The first partial derivatives are:
∂w/∂x = 8/x∂w/∂y = 9/y∂w/∂z = 1/zTo find the first partial derivatives of the function w = ln(x^8y^9z), we differentiate with respect to each variable separately while treating the other variables as constants.
∂w/∂x:
When differentiating with respect to x, we treat y and z as constants:
∂w/∂x = (∂/∂x) ln(x^8y^9z)
To differentiate ln(u), where u is a function of x, we apply the chain rule:
∂w/∂x = (1/u) * du/dx
In this case, u = x^8y^9z, so:
∂w/∂x = (1/(x^8y^9z)) * (∂/∂x) (x^8y^9z)
Differentiating x^8y^9z with respect to x gives us:
∂w/∂x = (1/(x^8y^9z)) * (8x^7y^9z)
Simplifying:
∂w/∂x = 8x^7y^9z / (x^8y^9z)
∂w/∂x = 8/x
Similarly, we can find the other partial derivatives:
∂w/∂y:
Treating x and z as constants, differentiate x^8y^9z with respect to y:
∂w/∂y = (1/(x^8y^9z)) * (∂/∂y) (x^8y^9z)
∂w/∂y = (1/(x^8y^9z)) * (9x^8y^8z)
∂w/∂y = 9x^8y^8z / (x^8y^9z)
∂w/∂y = 9/y
∂w/∂z:
Treating x and y as constants, differentiate x^8y^9z with respect to z:
∂w/∂z = (1/(x^8y^9z)) * (∂/∂z) (x^8y^9z)
∂w/∂z = (1/(x^8y^9z)) * (x^8y^9)
∂w/∂z = 1/z
Therefore, the first partial derivatives are:
∂w/∂x = 8/x
∂w/∂y = 9/y
∂w/∂z = 1/z
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Shayna and Arjun each improved their yards by planting grass sod and geraniums. They bought
their supplies from the same store. Shayna spent $16 on 2 ft² of grass sod and 1 geraniurn, Arjun
spent $73 on 1 ft² of grass cod
WL
fta of grass sod and the
So on solving the provided question, we can say that by equation Shayna : [tex];5x + 10y = $175[/tex] and Brenda : [tex]9x + 4y = $147[/tex]; cost = [tex]$11/ft^2[/tex] of grass, $12/geranium
What is equation?A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.
the equation, that will be formed are
Shayna : [tex];5x + 10y = $175[/tex]
Brenda : [tex]9x + 4y = $147[/tex]
[tex]5x + 10y = $175 \\ 5x = $175 - 10y\\(5x/5) = ($175/5) - (10y/5) = > x = $35 - 2y\\9x + 4y = $147 = > 9($35 - 2y) + 4y = $147\\ $315 - 18y + 4y = $147\\$315 - $147 = 18y - 4y \\ $168 = 14y = > y = $12\\ 5x + 10y = $175 = > 5x + 10($12) = $175 = > 5x + $120 = $175 \\5x = $175 - $120 = > 5x = $55 = > x = $11[/tex]
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You take out a loan of $4050, with an interest rate of 6.5% over a period of 2 years. What is the interest accumulated?
The interest accumulation on the loan is equals to $526.50.
What is interest accumulation?As a financial term, when an interest is added to the account balance, it is considered accrued. Interest accrues on loans such as a mortgage, savings accounts, student loans, and other investments.
In our calculation, we are using a simple interest because it is only an interest charge that borrowers pay lenders for a loan. It is calculated by using the principal only and does not include compounding interest.
Data
Loan = 4,050
Interest rate = 6.5%
Time = 2 years
To get our Interest, we will use the simple interest formula which is "S.I. = PRT/100". Where P is 4050, r = 6.5%, T= 2 years.
The interest accumulated is computed as follows:
= $4050 * 6.5% * 2 years
= $526.50
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This text describes a scenario where someone has taken out a loan of $4050. The loan has an interest rate of 6.5%, which means that the borrower will have to pay an additional 6.5% of the loan amount as interest. The loan is taken for a period of 2 years, which is the time frame during which the borrower has to repay the loan.
The question being asked is about the interest that will be accumulated on this loan. Interest accumulation refers to the increase in the amount of money that the borrower owes due to the interest that is being charged on the loan.
In this case, we need to calculate the total amount of interest that will be charged over the 2-year period, based on the loan amount and interest rate.
