Given that x, y, z are different and satisfy the equations:
x^3 + x^2 - x = 0
y^3 + y^2 - y = 0
z^3 + z^2 - z = 0
We can multiply the first equation by y, the second by z, and the third by x to obtain:
x^3y + x^2y - xy = 0
y^3z + y^2z - yz = 0
z^3x + z^2x - zx = 0
Adding these three equations together, we get:
x^3y + x^2y - xy + y^3z + y^2z - yz + z^3x + z^2x - zx = 0
Expanding and simplifying, we find that:
x^2y + xy^2 + y^2z + yz^2 + z^2x + zx^2 - xy - yz - zx = 0
Now, we use the fact that x, y, and z are roots of the equations given above to simplify the expression further:
x^2 - x + y^2 - y + z^2 - z = 0
We can also write this as:
(x - 1)(x + 1) + (y - 1)(y + 1) + (z - 1)(z + 1) = 0
This expression can be factorized as:
(x - 1)(x + 1)(y - 1)(y + 1)(z - 1)(z + 1) = 0
Since x, y, and z are all different, we have that:
(x - 1)(x + 1) ≠ 0
(y - 1)(y + 1) ≠ 0
(z - 1)(z + 1) ≠ 0
Therefore, we have:
(x - 1)(x + 1)(y - 1)(y + 1)(z - 1)(z + 1) = (x - 1)(x + 1)(y - 1)(y + 1)(z - 1)(z + 1) = 0
So, we can conclude that:
xyz = -1
And,
1 + xyz = 0
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The expression -210+ 12m represents a submarine that began at a depth of 210 feet below sea level and ascended at a rate of 12 feet per minute. What was the depth of the submarine after 9 minutes?
-108 is the depth of the submarine after 9 minutes.
What is Expression?An expression is combination of variables, numbers and operators.
The given expression is -210+ 12m
This expression represents a submarine that began at a depth of 210 feet below sea level and ascended at a rate of 12 feet per minute.
We need to find the depth of the submarine after 9 minutes
m=9
Now put m value in the expression
-210+12(9)
-210+108
-108
Hence, -108 is the depth of the submarine after 9 minutes.
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One side of a rectangle is 16 inches and the other side is x inches. What values of x will make the
perimeter more than 50 inches?
Any real value of x which is greater than 9 inches will make the perimeter of rectangle more than 50 inches
Let us assume that the length of the rectangle is represented by l and the width is represented by w.
Assuing l = 16 inches and width = x inches
Let P represents the perimeter of the rectangle.
We know that the formula for the perimeter of the rectangle is:
P = 2(l + w)
The perimeter should be more than 50.
So, we get an inequality,
P > 50
2(l + w) > 50
l + w > 25
16 + x > 25
16 + x - 16 > 25 - 16
x > 9
Thi means, that for any x > 9 will make the perimeter greater than 50 in.
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from january 2014 to january 2019 the doubling time for the number of wikipedia articles during this time period is
The doubling time for the number of Wikipedia articles during this period is 346.56 days.
So, based on the question, we need to find t. And the condition required to find t will be -
N(t) = 2 [tex] N_{0} [/tex]
So, keeping the value of N (t) in the equation mentioned in question.
[tex] N_{0} [/tex] [tex] {e}^{t/500} [/tex] = 2 [tex] N_{0} [/tex]
Cancel [tex] N_{0} [/tex] on both sides of the equation
[tex] {e}^{t/500} [/tex] = 2
Taking log in both sides of the equation and generating equation according to the rule
t/500 = ln 2
t = 500 × ln 2
Performing multiplication on Right Hand Side of the equation
t = 346.57
Thus, the time period is 346.57 days
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The complete question is attached in the figure.
Can someone answers these fast I need them rn
Step-by-step explanation:
3.
30 + 60+ x=l 180°( being sum of angle of triangle is 180°)
90+ x= 180°
x= 180 - 90
=90
The function V(x) = 4/3πx^3 represents the volume (in cubic inches) of the sphere. The function W(x) = V(27x) represents the volume (in cubic inches) of the sphere when x is measured in feet. Write a rule for W. Find W(6) in terms of π.
