(3)+1=x^2-2(x)(3)+3^2-3^2+2\\\\\text{Use}\ (a-b)^2=a^2-2ab+b^2\to x^2-2(x)(3)+3^2=(x-3)^2\\\\f(x)=(x-3)^2-9+2=(x-3)^2-7[/tex]
1) An apple drops off the apple tree from a height of 8 feet. How long does it take the apple to reach the ground? Use the function f(x) = -16x^2 + c, where c is the initial height of a falling object, to find the answer.
2) Trinette cut a square tablecloth into 4 equal pieces that she used to make two pillow covers. The area of the tablecloth was 3600 square inches. What is the side length of each piece Trinette used to make the pillow covers?
3) Elton earns x dollars per hour at the bookstore. His mother, Evelyn, earns x^2 dollars per hour as a career counselor. Twice Evelyn's wage equals $84.50. What is Elton's hourly wage? Round you answer to the nearest cent.
Answer:
Part 1) [tex]\frac{\sqrt{2}}{2}\ sec[/tex] or [tex]0.7\ sec[/tex]
Part 2) The side length of each piece Trinette used to make the pillow covers is [tex]30 in[/tex]
Part 3) Elton's hourly wage is [tex]\$6.50[/tex]
Step-by-step explanation:
Part 1)
Let
x------> the time in seconds
we have
[tex]f(x)=-16x^{2}+c[/tex]
In this problem the initial height is 8 ft
so
[tex]f(x)=-16x^{2}+8[/tex]
To find how long does it take the apple to reach the ground, equate the function to zero and solve for x
[tex]0=-16x^{2}+8[/tex]
[tex]16x^{2}=8[/tex]
[tex]x^{2}=1/2[/tex]
[tex]x=\frac{\sqrt{2}}{2}\ sec[/tex]
[tex]x=0.7\ sec[/tex]
Part 2)
step 1
The area of a square tablecloth is
[tex]A=3,600\ in^{2}[/tex]
Divided by 4
[tex]3,600/4=900\ in^{2}[/tex]
step 2
Find the length of each piece
[tex]A=b^{2}[/tex]
so
[tex]900=b^{2}[/tex]
[tex]b=30 in[/tex]
Part 3)
we know that
[tex]2x^{2} =84.50[/tex]
solve for x
[tex]x^{2} =42.25[/tex]
[tex]x=\sqrt{42.25}[/tex]
[tex]x=\$6.50[/tex]
Determine the unknown side of the similar triangle. A) 2 B) 3 C) 4 D) 5
Step-by-step explanation:
You didn't provide any context so I can't answer this. BUT!!
If you're looking at two triangles, look for the side corresponding to the unknown side. If it's an equilateral (all sides same length) triangle no need to worry. It's probably a scalene, all different lengths, because math is hard. So you'll have 3 different lengthy sides, and you can probably see which ones correspond with which based on length and position. You can use angles to help you.
Now use the sides you have the value of to determine scale factor.
Divide the length of one of the sides on one triangle by the corresponding side on the other. That's your scale factor. Now multiply it by the side that corresponds to the unknown side, and that is supposed to be your length.
Sorry couldn't help any more
Answer:
thwe answer is c
Step-by-step explanation:
Solve for x.
x = ___
Answer:
x = 2
Step-by-step explanation:
Similar triangles are triangles which have the same shape but not the same size. This means their angle measures are equal but their side lengths are not instead they are proportional. To solve, we will set up a proportion and solve.
A proportion is two equal ratios set equal to each other. We form the ratios by finding corresponding sides (sides which match to each other on both triangles by their position). Out ratios will be one side of the small triangle over the corresponding side on the big triangle as shown below:
[tex]\frac{little}{big} =\frac{little}{big}[/tex].
[tex]\frac{36}{36 + 12}=\frac{24}{24 + 6x - 4}[/tex]
To solve the proportion, we'll cross multiply by multiplying numerator with denominator across the equal sign.