To calculate the interest accumulated on this loan, we can use the following formula:
Interest = Principal * Rate * Time
Where:
- Principal = the amount borrowed (in this case, $4050)
- Rate = the interest rate (in this case, 6.5% or 0.065 as a decimal)
- Time = the length of the loan in years (in this case, 2 years)
Plugging in the values, we get:
Interest = $4050 * 0.065 * 2
Interest = $527.25
So the interest accumulated on this loan is $527.25. This means that the borrower will have to pay back a total of $4577.25 ($4050 + $527.25) over the 2-year period.
Given the equation 10r−9y=8, what is the value of r when y=18? r=
The value of r when y = 18 is 17.
What is an Equation?An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side.
Given the equation which is linear having the exponent of the variable equals 1.
10r - 9y = 8
We have to find the value of r when value of y = 18.
Substituting y = 18 in the equation,
10r - (9 × 18) = 8
10r - 162 = 8
10r = 170
r = 170/10
r = 17
Hence when the value of y = 18, then the value of r = 17.
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an elephant weight 1,495 kilograms if this number was rounded to the nearest hundred how many zeros would it have?
PLEASE PLEASE HELP ME
Two numbers have a sum of 18. The first number is 2 more than 3 times the second number. Find the two numbers.
Answer: Let one of the number be X and the other no will be x +18, this is as per given sum.
By sum,
Step-by-step explanation: x + x + 18 = 128
or, 2 x = 128 - 18 = 110
Therefore x = 55 & the other no will be 55 + 18 = 73
Hence, 55 & 73 are the two numbers.
A hot tub that hold 300 gallon of water drain at a rate of 8 gallon per minute write and equation that repreent how many gallon of water are left in the tub after it ha drained for x minute
The equation that represent how many gallon of water are left in the tub after it ha drained for x minute is A(n) = -8x + 300.
In the given question, we have to write and equation that represent how many gallon of water are left in the tub after it ha drained for x minute.
From the given question, a hot tub that hold 300 gallon of water.
Hot tub drain at a rate of 8 gallon per minute.
So the slope will be = -8 , the negative sign is there because the water is leaving the tub.
Let the amount of water left in the tub at time n minutes is A(n) gallons.
The initial value of A(n) is 300 gallons.
Thus when n = 0, the value A(n) will be 300.
So, we can write the relation as
A(n) = -8n + 300
We need to find the amount left after x minute. So n = x
Now the equation is A(n) = -8x + 300.
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PLEASE HELP!! When the function f(x)=x^4+2x^3+4x^2+x-1 is devided by (x-2), the results x^3+4x^2+12x+25, with a remainder of 49. Which conclusion is valid?
F(-2)=49
F(49)=-2
F(2)=49
F(49)=2
The remainder when the function f(x) = x⁴ + 2x³ + 4x² + x - 1 is divided by (x - 2) to get x³ + 4x² + 12x + 25 is 49. Therefore f(2) = 49
What is an equation?An equation is used to show the relationship between numbers and variables.
Polynomial is an expression composed of variables, constants and exponents using mathematical operations such as addition, subtraction, multiplication and division.
The remainder theorem states that when a polynomial f(x) is divided by (x - a), it gives a remainder equal to f(a).
Given that f(x) = x⁴ + 2x³ + 4x² + x - 1 is divided by (x - 2) to get x³ + 4x² + 12x + 25 with a remainder of 49
Applying the remainder theorem to the above polynomial:
x - 2 = 0
x = 2
Hence:
f(2) = 49
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How to solve factor by grouping. Step by step.
Answer:
(n+5)(n-2)
Step-by-step explanation:
n^2+3n-10
(n+5)(n-2)
Two vats of maple sap are boiling to make maple syrup. They are evaporating at different rates.
Vat A has a capacity of 35 L and is evaporating at rate of 0.25L/hr
Vat B has a capacity of 40 L and is evaporating at a rate of 0.35L/hr
How many hours with it take for both vats to contain the same amount of liquid?
Around 100 hours will take for both vats to contain the same amount of liquid.
How do you calculate the rate of evaporation of a liquid?
The equation indicates that the evaporation rate,
W = (95+0.425*V)*(Pw-Pa)/Y, where W is the evaporation rate, V is the air velocity at the water surface, Pw is the saturation vapor pressure at water temperature.
To find out how many hours it will take for both vats to contain the same amount of liquid, we need to compare the rates at which the liquid is evaporating from each vat.
We can set up the following equation:
(35 - x) / 0.25 = (40 - x) / 0.35
where x is the amount of liquid in both vats at the time they reach the same level.