W(x) = ?
W(6) = ? cubic inches.
The value of the function W(x) for x=6 is 17,799,730.56 cubic inches.
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
Given that, the function V(x)=4/3 πx³ represents the volume (in cubic inches) of the sphere.
The function W(x) = V(27x) represents the volume (in cubic inches) of the sphere when x is measured in feet.
W(x) = 4/3 π(27x)³
W(x) = 4/3 π(27x)³
= 4/3 π×19683×x³
= 4π×6561×x³
W(x)= 26244πx³
W(6)= 26244×3.14×(6)³
= 17,799,730.56 cubic inches
Therefore, the value of the function W(x) for x=6 is 17,799,730.56 cubic inches.
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Help……………………………………..
at 5:45 p.m., a jet is located 108 mi due east of a city. a second jet is located 214 mi due north of the city. to the nearest tenth of a mile, what is the distance between the two jets?
The distance between the two jet when one is in the 108miles of east and second one 214miles in the north is equal to 239.7 miles ( nearest tenth ).
Location of the first jet at 5:45 p.m. is :
= 108 miles in the east of the city.
Location of the second jet at 5:45 p.m. is :
= 214 miles in the north of the city.
Let 'x' miles be the distance between the jets.
East and north from a city forms a right angled triangle.
'x' represents the hypotenuse.
Using Pythagoras theorem we have,
x² = 108² + 214²
⇒ x² = 11664 + 45796
⇒x² = 57460
⇒ x = 239.708 miles
⇒ x = 239.7 miles (nearest tenth )
Therefore, the distance between the two jets as per given condition is equal to 239.7 miles
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The curve above is the graph of a sinusoidal function. It goes through the points
(
-
9
,
0
)
and
(
5
,
0
)
. Find a sinusoidal function that matches the given graph. If needed, you can enter
π
=3.1416... as 'pi' in your answer, otherwise use at least 3 decimal digits.
Answer:
Step-by-step explanation:
To find a sinusoidal function that matches the given graph, we need to find the amplitude (A), period (T), phase shift (h), and vertical shift (k) of the function.
Since the graph goes through the points (-9,0) and (5,0), we know that the maximum value of the function is located at the average of the x-values, which is -2, and that the minimum value of the function is located at the average of the x-values, which is (9+5)/2 = 7. The amplitude of the function can be found by taking half the difference of the maximum and minimum y-values, which is 0 - (-2)/2 = 1.
The period of the function can be found by taking the difference of the x-values of two consecutive maxima or minima, which is 5-(-9) = 14.
The phase shift can be found by noting the horizontal shift of the graph from its standard form, which is usually given by the equation y = A sin(x). In this case, the graph is shifted 2 units to the right, so the phase shift is 2.
The vertical shift can be found by noting the location of the midline of the graph, which is half way between the maximum and minimum y-values. In this case, the midline is at y = -1.
Putting all these values together, we can write a sinusoidal function that matches the given graph as:
f(x) = 1 sin(x + 2π/14) - 1
This function has amplitude 1, period 14, phase shift 2π/14 to the right, and a vertical shift of -1 down.
Which of the following are roots of the polynomial function below?
Check all that apply.
F(x)=x²-3x²+2
A. 3-√17
4
B. 2+√/12
C. 1
D. 3+√17
4
□ E. 2-12
SUBMIT
Answer:
[tex]\textsf{B.} \quad \dfrac{2 +\sqrt{12}}{2}[/tex]
[tex]\textsf{C.} \quad 1[/tex]
[tex]\textsf{E.} \quad \dfrac{2-\sqrt{12}}{2}[/tex]
Step-by-step explanation:
Factor TheoremIf f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).
If the coefficients in a polynomial add up to 0, then (x - 1) is a factor.
Given polynomial function:
[tex]f(x)=x^3-3x^2+2[/tex]
Sum the coefficients:
[tex]\implies 1-3+2=0[/tex]
As the sum of the coefficients equals one, (x - 1) is a factor the polynomial.
