Answer:
True
Step-by-step explanation:
If ab < 0, then ab = negative #.
In order for ab to be a negative #, one of them has to be negative while the other one needs to be positive.
Example:
a = -2, b = 1
ab < 0
(-2)(1) < 0
-2 < 0, TRUE
a < 0
-2 < 0, TRUE
b > 0
1 > 0, TRUE
If I switch a = -2 to 1 and b = 1 to -2, a > 0 and b < 0 is true too.
A full can of milk weighs 70 pounds. If exactly half of the milk is poured out, it weighs 38 pounds. How much does the empty can weigh?
Answer:
6 pounds
Step-by-step explanation:
Let the can weigh x pounds
Let the milk weigh y when full, therefore half is [tex]\frac{y}{2}[/tex]
Now we for 2 simultaneous equations:
[tex]x + y =70[/tex] ....(i)
[tex]x+ \frac{y}{2}= 38[/tex] .... (ii)
we solve by eliminating the x in both equation throug subtraction
[tex]x - x + y- \frac{y}{2} =70-38\\y- \frac{y}{2} = 32\\\frac{y}{2} = 32\\y=2(32)\\y=64\\x+y=70\\x+64=70\\x=6[/tex]
By definition, when a test or technique measures what it intends to, it is a. valid. b. reliable. c. significant. d. generalizable.
By definition, when a test or technique measures what it intends to, it is valid.
The answer is option A.
The criterion-related validity of a take-a-look is measured by the validity coefficient. Its miles said as a number between zero and 1.00 that indicates the significance of the relationship, "r," among the take a look at and a measure of task overall performance (criterion).
Reliability and validity are both approximately how nicely a technique measures something: Reliability refers to the consistency of a measure (whether or not the consequences can be reproduced below identical situations).
Validity refers back to the accuracy of a measure (whether or not the results really do constitute what they may be speculated to a degree).
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Answer the following questions that follow up with the picture.
The coordinates of the pint of tangency of the two circles is(_,_)
The radius of the circle with the center N has a length of ___ units.
The radius of the circle with center P has a length of ___ units.
The diameter of the circle with center N has a length of ___ units.
which vectors are unit vectors?
The unit vectors are u = {1, 1}. Option C
What are unit vectors?Unit vectors can be defined as vector with magnitude of 1.
It is important to note that a unit vector is a vector quantity. Vector quantities are known to have magnitude and direction.
They are represented with the sign 'u'
Thus, the unit vectors are u = {1, 1}. Option C
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What is the value of X
Answer:
x = 10
Step-by-step explanation:
BF is two times AX so
2 ( 2x+10) = 5x+10
4x + 20 = 5x + 10
x = 10
A team of 10 volunteers is visiting families in a local community to deliver canned goods. It takes one volunteer 24 hours to complete one visit. What is the capacity of the team over the course of an 8-hour workday?
1/3 of the volunteers can complete the 8-hour workday
How to determine the capacity of the team
Total number of volunteers = 10
If it takes 1 volunteer = 24 hours = 1 day
It would take x number of volunteers = 8 hours
To find 'x' , cross multiply
We have,
x * 24 hours = 8 hours * 1
24x = 8
Make 'x' the subject of formula
x = 8/ 24
x = 1/3
Therefore, 1/3 of the volunteers can complete the 8-hour workday
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The volume of a rectangular prism is given by the expression 2x^4+2x^3-4x^2-4x. Write the volume as the product its dimensions. Remember, V = lwh.
=========================================================
Explanation:
First factor out the GCF 2x
2x^4+2x^3-4x^2-4x
2x*x^3+2x*x^2-2x*2x-2x*2
2x(x^3 + x^2 - 2x - 2)
Then let's factor the expression inside the parenthesis using the factor by grouping method
x^3 + x^2 - 2x - 2
(x^3 + x^2) + (- 2x - 2)
x^2(x + 1) - 2(x + 1)
(x^2 - 2)(x+1)
We see that x^3 + x^2 - 2x - 2 factors to (x^2-2)(x+1)
Overall, the original expression fully factors to 2x(x^2-2)(x+1)
length = 2x
width = x^2-2
height = x+1
The order of length, width, and height doesn't matter.
The volume as the product its dimensions 2x*( x²-2)*(x+1).