We can then solve for x by multiplying both sides by 0.25 and 0.35 and then subtracting one equation from the other:
35 - x = 40 - x
x = 5
So, both vats will contain 5 liters of liquid at the time they reach the same level.
To find out how many hours it will take to reach this point, we can use the rate of evaporation for either vat.
For Vat A:
35 - 5 = 30 liters
30 liters / 0.25 liters/hour = 120 hours
For Vat B:
40 - 5 = 35 liters
35 liters / 0.35 liters/hour = 100 hours
So, It will take around 100 hours for both vats to contain the same amount of liquid.
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What is the slope of y =- 5x 5?
The slope of the given equation [tex]y = -5x^5[/tex] is [tex]-25x^4[/tex].
To find the slope of the equation [tex]y = -5x^5[/tex], you need to take the derivative of the equation with respect to x. The slope of the equation represents the rate of change of y with respect to x, that is how y changes as x changes.
The derivative of [tex]y = -5x^5[/tex] can be calculated using the power rule, which states that if f(x) = [tex]x^n[/tex] then f'(x) = [tex]nx^{(n-1)}[/tex], the slope of the equation is [tex]-55x^4[/tex].
So, the slope of the equation [tex]y = -5x^5[/tex] is [tex]-25x^4[/tex].
Here, note that this equation represents a polynomial function of degree 5, which means that it is a function that has the highest power term of [tex]x^5[/tex]. Also, the slope of a polynomial function is always a polynomial function of one degree lower than the original function.
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5) Which inequality represents the solution for 33 < y - 8?
Oa.) y < 25
Ob.) y < 41
Oc.) y > 25
Od.) y > 41
Answer:
d.) y > 41
Step-by-step explanation:
33 < y - 8
41 < y
Use a trigonometric ratio to solve for b. Round to two decimal places as necessary!!
HELP, and can someone explain how to get the answer?
When one side and one angle of a right angled triangle are given as 19 and 35°, respectively, the value of b is calculated using trigonometric ratios as 13.03.
what is trigonometric ratios ?Trigonometric ratios are defined as the values of all trigonometric functions based on the right-angled triangle's side ratio. The trigonometric ratios of any acute angle in a right-angled triangle are the ratios of its sides to that angle.
given
one side = 19
angle = 35°
one angle = 90° as right angle triangle
another angle = 35°+ 90° + x = 180°
= 125° + x = 180°
x = 55°
[tex]\frac{a}{sinA} = \frac{b}{sinB}[/tex]
19 / sin55° = b / sin35°
b = 13.03
When one side and one angle of a right angled triangle are given as 19 and 35°, respectively, the value of b is calculated using trigonometric ratios as 13.03.
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Where is midpoint theorem used?
The midpoint theorem is used in a variety of mathematical fields, but it is primarily used in geometry and algebra.
In geometry, the midpoint theorem is used to find the midpoint of a line segment, which is the point that is exactly halfway between the two endpoints of the segment. The theorem states that the coordinates of the midpoint of a line segment can be found by averaging the coordinates of the endpoints.
For example, if the endpoints of a line segment have coordinates (x1, y1) and (x2, y2), the coordinates of the midpoint would be ((x1 + x2)/2, (y1 + y2)/2).
In algebra, the midpoint theorem can be used to find the equation of a line when the coordinates of the endpoints of a line segment are known. The equation of a line can be represented in the form of y = mx + b, where m is the slope and b is the y-intercept.
The slope of the line can be found by taking the difference of the y-coordinates of the two endpoints divided by the difference of the x-coordinates of the two endpoints. The y-intercept of the line can be found by using the midpoint of the line segment and the slope of the line.
In Calculus, the midpoint theorem is used in optimization problems. Optimization problems are used to find the maximum or minimum value of a function.
The midpoint theorem is used to find the critical points of the function, which are the points where the function reaches its maximum or minimum value. The critical points are found by taking the derivative of the function and equating it to zero.
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.
Unit: Personal Financial Literacy
Homework 1
COMPARING SALARIES
Answer each of the questions below. Be sure to show your work.
Ariana has determined that she either wants to
study public relations or accounting.
.
Public Relations Specialist: $58,020 annual
salary
Accountant: $68,150 annual salary
1. After working one year, how much more will
she earn as an accountant than a public
relations specialist?
10,13 0
2. If Ariana decides to become a CPA, she will
need to continue her education. Her master's
degree will cost $28,500, but as a CPA she will
earn $73,850 each year. How much more will
Ariana earn each year as a CPA than as an
accountant each year?
"5,700
Name
Date 1/1/23
3. Would you recommend that Ariana continue
her education? Why or why not?
Grottlin
Yu is considering two different education
routes.
BACHELOR'S DEGREE
$42,800
entry salary
Pd2
MASTER'S DEGREE
$58,400
entry salary
4. Over a period of 10 years, how much more
will Yu earn by having his master's degree?
5. Over a period of 30 years, how much more
will Yu earn by having his master's degree?
0. Describe some of the advantages and disadvantages to additional schooling
She will make $ 10130 more per year as an accountant than a public relations expert by frequency .
what is frequency distribution ?Frequencies are arranged into visual displays known as frequency distributions in order to make the information more understandable. Absolute or relative frequencies, such as ratios or percentages, can be shown in a frequency distribution. A frequency distribution can be shown using a graph or a frequency table. In a frequency distribution, the precise number of observations and the percentage of observations falling inside each range may both be shown. The distribution in the latter case is known as a relative frequency distribution. An observation's frequency is the frequency with which it appears in the data. For example, the frequency of the number 9 in the following list of integers is 5. (because it occurs 5 times). 1,
given
$58,020 Public Relations Specialist
$68,150 for an accountant
How much more will she make working as an accountant compared to a public accountant? $68,150 - $58,020, or $ 10130.
She will make $ 10130 more per year as an accountant than a public relations expert.
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A box meaure 84cm by 60cm by 36cm
the box i completely filled with identical cube
find the minimum number of cube required
840 cubes can be placed in the given cuboid.
Now, According to the question:
We know that,
Volume of cuboid = l × b × h
Volume of cuboid = 84cm × 60cm × 36cm
Volume of cuboid = 181,440[tex]cm^3[/tex]
Side of the cube = 6 cm
Volume of the cube = side³
= 6³
= 216 cm³
Required number of cubes = volume of cuboid / volume of the cube
= 181,440/216
= 840
Thus, 840 cubes can be placed in the given cuboid.
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The complete question is this:
A box measure 84cm by 60cm by 36cmthe box is completely filled with identical cube. How many small cubes with side 6 cm can be placed in the given cuboid?
Which is the equation of the line, in point-slope form, that has a slope of 5 and passes through the point (-2, 4)? Oy - 2 = -5(x-4) Oy - 2 = 5(x+2) Oy - 4 - 5(x+2) Oy - 4 = 5(x-2)
The sorrect option: y - 4 = 5x - 10, is the equation of the line, in point-slope form.
What is meant by the term point-slope form?A line's equation written using a single point on the line and also the line's slope is referred to as the point-slope form. The slope is the ascent over run, or indeed the ratio of said change mostly in y values well over change in the x values, and the point form is denoted as (x,y).The slope of the line;
Slope m = 5.
Passing points: (x1, y1) = (-2, 4).
So, general equation of the line,
y - y1 = m(x - x1)
y - 4 = 5(x + 2)
y - 4 = 5x - 10
Thus, the equation of line, in point-slope form, that has a slope of 5 is found as : y - 4 = 5x - 10.
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what is 3w-4z= 8 and 2w+3z=-6
w = 0 and z= -2 are the value in linear equation.
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
3w-4z= 8
2w+3z=-6
multiply both sides of the equation of a
6w - 8z = 16
3(2w +3z ) = - 6 * 3
calculate the quotient
6w - 8z = 16
3(2w +3z ) = - 6 * 3
6w - 8z = 16
6z + 3z = -18
substract the two equation
6w - 8z - ( 6z + 3z) = 16 - (-18)
-17z = 34
z = -34/17
z = - 2
put the value of z in equation
6w - 8 * -2 = 16
6w + 16 = 16
6w = 0
w = 0
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What is the value of m?
79°
m
The value of m in the linear pair with an array is 101°, using supplementary rule.
What is Linear pair?Two lines converge at a single point to generate a pair of linear angles. If, following the junction of the two lines, the angles are next to one another, they are said to be linear. A linear pair always has an angle total of 180 degrees.
When two lines cross, they generate a linear pair, which is two nearby angles. The numbers 1 and 2 form a linear pair in the figure. The same goes for 2 and 3, 3 and 4, and 1 and 4. In a linear pair, the two angles are always supplementary, which implies that their sum total is 180 degrees.
79° + m = 180°
m = 180° - 79°
m = 101°
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Complete question:
What is the value of m?