Find the other factor by dividing the polynomial by (x - 1):
[tex]\large \begin{array}{r}x^2-2x-2\phantom{)}\\x-1{\overline{\smash{\big)}\,x^3-3x^2+2\phantom{)}}}\\{-~\phantom{(}\underline{(x^3-x^2)\phantom{-)..)}}\\-2x^2+2\phantom{)}\\-~\phantom{()}\underline{(-2x^2+2x)\phantom{}}\\-2x+2\phantom{)}\\\phantom{)}-~\phantom{()}(-2x+2)\\\end{array}[/tex]
Therefore, the factored form of the polynomial is:
[tex]f(x)=(x-1)(x^2-2x-2)[/tex]
To find the roots, set the function to zero and solve for x.
Set the first factor to zero and solve for x:
[tex]\implies (x-1)=0 \implies x=1[/tex]
Set the second factor to zero and solve for x using the quadratic formula:
[tex]\implies x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\implies x=\dfrac{-(-2) \pm \sqrt{(-2)^2-4(1)(-2)}}{2(1)}[/tex]
[tex]\implies x=\dfrac{2 \pm \sqrt{12}}{2}[/tex]
Find the volume of the composed figure
Enter the correct number in the box
The required volume of the composite figure is given as 18,300 cm³.
What is volume?Volume is defined as the mass of the object per unit density while for geometry it is calculated as profile area multiplied by the length at which that profile is extruded.
Here,
The volume of the composed figure is given as,
Split the figure into two,
So,
Volume = Volume of 1 + Volume of 2
Volume = 15 × 15 × 30 + 35 × 22 × 15
Volume = 6750 + 11550
Volume = 18,300 cm³
Thus, the required volume of the composite figure is given as 18,300 cm³.
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(HELP!!!) 1. 20 % of the total number of pupils in a class is 6, How many pupild are there in the class altogether? (with explanation pls)
Answer:
30
Step-by-step explanation:
this is the explanation
6÷20% = 30
height of a cuboid is three-fourths of its length,breadth is 4 cm and total surface area is 608,finds the length and height of the cuboid.
Height of a cuboid is three-fourths of its length,breadth is 4 cm and total surface area is 608.
the length of the cuboid is 24 cm and its height is 18 cm.
Let us denote the length of the cuboid as "l" and the height of the cuboid as "h".
Based on the facts provided, we may conclude:
h = (3/4)l
Furthermore, the total surface area is 608, which may be computed as follows:
608 = 2(lb + bh + lh)
Extending the formula and changing the value of h:
608 = 2(lb + bh + l(3/4)l)
2(lb + 4h + (3/4)l^2) = 608
Now that we have the equation, we can solve for l by substituting the values of b (breadth) and h (height):
2(4l + 4h + (3/4)l^2) = 608
8l + 8h + (3/2)l^2 = 304
(3/2)l^2 + 8l + 8h = 304
We can now solve for l using the quadratic formula:
l = (-8 ± √(8^2 - 4 * (3/2) * (304 - 8h))) / (2 * (3/2))
Because h = (3/4)l, we may simplify by substituting this formula for h:
l = (-8 ± √(8^2 - 4 * (3/2) * (304 - 3l))) / (2 * (3/2))
Expanding the square root:
l = (-8 ± √(64 + 12l)) / 3
When we solve for l, we get:
l = 24
Finally, for h, substitute the value of l into the equation:
h = (3/4) * 24 = 18
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Which equation represents a function?
Responses
A y = ±2x + 3y = ±2x + 3
B x2
+ y2
= 16x 2 + y 2 = 16
C y = ±3x+8−−−−−√
y = ± 3 x + 8
D y2
= -5x - 8y 2 = -5x - 8
E x = 4y + 7
Option C is correct : y = ±3x + 8, represents a function because it satisfies the vertical line test.
"y = ±3x + 8" represents a function. A function is a mathematical relationship between a set of inputs (in this case, x) and a set of outputs (in this case, y). In this equation, for each value of x there is exactly one corresponding value of y, which means that it represents a function.