What is Volume?Volume is a mathematical quantity that shows the amount of three-dimensional space occupied by an object or a closed surface.
Given:
volume of a rectangular prism is given by the expression 2x^4+2x³-4x²-4x
=2x*x^3+2x*x^2-2x*2x-2x*2
=2x(x³ + x² - 2x - 2)
Now, we get the factor 2x(x³ + x² - 2x - 2).
Keeping aside 2x we will now focus on (x³ + x² - 2x - 2)
We already get the one factor 2x and now using (x³ + x² - 2x - 2) to get the other two factors.
Now solving x³ + x² - 2x - 2
=(x³ + x²) + (- 2x - 2)
= x²(x + 1) - 2(x + 1)
=( x² - 2)(x+1)
Now we get the two factors ( x² - 2)(x+1).
The expression for volume as the product its dimensions is 2x*( x²-2)*(x+1)
Hence, volume as the product its dimensions is 2x*( x²-2)*(x+1).
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Using the following image, solve the problems below given that B is the midpoint of AC.
How would you set this problem up to find x? (Hint: Enter in the equation to find x.)
Using your setup, what are the values of x, AC, BC, and AC?
x=?
AB=?
BC=?
AC=?
(Giving 60 points and brainly!)
Answer:
AB = 4
BC = 4
AC = 8
Step-by-step explanation:
Given, B is the midpoint of AC which means it divided the AC into two equal parts. Using this information we can write the following equation:
5x - 6 = 2x add 6 to both sides
5x = 2x + 6 subtract 2x from both sides
3x = 6 divide both sides by 3
x = 2 to find the length of AB, AC, and BC replace x with 2
AB = 5*2 - 6
BC = 2*2
AC = 8 (sum of AB and BC)
[tex]\huge \boxed{\sf x=2}\\\\\huge \boxed{\sf AB=4}\\\\\huge \boxed{\sf BC=4}\\\\\huge \boxed{\sf AC=8}[/tex]
[tex]\displaystyle \sf AB=BC\\\\5x-6=2x\\\\Subtracting\ 2x\ and\ adding\ 6\ to\ each\ side \\\\5x-6+6-2x=2x-2x+6\\\\5x-2x=6\\\\3x=6\\\\Divide\ each\ side\ by\ 3\\\\\frac{3x}{3} =\frac{6}{3} \\\\x=2[/tex]
[tex]\sf Substituting\ x=2\\\\AB=5(2)-6=10-6=4 \\\\BC=2(2)=4\\\\AC=AB+BC=4+4=8[/tex]
Please help with this and pls give all steps!!!
Answer:
No solutions.
Step-by-step explanation:
Problem:
Solve y−4x=−3;y=4x+7
Steps:
I will solve your system by substitution.
y=4x+7;−4x+y=−3
Step: Solve y=4x+7 for y:
Step: Substitute 4x+7 for y in −4x+y=−3:
−4x+y=−3
−4x+4x+7=−3
7=−3(Simplify both sides of the equation)
7+−7=−3+−7(Add -7 to both sides)
0=−10
Answer:
No solutions.
Answer:
No solutions
Step-by-step explanation:
It is easiest to determine whether the equations are consistent by putting both of them in the same form.
Adjusting the formThe equation on the left is in "standard form." The one on the right is in "slope-intercept form." We can rewrite either one of them to put it into the other form.
Rewriting the left equation to slope-intercept form, we have ...
y = 4x -3 . . . . . . . add 4x to both sides
Comparing this to the right equation ...
y = 4x +7
we see that the slopes are the same, and the intercepts are different. These equations describe parallel lines. Parallel lines can never meet, so cannot have any point in common. The equations are inconsistent and have no solution.
Rewriting the right equation to standard form, we have ...
y -4x = 7 . . . . . . . subtract 4x from both sides
Comparing this to the left equation ...
y -4x = -3
we see that the coefficients are the same, but the constants are different. There can be no values of x and y that will satisfy both equations. Any values that make the terms have one sum cannot also make them have a different sum.
There are no solutions.
Find the altitude of this equilateral triangle - geometry
Answer:
4√3.
Step-by-step explanation:
If the height is h then
tan 60 = h / (1/2*8)
tan 60 = h/4
h = 4 tan 60
= 4 * √3.
what is the value of x when (fog)(x)=-8
The value of x when (f o g)(x) = -8 is 3
How to solve for x?The composite function is given as:
(f o g)(x) = -8
The above is calculated as:
(f o g)(x) = f(g(x))
So, we have:
f(g(x)) = -8
From the table, we have:
f(-4) = -8
This means that:
g(x) = -4
From the ordered pair, we have:
g(x) = -4 when x = 3
Hence, the value of x is 3
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The value of x when (f o g)(x) = -8 is 3
How to solve for x?
The composite function is given as:
(f o g)(x) = -8
The above is calculated as:
(f o g)(x) = f(g(x))
So, we have:
f(g(x)) = -8
From the table, we have:
f(-4) = -8
This means that:
g(x) = -4
From the ordered pair, we have:
g(x) = -4 when x = 3
Hence, the value of x is 3
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Suppose the population of a certain city increases at a rate proportional to the number of Inhabitants at any time. If the population doubles in 30 years, in how many years will it triple
Answer:
It will triple in approximately 48 years.
Step-by-step explanation:
t=30*ln3/ln2=30*1.5849= 48 years (approx.)
PLEASE HELP ME WITH THIS QUESTION TT ITS A EQUIVALENT QUESTION!!!
thank you!
Answer: [tex]\boldsymbol{\frac{5\text{x}-2}{\text{x}^2-\text{x}}}[/tex] (choice D)
You have the correct answer.
=====================================================
Work Shown:
[tex]\frac{2}{\text{x}} + \frac{3}{\text{x}-1}\\\\\frac{2(\text{x}-1)}{\text{x}(\text{x}-1)} + \frac{3\text{x}}{\text{x}(\text{x}-1)}\\\\\frac{2\text{x}-2}{\text{x}^2-\text{x}} + \frac{3\text{x}}{\text{x}^2-\text{x}}\\\\\frac{2\text{x}-2+3\text{x}}{\text{x}^2-\text{x}}\\\\ \boldsymbol{\frac{5\text{x}-2}{\text{x}^2-\text{x}}}\\\\[/tex]
--------
Explanation:
The idea I used is that we cannot add the fractions unless the denominators are the same. The denominators x and (x-1) lead to the lowest common denominator, aka LCD, of [tex]\text{x}(\text{x}-1) = \text{x}^2 - \text{x}[/tex]. We multiply the denominators together to get the LCD.
The first original fraction [tex]\frac{2}{\text{x}}[/tex] is missing (x-1) in the denominator. This is why I multiplied top and bottom by (x-1) in the 2nd step. Similarly, the fraction [tex]\frac{3}{\text{x}-1}[/tex] is missing an x out front to get to x(x-1). This is why I multiplied top and bottom of that fraction by x.
After we get the denominators to be the same, we can then add the numerators like any other algebraic expression. The denominator stays the same the entire time.
It's similar to how [tex]\frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7}[/tex] has the numerators add like you'd expect while the denominator stays at 7 the entire time.
You can think of it like this: "2 sevenths + 3 sevenths = 5 sevenths" or "2s+3s = 5s" for short. The term "sevenths" is effectively a unit such as cm or meters. We must have common units if we want to add them.
ASAP please
The sequence of transformations that can be performed on quadrilateral ABCD to show that it is congruent to quadrilateral GHIJ is followed by a--------------.
The sequence of transformations that can be performed on quadrilateral ABCD to show that it is congruent to quadrilateral GHIJ is a translation followed by a rotation.
What is a transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformation are reflection, translation, rotation and dilation.
Translation is the movement of a point either up, down, left or right on the coordinate plane.
The sequence of transformations that can be performed on quadrilateral ABCD to show that it is congruent to quadrilateral GHIJ is a translation followed by a rotation.
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How long is the blue ribbon if the red ribbon is 12 inches?
Tyler has 3 times as many books as Mai.
How many books does Mai have if Tyler has:
The length of the blue ribbon given the length of red ribbon is 8 inches
AlgebraLength of red ribbon = 12 incheslength of blue ribbon = 2/3 of red ribbon
= 2/3 × 12
= (2×12)/3
= 24/3
= 8 inches
Let
Number of books Mai has = xNumber of books Tyler has = 3x = 15 books3x = 15
x = 15/3
x = 5 books
Therefore, the number of books Mai has is 5 books
Complete question:
The length of the blue ribbon is two-thirds the length of the red ribbon. How long is the blue ribbon if the red ribbon is 12 inches?