79° + m = 180°
The circumference of a circle is 91.688 millimeters. What is the radius of the circle? Use 3.14 for π.
A circle has a circumference of 91.688 millimeters. The above statement indicates that the circle's radius is 14.6 millimeters.
What is its diameter?The radius of a circle or other curvy geometric object pertains to its length. It is the boundaries across another two-dimensional circular surface measured linearly in one dimension. The length of something like a circle's contour is referred to as its circumference, while the length of a form with straight sides is referred to as its perimeter.
Circumference of a circle = 91.688 millimeters
π = 3.14
Radius = ?
Therefore, cicumference of a circle = 2 * π * r
91.688 = 2 * 3.14 * r
r = 91.688/6.28
r = 14.6 millimeters.
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I KNOW IT LOOKS WIERD BUT PLEASE HELP
Answer: Your answer is x = 8 for c and m = -12 for b
Step-by-step explanation:
c.
-8x = -64
-8x - 8 these cancel out
-64 - 8
64/8 = 8
x = 8
b.
-8 + m = -20
m - 8 = -20
m - 8 + 8 = -20 + 8 (add the -20 + 8 because the 8's cancel out)
m = -12
Tyler atex fruit snacks, and Han ate 34 less than that. Write an equation to represent the relationship between the number Tyler ate (x) and the number Han ate (y).
The equation that represents the relationship between the number of fruit Tyler ate and Han ate is x = y + 34
What is a linear equation?A linear equation is an algebraic equation where each term has an exponent of 1 and when this equation is graphed, it always results in a straight line.
Also a linear equation can be defined as an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept.
The number of fruits Tyler ate is represented by x
The number of fruits Han ate is represented by y
Han ate 34 less than tyler, this means that y = x-34
and also making x the subject,
x = y + 34
Therefore the equation that represents the relationship between the number of fruits Tyler ate and Han ate is x = y+34
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Letica invests $200 at 5% interest. If y represents the amount of money after x time periods, which describes the graph of the exponential function relating time and money?
The graph of the exponential function y = 200(1 + 0.05)ˣ starts with a curve and increases non-linearly or exponentially at any given time
What is an Exponential FunctionMathematical functions that rise or decrease in value at a rate proportional to their existing value are known as exponential functions. An exponential growth or decay function is another name for it. An exponential function is represented by the equation y = bx, where b denotes the base and x the exponent.
In this problem, we can use the basic form of an exponential function to represent this.
y = p(1 + r)ˣ
r = ratex = time periodSubstituting the values into the formula above;
y = 200(1 + 0.05)ˣ
The equation above is an exponential function that represents the growth over a period of time "x"
The graph of an exponential function curves from it's starting point and increases non-linearly steadily till it reaches it's peak.
Kindly find attached graph below
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Teresa is maintaining a camp fire. She can keep the fire burning for 444 hours with 666 logs. She wants to know how many logs (y)(y)left parenthesis, y, right parenthesis she needs to keep the fire burning for 181818 hours. She assumes all logs are the same.
How many logs does Teresa need to maintain the fire for 181818 hours?
logs
27 logs Teresa can keep the fire going for 444 hours with 666 logs instead of having to maintain it for 181818 hours.
what is unitary method ?The unit technique is a strategy for addressing issues that entails first figuring out the price of a specific unit, then multiplication that value to figure out the needed value. Simply put, the unit method is employed to derive a single unit value from a multiple that is supplied. For example, 40 pens could cost 400 rupees, which is equal to the cost of one pen. This procedure could follow a regular procedure. a single nation. anything has a distinct identity. Its adjoint and reciprocal are equal in terms of mathematics, algebra, abstract algebra, mathematical analysis, or mathematical skills of matrices or operators.
given
6 logs in 4 hours
18hrs= ?x
then you multiply and cross.
so,18x6= 4x
4x/4=108/4
x=27 logs
27 logs Teresa can keep the fire going for 444 hours with 666 logs instead of having to maintain it for 181818 hours.
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It takes an ant habitat 4 days to consume 13 of a banana. At that rate, in how many days will the ant habitat consume 4 bananas? Type your answer in the box below.
Using the concept of rates and ratio, it will take approximately 0.12 days to finish 4 bananas
What is RatesWhen comparing two or more similar amounts or numbers using the same units, a ratio is utilized. In spoken language, we state the ratio of one quantity "to" the second quantity, and it is frequently written with a colon. For instance, a class has a 3:4 (or 3:4) ratio of boys to girls. When comparing two quantities, for example, some issues may only give you two numbers, but other problems might have more than two. For example, the proportions of the various ingredients used in a recipe.
We can use the concept of ratio or proportions to determine the number of days they will consume 4 banana
4 days = 13 banana
1 day = x banana
Cross multiply both sides and solve for x
x * 4 = 13 * 1
4x = 13
x = 13 / 4
x = 3.25
Now, we can find the number of days it will take to consume 4 banana
1 day = 3.25 banana
x days = 4 banana
Cross multiply both sides and solve for x
4 * 1 = 3.25x
x = 4 / 3.25
x = 0.12 days
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Which of the following are integer solutions to the inequality below? − 2 < x ≤ − 1
Helpppo meeeee!!!please I really need help
When a polynomial is to be divided by a linear expression and the leading coefficient (first number) must be a 1, synthetic division is used. Any polynomial equation, regardless of degree, can, for instance, be divided by x + 1, but not by x2+1.
When is the use of synthetic division prohibited?Additionally, the binomial must be of degree 1; synthetic division cannot be used to divide by a binomial such as x2 + 1. Here are the steps for performing synthetic division on a polynomial by a binomial.
What is the synthetic chemical formula?To get the quotient and the remainder, write down the coefficients and divide them by the linear factor's zero. Q(x) + (R/(x - a)) = (P(x)/(x - a)).
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Find the first four terms of the sequence given by the following. an =54 +5(n - 1), n f1, 2, 3...
Answer:
[tex]54,59,64,69[/tex]
Step-by-step explanation:
[tex]a_{n}=54+5(n-1)=54+5n-5=49+5n \\ \\ a_{1}=49+5(1)=54 \\ \\ a_2=49+5(2)=59 \\ \\ a_3=49+5(3)=64 \\ \\ a_3=49+5(4)=69[/tex]
What are the first 20 prime numbers?
First twenty prime numbers among the set of prime numbers are as follow :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
As given in the question,
Prime numbers are defined as the numbers which has only two factors one and the number itself.
Prime number does not exactly get divided by any other integer by leaving remainder equals to zero.
Prime numbers present in the number between 1 to 100 are equal to :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
As prime numbers are already written in ascending order.
First twenty prime numbers are as follow :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
Therefore, the first twenty prime numbers are given by :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
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A laboratory that uses radioactive substances received a shipment of 122 g of bismuth-210. Only 6.64 g of the bismuth-210 remained 21.0 days later. Determine the half-life of bismuth-210 algebraically using logarithms, to the nearest tenth of a day. The half-life equation is ID: A A = 4₂ (²) where A represents the amount of the substance remaining, Ao represents the initial amount of the substance, t represents the time, and h represents the time at which only half of the substance remains.
The half-life of bismuth-210 is given as follows:
5 days.
How to model an exponential function?A decaying exponential function is modeled as follows:
y = a(1 - r)^t.
In which the parameters are given as follows:
a represents the initial value.r represents the decay rate, as a decimal.For this problem, we have that:
a = 122.y(21) = 6.64.Hence the decay rate is obtained as follows:
[tex]6.64 = 122(1 - r)^{21}[/tex]
[tex](1 - r)^{21} = \frac{6.64}{122}[/tex]
[tex]((1 - r)^{21})^{\frac{1}{21}} = \left(\frac{6.64}{122}\right)^{\frac{1}{21}}[/tex]
1 - r = 0.8706
r = 1 - 0.8706
r = 0.1294.
Hence the function is given as follows:
y = a(0.8706)^t.
The half-life is the value of t for which:
y = 0.5a.
Hence:
0.5a = (0.8706)^t.
(0.8706)^t = 0.5
log((0.8706)^t) = log(0.5)
tlog(0.8706) = log(0.5)
t = log(0.5)/log(0.8706)
t = 5 days.
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Find the measure of arc GH.
The measure of arc GH is 102°.
What is inscribed angle?
In terms of geometry, an inscribed angle is the angle created when two chords intersect within a circle. It can also be described as the angle formed by two specified points on a circle at a given point on the circle. Two chords of the circle sharing an endpoint are equivalently used to define an inscribed angle.
Given:
The measure of inscribed angle ∠FHG is 39°.
We have to find the measure of arc GH.
Since,
FG = 2*m ∠FHG
FG = 2(39°)
FG = 78°
Let FH = 180°
⇒ GH = FH = FG
GH = 180° - 78°
GH = 102°
Hence, the measure of arc GH is 102°.
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