[tex]36(24 + 6x - 4)=(36+12)(24)\\36(6x + 20)=48(24)\\216x + 720 = 1152 \\216x = 432\\\frac{216x}{216}=\frac{432}{216} \\x = 2[/tex]
The vertex of this parabola is at (-2,5) which of the following could be its equation (apex)
Answer:
Option D. [tex]y=3(x+2)^{2}+5[/tex]
Step-by-step explanation:
we know that
The equation of a vertical parabola in vertex form is equal to
[tex]y=a(x-h)^{2}+k[/tex]
where
(h,k) is the vertex of the parabola
If a> 0 then the parabola open up and the vertex is a minimum
If a< 0 then the parabola open down and the vertex is a maximum
In this problem the vertex is the point (-2,5)
so
the equation must be equal to
[tex]y=a(x+2)^{2}+5[/tex] and the value of a is positive
therefore
the answer is the option D
Answer:
y=3(x+2)^2+5
Step-by-step explanation:
y = 3(x+2)^2 + 5 Is the correct answer
Write this ratio as fraction in simplest form 20 ounces to 2 pounds
Answer:
10oz:1lb
Step-by-step explanation:
divide by two
Solve the system graphically:
b
x−y=0
2x+3y=−5
x=
y=
Answer:
(-1,-1)
Step-by-step explanation:
x=y
2x+3x=-5
x=-1
y=-1
What is the 9th term of the sequence? 3,-12,48,-192,
A. 786,432
B. -196,608
C. -786,432
D. 196,608
D, 196,608.
The sequence is multiplying by -4 every time.
Answer: The correct option is (D) 196608.
Step-by-step explanation: We are given to find the 9th term of the following sequence :
3, -12, 48, -192, . . .
Let a(n) denote the n-th term of the given sequence.
Then, a(1) = 3, a(2) = -12, a(3) = 48, a(4) = -192, . . .
We see that
[tex]\dfrac{a(2)}{a(1)}=\dfrac{-12}{3}=-4,\\\\\\\dfrac{a(3)}{a(2)}=\dfrac{48}{-12}=-4,\\\\\\\dfrac{a(4)}{a(3)}=\dfrac{-192}{48}=-4,~~.~~.~~.[/tex]
So, we get
[tex]\dfrac{a(2)}{a(1)}=\dfrac{a(3)}{a(2)}=\dfrac{a(4)}{a(3)}=~~.~~.~~.~~=-4.[/tex]
That is, the given sequence is a GEOMETRIC one with first term a = 3 and common ratio d= -4.
We know that
the n-th term of an geometric sequence with first term a and common ratio r is given by
[tex]a(n)=ar^{n-1}.[/tex]
Therefore, the 9th term of the given sequence is
[tex]a(9)=ar^{9-1}=3\times(-4)^8=3\times 65536=196608.[/tex]
Thus, the 9th term of the given sequence is 196608.
Option (D) is CORRECT.
Find the equation of the circle in standard form for the given center (h, k) and radius r: (h, k) = (0, 0), r = 4
Answer:
[tex]x^2+y^2=16[/tex]
Step-by-step explanation:
The equation of a circle with center (h,k) and radius r is given by the formula;
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Given (h,k)=(0,0) and r=4, we substitute the values to obtain;
[tex](x-0)^2+(y-0)^2=4^2[/tex]
The required equation is
[tex]x^2+y^2=16[/tex]
The standard form of the equation for a circle centered at the origin with a radius of 4 is x2 + y2 = 16.
The equation of a circle in standard form with the center at the origin (0, 0) and a radius of 4 is given by x2 + y2 = r2, where x and y are the coordinates of any point on the circle, and r is the radius.
Plugging the given radius into the equation, we have x2 + y2 = 42. Simplifying this, we get the equation x2 + y2 = 16. This is the standard form of the equation for the given circle.