"y = ±2x + 3" is not a function because for each value x, there are two possible values y (one positive and one negative). This is not the definition of a function, as a function must have exactly one output value for each input value.
Options B, D, and E do not represent functions because they do not have an equal sign between the x and y variables, which is the defining characteristic of a mathematical function.
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what is coefficient?
Answer:
a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g. 4 in 4x y).
Answer:
A coefficient is a numerical or constant quantity placed before and multiplying variable.
Step-by-step explanation:
For example, 4x+8=12. 4 is the coefficient.
the polygon exterior angle-sum theorem states that the exterior angles of any polygon add up to 360 degrees. *write a formula that can help you find the measure of each individual exterior angle in any polygon. use n for the number of sides.
The formula to calculate the measure of each individual exterior angle in any polygon is: Exterior angle measure = 360°/n.
To find the measure of each individual exterior angle in any polygon, first, you must understand the Polygon Exterior Angle-Sum Theorem, which states that the exterior angles of any polygon add up to 360 degrees.
With this information, the formula for the measure of an individual exterior angle is simple: Exterior angle measure = 360°/n, where n is the number of sides in the polygon.
By dividing the total sum of the exterior angles (360°) by the number of sides (n) in the polygon, you can determine the measure of each individual exterior angle.
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Solve 2*x2+3xy if x=6 and y=5
The solution to the given algebraic expression is 162.
Solving algebraic expressions.The process of solving algebraic expressions can sometimes be the process whereby numbers are used as a substitute or to replace a variable(s) in order to get a desired solution.
In the given question, we have 2x² + 3xy, where x = 6 and y = 5. It means that we are going to replace the value of x with 6 where ever we see x in the expression and y with 5.
Thus, 2x² + 3xy becomes:
= 2(6)² + 3(6)(5)
= 2(36) + 3(30)
= 72 + 90
= 162
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divide the amount 300 in the ratio 1;5 in the correct ratio
Answer:
$50, and $250
Step-by-step explanation:
5x + 3y = 12
x - 4y= 7
Answer:
x = 3
y = -1
Step-by-step explanation:
5x + 3y = 12
x - 4y= 7
Times the second equation by -5
5x + 3y = 12
-5x + 20y = -35
23y = -23
y = -1
Now put -1 back in for y and solve for x
x - 4(-1) = 7
x + 4 = 7
x = 3
Let's check
3 - 4(-1)= 7
3 + 4 = 7
7 = 7
So, x = 3 and y = -1 is the correct answer.
Solve for X. Find the length of angle
Answer:2
Step-by-step explanation:
4*(4+2)=2*(2x+5+3)
4*6=2*(2x+8)
24=4x+16
4x=8
x=2
Triangle ABC with vertices at A(−3, −3), B(3, 3), C(0, 3) is dilated to create triangle A′B′C′ with vertices at A′(−1, −1), B′(1, 1), C′(0, 1). Determine the scale factor used.
1
1/2
3
1/3
Since triangle ABC with vertices at A(−3, −3), B(3, 3), C(0, 3) is dilated to create triangle A′B′C′ with vertices at A′(−1, −1), B′(1, 1), C′(0, 1), the scale factor used is: D. 1/3.
What is scale factor?In Geometry, the scale factor of a geometric figure can be calculated by dividing the dimension of the image (new figure) by the dimension of the pre-image (original figure):
Scale factor = Dimension of image (new figure)/Dimension of pre-image(original figure)
Substituting the given parameters into the scale factor formula, we have the following;
Scale factor = Dimension of image (new figure)/Dimension of pre-image(original figure)
Scale factor = -1/-3
Scale factor = 1/3.
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Answer: D = 1/3
Step-by-step explanation: Took the test and got it right :>
if 5 sina =3 then find tana/cosa=5/13
Answer:
15/16
Step-by-step explanation:
PLEASE HELP SUPER EASY
Answer: isosceles, because each side is accurate
Answer:
Scalene, right.
Step-by-step explanation:
Its scalene because all the sides are not equal.
It is a right angled triangle.