Tyler has 3 times as many books as Mai. How many books does Mai have if Tyler has 15 books?
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There are 480 students in a school.
and
1
of the students wear glasses.
5
How many of the students do not wear glasses
Answer:
96 students
Step-by-step explanation:
I am not exactly sure if this question is talking about a 1:5 ratio, but that's what I am assuming. To get 96, you need to replace the 5 with 480 because the problem seems to mean 1 to every 5 people wear glasses. Next, you do 480/5 to get the number on the left side of the ratio. You get 96 when you do that. I am a little confused with the question because it isn't exactly formatted correctly, but this is what I am guessing. Hope this helps!
For the arithmetic sequence beginning with the terms {7, 11, 15, 19, 23 ...}, what is the sum of the first 31 terms?
a.)
1827
b.)
1950
c.)
2077
d.)
2208
Step-by-step explanation:
here a= 7 ,d=4
then S31= 31/2(7+7+30*4)= 31/2(14+120)= 31/2(134)= 31*67= 2077
hope it helps
The sum of the first 31 terms is 2077.
What is Arithmetic Sequence?An arithmetic sequence is one in which each phrase grows by adding or subtracting some constant k. In contrast, in a geometric sequence, each term grows by dividing/multiplying some constant k.
We have,
Sequence: {7, 11, 15, 19, 23 ...}
First term = 7
Common difference, d= 11- 7 = 4
Now, the sum of first 31 st term of sequence
= 31/2 [ 2(7) + (31-1)4]
= 31/2 [ 14 + 120]
= 31/2 x 134
= 31 x 67
= 2077
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Draw a tree diagram to represent the following scenario. You choose a random letter from the word SANDWICH. How many possible outcome are there?
Probability helps us to know the chances of an event occurring. There are 8 different scenarios possible.
What is Probability?Probability helps us to know the chances of an event occurring.
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
For the given word SANDWICH, there are 8 different scenarios possible, which can be represented as shown below.
Hence, there are 8 different scenarios possible.
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Determine the value of K that will cause f(x)=Kx^2+4x-3 to intersect the line g(x)=2x-7 at 2 points
By using the definition of second order polynomials and the discriminant from quadratic formula, we conclude that the values of k must be less than 1/4.
How to determine the value of k such that a line intersects a quadratic equation
In this question we must determine the set of values of k such that the function g(x), a linear function, intersects a quadratic function f(x) at two points. In this case, we must solve the following second order polynomial:
k · x² + 4 · x - 3 - g(x) = 0
k · x² + 4 · x - 3 - 2 · x + 7 = 0
k · x² + 2 · x + 4 = 0
In this case, the discriminant of the equation described above must be positive:
2² - 4 · k · 4 > 0
4 - 16 · k > 0
4 > 16 · k
16 · k < 4
k < 1/4
By using the definition of second order polynomials and the discriminant from quadratic formula, we conclude that the values of k must be less than 1/4.
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Use any method to solve the equation. If necessary, round to the nearest hundredth.
7x^2-16x=8
Hello,
Answer:
S = { -0,42 ; 2,71 }
Step-by-step explanation:
7x² - 16x = 8 ⇔ 7x² - 16x - 8 = 0
a = 7 ; b = -16 ; c = -8
Δ = b² - 4ac = (-16)² - 4 × 7 × (-8) = 480 > 0
x₁ = (-b - √Δ)/2a = (16 - √480)/14 ≈ -0,42
x₂ = (-b + √Δ)/2a = (16 + √480)/14 ≈ 2,71
Answer:
The two answers:
2.707778736
0.42206445
Step-by-step explanation:
Here is the equation:
[tex] {7x}^{2} - 16x = 8[/tex]
Take away the 8 from the right hand side, so that we are left with this quadratic equation:
[tex] {7x}^{2} - 16x - 8 = 0[/tex]
This equation is too complex to solve with the factorizing method, so let's use the quadratic formula, which is as follows:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
In this equation, a = 7, b = -16, and c = -8. So let's substitute in:
[tex]x = \frac{-( - 16) \pm \sqrt{ { - 16}^{2} - 4 \times 7 \times - 8}}{2 \times 7}[/tex]
[tex]x = \frac{ - ( - 16) \pm \sqrt{256 + 224} }{14}[/tex]
[tex]x = \frac{ 16 \pm \sqrt{480}}{14}[/tex]
And let's work out the two possible answers:.
2.707778736
0.42206445
Heheh be svdbtjdjsbddb
What is the horizontal asympote of this graph?
Using it's concept, it is found that the graph has no horizontal asymptote.
What are the horizontal asymptotes of a function f(x)?The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.
In this problem, we have that:
The function is undefined for x < 0, hence [tex]\lim_{x \rightarrow -\infty} f(x)[/tex] is undefined.For x > 0, the funciton goes to infinity, hence [tex]\lim_{x \rightarrow \infty} f(x) = \infty[/tex].Thus, the graph has no horizontal asymptote.
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Stuart wants to raise $100 for the Rainbow Club charity. He already has three donations of $30.20.
$10.50 and $5.00. How much does Stuart still need to raise?
Answer:
$54.30
Step-by-step explanation:
Given Total Donations already raised
= $30.20 + $10.50 + $5.00
= $45.70
Donations still required = Total Donations needed - Donations already raised
= $100 - $45.70 = $54.30
3
Which expression is equivalent to x-2
O
O O
O
2x-8
13–5x
-5r-8
2x-8
-5x-4
x-2
13–5x
2x-8
-
5
2- 4
x-2
?
Step-by-step explanation:
[tex] = \frac{ \frac{3}{x - 2} - 5 }{2 - \frac{4}{x - 2} } [/tex]
[tex] = \frac{ \frac{3 - 5.(x - 2)}{x - 2} }{ \frac{2.(x - 2) - 4}{x - 2} } [/tex]
[tex] = \frac{ \frac{3 - 5.(x - 2)}{ \cancel{x - 2}} }{ \frac{2.(x - 2) - 4}{ \cancel{x - 2} }} [/tex]
[tex] = \frac{3 - 5.(x - 2)}{2.(x - 2) - 4} [/tex]
[tex] = \frac{3 - 5x + 10}{2x - 4 - 4} [/tex]
[tex] = \frac{ - 5x + 13}{2x - 8} [/tex]
[tex] = \frac{13 - 5x}{2x - 8} [/tex]
The answer is D.
The equivalent value of the expression is A = ( 13 - 5x ) / ( 2x - 8 )
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
In an equation, the expressions on either side of the equals sign are called the left-hand side (LHS) and the right-hand side (RHS), respectively. The equals sign (=) indicates that the two expressions have the same value, and that the equation is true for certain values of the variables involved.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. It typically consists mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation.
Given data ,
Let the equation be represented as A
Now , the value of A is
[tex]A = \frac{\frac{3}{x-2}-5}{2\:-\:\frac{4}{x-2}}[/tex]
On simplifying , we get
[tex]A = =\frac{\frac{-5x+13}{x-2}}{2-\frac{4}{x-2}}[/tex]
On further simplification , we get
[tex]A = =\frac{\frac{-5x+13}{x-2}}{\frac{2x-8}{x-2}}[/tex]
Therefore , the value of A is A = ( 13 - 5x ) / ( 2x - 8 )
Hence , the equation is A = ( 13 - 5x ) / ( 2x - 8 )
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Let $m$ and $n$ be positive integers such that $m$ has exactly 5 positive divisors, $n$ has exactly 6 positive divisors, and $mn$ has exactly 14 positive divisors. How many distinct prime factors does $mn$ have
If [tex]A[/tex] is the set of positive divisors of [tex]m[/tex] and [tex]B[/tex] the set of positive divisors of [tex]n[/tex], then [tex]A\cup B[/tex] is the set of positive divisors of [tex]mn[/tex].
Use the inclusion/exclusion principle:
[tex]|A \cup B| = |A| + |B| - |A \cap B|[/tex]
where [tex]|\cdot|[/tex] denotes set cardinality (the number of elements the set contains). The set [tex]A\cap B[/tex] is the set of common divisors of [tex]m[/tex] and [tex]n[/tex]. Then
[tex]14 = 5 + 6 - |A\cap B| \implies |A\cap B| = 3[/tex]
so that [tex]m[/tex] and [tex]n[/tex] share 3 divisors [tex]d_1,d_2,d_3[/tex]; let [tex]k=d_1d_2d_3[/tex] be their product. They must be prime
This means we can write
[tex]m = p_1 p_2 k[/tex]
[tex]n = p_3 p_4 p_5 k[/tex]
[tex]\implies mn = p_1 p_2 p_3 p_4 p_5 k^2 = p_1 p_2 p_3 p_4 p_5 {d_1}^2 {d_2}^2 {d_3}^2[/tex]
so that [tex]mn[/tex] has up to 8 distinct prime factors.
Identify the segments that are parallel, if any, if
Answer:
The answer is C
Step-by-step explanation:
it is C because parallel lines don't intercept the go up or across but they don't touch, just imagine it isn't graphed.
What is the equation of the function that is graphed as line a?
Answer:
Option 1
Step-by-step explanation:
The equations in the options are in the slope-intercept form (y= mx +c, where m is the slope and c is the y-intercept).
In the graph, the line cuts through the y-axis at y= -2. Thus, c= -2 and the 2nd option can be eliminated.
The slope can be found using the formula below.
[tex]\boxed{\text{slope}=\frac{y_1-y_2}{x_1-x_2} }[/tex]
Let's choose 2 points on the line: (0, -2) and (-2, 4)
Slope
[tex]=\frac{4-(-2)}{-2-0}[/tex]
[tex]=\frac{4+2}{-2}[/tex]
[tex]=\frac{6}{-2}[/tex]
= -3
The equation of the line is thus y= -3x -2.
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https://brainly.com/question/14200719I’m having issues with finding the restriction of the equation in the photo
Answer:
x = 9
Step-by-step explanation:
Restricted value of the expressionSimplify the equation.Set the denominator to 0.Solve and find the solution.The solution is the restricted value of the equation.[tex]\sf \dfrac{(x +7)(x+8)}{(x -9(x + 6)} \ \div \ \dfrac{(x+8)(x - 2)}{(x+ 6)(x - 2)} =\dfrac{(x +7)(x+8)}{(x -9)*(x +6)}*\dfrac{(x + 6)(x - 2)}{(x +8 )(x - 2)}[/tex]
[tex]\sf =\dfrac{x +7}{x - 9}[/tex]
x - 9 = 0
x = 9
If we plugin x = 9, then the denominator would become 0 and hence the expression will become undefined.
So, x = 9 is the restricted value of the expression.
Suppose a batch of metal shafts produced in a manufacturing company have a variance of 9 and a mean diameter of 207 inches. If 72 shafts are sampled at random from the batch, what is the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches
The probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches is 39.54%.
Given mean diameter of 207, variance=9, sample size of 72.
We have to calculate the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches.
The sample mean may be greater than or less than from population mean than 0.3 inches.
Either greater than 207+0.3=207.3 inches,
Smaller =207-0.3=206.7
Since the normal distribution is symmetric these probabilities are equal. So we find one of them and multiply by 2.
Probability of being less than 206.7
P value of z when X=206.7. So
Z=(X-μ)/s
=(206.7-207)/0.35
=-0.3/0.35
=-0.857
p value =0.1977
Probability of differing from population mean greater than 0.3 inches=2*0.1977
=0.3954
=39.54%
Hence the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches is 39.54%.
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Greetings. As a beginner, I'm struggling a bit to learn calculus. May I know what is the derivative of x to the power 4 step by step?
Thanks in advance!
If you're just starting calculus, perhaps you're asking about using the definition of the derivative to differentiate [tex]x^4[/tex].
We have
[tex]\dfrac{d}{dx} x^4 = \displaystyle \lim_{h\to0} \frac{(x+h)^4 - x^4}h[/tex]
Expand the numerator using the binomial theorem, then simplify and compute the limit.
[tex]\dfrac{d}{dx} x^4 = \displaystyle \lim_{h\to0} \frac{(x^4+4hx^3 + 6h^2x^2 + 4h^3x + h^4) - x^4}h \\\\ ~~~~~~~~ = \lim_{h\to0} \frac{4hx^3 + 6h^2x^2 + 4h^3x + h^4}h \\\\ ~~~~~~~~ = \lim_{h\to0} (4x^3 + 6hx^2 + 4h^2x + h^3) = \boxed{4x^3}[/tex]
In general, the derivative of a power function [tex]f(x) = x^n[/tex] is [tex]\frac{df}{dx} = nx^{n-1}[/tex]. (This is the aptly-named "power rule" for differentiation.)