Hope this helps!
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Priya’s Pet Store never has more than a combined total of 20 cats and dogs and never more than 8 cats. This is represented by the inequalities x+y≤20
and x≤8
. Represent the number of cats and dogs that can be at the store on a graph. Solve the system of inequalities by graphing.
Need help graphing it and finding where the two inequalities meet.
The common region or overlapped region in the graph of both the inequalities represents the number of cats and dogs that can be at the store.
What is an inequality? What are algebraic expressions?An inequality is used to make unequal comparisons between two or more expressions. For example → ax + b > cIn mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that Priya’s Pet Store never has more than a combined total of 20 cats and dogs and never more than 8 cats. This is represented by the inequalities x + y ≤ 20 and x ≤ 8.
The number of cats and dogs that can be at the store on a graph can be represented by the common region in the graph of both the inequalities.
Therefore, the common region or overlapped region in the graph of both the inequalities represents the number of cats and dogs that can be at the store.
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tay practices piano for x hours. Omar practices for 2/5 less than that
Answer: Let x be the number of hours Tay practices the piano. Then, Omar practices for 2/5 less than that, or x - 2/5x = 3/5x hours.
Step-by-step explanation:
when does a simple correlation between two variables justify concluding that a definitely causes b? question 9 options: when the correlation is statistically significant. when there is no obvious third variable that could be causing changes in both a
A simple correlation between two variables when there is no obvious third variable that could be causing changes in both a and b.
A measure of the degree to which two variables vary together or the strength of the link between two variables is simple linear correlation. Correlation is frequently misused. You must provide evidence that one variable is really having an impact on another. and when we compare r to the relevant crucial value in the table, the association is statistically significant. The correlation coefficient is significant if it is greater than zero and does not fall between the positive and negative critical values.
the possibility that a reported correlation between two variables may not actually reflect any underlying relationship (in a causal sense) between the two variables, but rather the shared correlation between each of the variables and a third variable.
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Describe the possible lengths of the third side of the triangle if the lengths of the other two sides are 6 meters and 9 meters.
The triangle third side may be 6 or 9 metres long, or it may be between 9 and 15 metres long.
Depending on the lengths of the other two sides, the triangle's third side can have a variety of lengths. The third side's length could range from 9 to 15 metres if the first two sides are 6 and 9 metres, respectively. The reason for this is that the length of any two sides of a triangle must always be greater than the length of the third side; in this instance, the sum of the two sides is 15 metres, or 6 and 9 metres. As a result, the triangle's third side must be either less than or equal to 15 metres. A triangle can also have two equal sides, so the third side could also be equal to 6 or 9 metres. In conclusion, the third side of the triangle may be anywhere between 9 and 15 metres long, or it could be 6 or 9 metres long.
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What percent of 125 is 32?
ames needs to make a total of 50 deliveries this week. so far he has completed 32 of them. what percentage of the total deliveries has lamar completed?
Answer:
64%
Step-by-step explanation:
"Percent" means per hundred. So Lamar has completed 32 out of 50 deliveries.
32/50 is the same as 64/100.
64/100 is 64%
Without mental math, if you can use a calculator, divide
32÷50
You'll get 0.64, times by 100 (or move the decimal point two places to the right) to get 64%
The length of an altitude of an equilateral triangle is √3/2 feet. Find the length of one side of the triangle
Answer: Let's call the length of one side of the equilateral triangle x.
Then, the height of the triangle can be represented as √3/2.
By the Pythagorean theorem, we have:
(x/2)^2 + (√3/2)^2 = (x/2)^2 + 3/4 = x^2/4 + 3/4 = x^2/4 + 3/4 = x^2/4
So:
x^2 = 4 * (x^2/4 + 3/4) = 4x^2/4 + 4 * 3/4 = 4x^2/4 + 3
Solving for x:
3x^2/4 = 3
x^2 = 4
x = 2
So the length of one side of the equilateral triangle is 2 feet.
Step-by-step explanation:
−n+(−3)+3n+5 ????????
Answer:
2n+2
Step-by-step explanation: