The derivative of the summation with lower limit that equal to 0 is [tex]$$y^{\prime}=\sum_{n=0}^{\infty}(-1)^n \cdot \frac{(2 n+1)}{(n !)^2} \cdot\left(\frac{x}{3}\right)^{2 n}$$[/tex].
From the given data,
The derivative of the summation from n equals 0 to infinity of the quotient of the product of negative 1 raised to the nth power and x raised to the quantity 2 times n plus 1 power and the product of 3 to the 2 times n power and the square of n factorial is [tex]$y=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2 n+1}}{3^{2 n} \cdot(n !)^2}$[/tex]
The process of finding the derivative is called differentiation. The inverse process is called anti-differentiation. Let’s find the derivative of a function y = f(x). It is the measure of the rate at which the value of y changes with respect to the change of the variable x. It is known as the derivative of the function “f”, with respect to the variable x.Derivatives can be classified into different types based on their order such as first and second order derivatives.
Derivative⇒ [tex]$y^{\prime}=\sum_{n=0}^{\infty} \frac{(-1)^n \cdot(2 n+1)(x)^{2 n}}{3^{2 n} \cdot(n !)^2}$[/tex]
⇒ [tex]$$y^{\prime}=\sum_{n=0}^{\infty}(-1)^n \cdot \frac{(2 n+1)}{(n !)^2} \cdot\left(\frac{x}{3}\right)^{2 n}$$[/tex].
Therefore, the summation with lower limit that equal to 0 is [tex]$$y^{\prime}=\sum_{n=0}^{\infty}(-1)^n \cdot \frac{(2 n+1)}{(n !)^2} \cdot\left(\frac{x}{3}\right)^{2 n}$$[/tex]
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help me please asap!!
Answer:
The length of the longer ladder is 36ft
Step-by-step explanation:
30ft/10ft = 3
12ft x 3 = 36ft
Another way to solve this is by cross multiplying
[tex]\frac{10}{12}[/tex] = [tex]\frac{30}{x}[/tex]
10x = 360
x = 36ft
Which of the following is a property of an isosceles trapezoid? Choose ALL that apply.
Four right angles
Two pairs of parallel sides
Diagonals are congruent
Diagonals are perpendicular
The property of an isosceles trapezoid is that the diagonals are congruent. That is option C.
What is an isosceles trapezoid?An isosceles trapezoid is defined as the type of quadrilateral whereby line of symmetry bisects one pair of opposite sides.
The unique characteristics of an isosceles trapezoid include the following:
Only one pair of sides are parallel. Non-parallel sides (legs) are equal in measure.The diagonals are equal in measure. The base angles are equal in measure. The opposite angles are supplementary.Therefore, the property of an isosceles trapezoid that is listed above is that Diagonals are congruent which means that they are equal.
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The properties of isosceles trapezoid are
Two pairs of parallel sides
Two pairs of parallel sides
What are features of Isosceles trapezoid?An isosceles trapezoid has left and right sides that are equal in length or identical in length, therefore the base angles are also equal.
The diagonals are identical because the lengths of the left and right sides are equal and the base angles are equal.
Additional varieties of trapezoids include
There are two right angles on the right trapezoid.It is an isosceles trapezoid because its parallel and non parallel sides are of the same length.It is a scalene trapezoid since neither its angles nor its sides are equal.Generally, all trapezoids have two parallel sides.
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PLEASE DONT IGNORE :( I NEED HELP THIS IS DUE IN A COUPLE MINUTES
Lines AB, BC, and AC have an equal slope, with a slope value of 2/3.
What is a Linear equation?A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. Sometimes, the aforementioned is referred to as a "linear equation of two variables," with y and x serving as the variables.
Given a line AC that has an x-intercept of "2"
Since,
AB is part of line AC thus the slope of AB will be the same as the slope of line AC and the Slope of line BC.
For the slope of line AB:
Since there is no Additional data given on the problem we will assume that every square is a unit square in the graph
Thus, Coordinates of A = (-5, -2)
Since Point B is the x-intercept of line AB thus its coordinates = (-2, 0)
from the formula of the slope of a line:
m = (y₂ -y₁)/(x₂ - x₁)
Thus, the slope of line AB:
m = (0+2)/-2 +5)
m = 2/3
therefore, the slope of the line AB, BC, and AC are equal and the value of the slope is 2/3.
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Factor the expression using the GCF.
49 - 28 = ?
By calculating, The GCF of the given numbers 49 and 28 is 7.
What is Algebraic expression ?
An algebraic expression is an expression that consists of numbers, variables and operators such as addition, subtraction, multiplication and division. These expressions can be used to solve equations and other mathematical problems. Algebraic expressions can also be used to represent relationships between different variables. For example, the expression 4x + 2y can be used to represent the relationship between two variables, x and y.
So lets first find the prime factorization of 28
prime factorization of 28 = 2 × 2 × 7
So lets first find the prime factorization of 49
prime factorization of 49 = 7 × 7
To find the GCF, multiply all the prime factors common to both numbers:
Therefore, The GCF of the given numbers 49 and 28 is 7.
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Which of these is a way to show that two figures are congruent
If two figures can be mapped onto one another or if all pairings of angles and side lengths are congruent, the figures are said to be congruent.
What is Triangle?
A polygon with three edges and three vertices is called a triangle. It is one of the fundamental geometric shapes. Triangle ABC is the designation for a triangle with vertices A, B, and C. In Euclidean geometry, any three points that are not collinear produce a distinct triangle and a distinct plane.
What is the condition to be congruent?
If two triangles are the same size and shape, they are said to be congruent. To establish that two triangles are congruent, not all six matching elements of either triangle must be located. There are five requirements for two triangles to be congruent, according to studies and trials. The congruence properties are SSS, SAS, ASA, AAS, and RHS.
Both triangles are said to be congruent if the three angles and three sides of one triangle match the corresponding angles and sides of the other triangle.
In Δ PQR and ΔXYZ, as shown in figure,
we can identify that PQ = XY, PR = XZ,
and QR = YZ
and ∠P = ∠X,
∠Q = ∠Y and ∠R = ∠Z.
Then we can say that Δ PQR ≅ ΔXYZ.
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Find the least positive integer m such that m^2 −m+11 is a product of at least four not necessarily distinct primes.
The least positive integer is m=132.
m(m−1)+11=N
It is clear that N must be odd and that can be divisible by the prime 11.
However, the following congruences have no solution as it is easy to prove directly by checking the few possibilities with m(m−1) not divisible by 3,5 and 7 respectively
{m(m-1)+2=0(mod 3)
{m(m-1)+1=0(mod 5)
{m(m-1)+4=0(mod 7)
Thus N=∏ with pi≥11
A necessary condition is that (solving the quadratic in m)
√4N-43=x² and from this, we need that
N≡1,3,7(mod10)
By chance, we find the minimum N=113⋅13 which corresponds to m=132
and that could have been calculated by direct trials but the next solution seems to be far from 132.
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Mrs moth picked 24 flowers.Six of them each had 9 petals,7 of them had 8 petals, 5 of them each had 3 petals, and the rest had 10 petals.How many flowers had 10 petals?
Using the subtraction operation, the number of flowers with 10 petals that Mrs. Moth picked was 6.
What is a subtraction operation?A subtraction operation is one of the four basic mathematical operations, including addition, division, and multiplication.
The subtraction operation involves the minuend, the subtrahend, and the difference.
The minuend is the number or value being subtracted from by the subtrahend. The result of a subtraction operation is known as the difference.
The total number of picked flowers = 24
The number of flowers with 9 petals = 6
The number of flowers with 8 petals = 7
The number of flowers with 3 petals = 5
The total number of flowers identified = 18 (6 + 7 + 5)
The number of flowers with 10 petals = x
x = 24 - 18
= 6
Thus, 6 flowers had 10 petals out of the 24 Mrs. Moth picked.
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What is the value or x?
Answer:
x = 25
Step-by-step explanation:
Since the given triangle is a right triangle and we are also given two sides which are opposite (height) = 24 units and adjacent (base ) = 7 units. We can apply the Pythagoras Theorem where:
[tex]\displaystyle{\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2}[/tex]
In simpler words, it is:
[tex]\displaystyle{\text{height}^2 + \text{base}^2 = \text{slope}^2}[/tex]
Substitute in given values:
[tex]\displaystyle{24^2+7^2=x^2}[/tex]
Evaluate and solve for x:
[tex]\displaystyle{576+49=x^2}\\\\\displaystyle{625=x^2}\\\\\displaystyle{\sqrt{625}=\sqrt{x^2}}\\\\\displaystyle{\pm \sqrt{625}=x}\\\\\displaystyle{\pm 25=x}[/tex]
However, x is a length value and length's value cannot be negative so x can only be 25.
Hence, x = 25 units
The sides of the rectangle in Fig. 14.4 are given in cm. 4x - y, 4x + 3, x + 6y, 3x + 1 Fig. 14.4 Find x and y and the area of the rectangle.
Variable x and y for the rectangle is given by
x = (3x + 3)/6 + 1
y = (3x + 3)/6
How to find x and y and the area o the rectangle?Yes that's right. You have to equate the expressions for widths and for the lengths, since left and right widths (and lengths) have the same dimensions.
You may rewrite the equation for y=3x+3÷6 as
y = (3x + 3)/6
And now you have an expression for y, we can substitute it to the other equation you got:
x = y + 1
x = (3x + 3)/6 + 1
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help someone please!!!!!
62.8363 cm is the total circumference of the given circle.
What is Fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a correct fraction is smaller than the denominator.
Given a circle that has an arc of 165 degrees and also given that arc has a circumference of 28.8cm
Circumference of 165 Degrees arc = 28.8cm
Circumference of 1 Degree arc = 28.8/165
Circumference of 1 Degree arc =0.17454
Since the total angle in a circle is 360 degrees
Thus from the fraction,
Circumference of 360 Degrees arc = 0.17454*360
Circumference of 365 Degrees arc = 62.8363 cm
Therefore, the total circumference of the given circle is 62.8363 cm.
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Find the inverse of y=2x²+3x+5
According to the given information in the question the inverse of y=2x²+3x+5 is x = (y - 5 - 3x)/2.
What does the math term "inverse" mean?Inverse operations, such as subtraction and addition multiplication and division, are pairs of mathematical operations where one operation reverses the effect of the other. The opposite of a number, or "inverse," is what is meant by the phrase. i.e., x - 1 = 1 / x.
Does inverse imply a negative value?Negative correlation and inverse relationships are interchangeable terms. Any two or more variables that consistently move in the opposing directions can be described by both. The outcome of measuring an inverse association will result in a negative sign.
To find the inverse of a function, you first need to switch the x and y variables, and then solve for y.
So, the inverse of y=2x²+3x+5 is x = (y - 5 - 3x)/2
Note that the inverse of a function is only possible if the function is a one-to-one function.
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Question 3 Multiple Choice Worth 2 points)
(04.05 MC)
B
A random sample of 75 juniors are asked whether they plan to attend Homecoming. Of these, 62 juniors said they will attend. What is the margin of error at 90% confidence and
its interpretation?
0.072; 90% of the time we will capture the true proportion of juniors who will attend Homecoming within 0.072 of the population proportion
0.072; we are 90% confident that the true proportion of juniors who will attend Homecoming is within 0.072 of the sample proportion
0.086; we are 90% confident that the true proportion of juniors who will attend Homecoming is within 0.086 of the sample proportion
0.086; 90% of the time we will capture the true proportion of juniors who will attend Homecoming within 0.086 of the population proportion
The margin of error at a 90% confidence level for this sample is approximately 0.086.
What is meant by margin of error?
In statistics, the margin of error is the level of inaccuracy in the findings of surveys using random samples.A wider margin of error in statistics denotes a lower chance of relying on a survey's or poll's findings, i.e., a lower level of confidence in the findings' ability to accurately reflect a population.When 62 out of 75 juniors in a random sample were asked if they would be attending homecoming, the margin of error at 90% confidence was roughly 0.086.This suggests that the true proportion of juniors who will attend homecoming is within 0.086 of the sample proportion, with a 90% confidence level.The standard error of the sample proportion is calculated using the following formula:
Standard error = √((sample proportion * (1 - sample proportion)) / sample size)
Plugging in the values for the sample proportion (62/75) and sample size (75), you get a standard error of approximately 0.072.
Multiplying the standard error by the critical value gives the margin of error:
The margin of error = critical value * standard error
= 1.645 * 0.072
= 0.086
Hence, the margin of error at a 90% confidence level for this sample is approximately 0.086.
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Mr. Stans bought 4.65 pounds of dog food, 2.28 pounds of cat food, and 1.4 pounds of rabbit food. How many pounds of pet food did Mr. Stans buy?
A. 7.33
B. 7.07
C.6.91
D.8.33
Below is the seating chart of 50 students in a STP226 class. You are tasked with selecting a sample of 5 students from the class in order to accurately estimate the average Exam 1 score for the entire class.
Front Desk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Post
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Post
Door
1) Considering the STP 226 class seating chart above, suppose you randomly selected a student from each row: students in seats 3, 18, 21, 34, and 50 were selected for your sample.
a) What was the sampling method used?
b) Is this sample going to be representative or do you see any issues with the sample? Answer briefly.
The sampling method used in this case is random sampling and the sample is representative.
What is a sample and what makes a sample to be representative?A sample is a small proportion of individuals taken from the general population to be studied. For a sample to be representative it is expected that all the individuals represent the population, for example, if the population includes people of different ages the sample should include this too.
Is the sample representative?The random sample selected is representative because the seats of the students selected are distributed within the class rather than in only one row or line.
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y÷15= 6 R 8
314÷y= 18 R8
What is y how can I solve this on my own for other problems?
Step-by-step explanation:
y÷15 or y/15 = 6
let's ignore the remainder for a moment.
y/15 = 6
y = 6×15 = 90
now, what does "remainder" mean ?
it means after a clean division there is a result, and there is still this additional remainder left.
the actual result of
a/b is
a/b = c + remainder/b
where c is an integer, and remainder is also an integer.
so, a = b×c + b×remainder/b = b×c + remainder
therefore,
y = 6×15 + 8 = 98
to check
98/15 = 6 remainder 8
so, it is correct
314÷y = 18 remainder 8
314 = 18y + 8
306 = 18y
y = 306/18 = 17
check
314 / 17 = 18 remainder 8
so, it is correct.
The following is a list of various costs of producing sweatshirts. Classify each cost as either a variable, fixed, or mixed cost for units produced and sold.a. Electricity costs of $0.025 per kilowatt-hour.b. Warehouse rent of $6,000 per month plus $0.50 per square foot of storage used.c. Thread.d. Lubricants used to oil machinery.e. Janitorial costs of $2,000 per month.f. Advertising costs of $10,000 per month.g. Sales salaries. Color dyes for producing different colors of sweatshirts.h. Salary of the production supervisor.i. Straight-line depreciation on sewing machines.j. Patterns for different designs. Patterns typically last many years before being replaced.k. Maintenance cost with sewing machine company. The cost is $2,000 per year plus $0.001 for each machine hour of use.l. Property taxes on factory, building, and equipment.m. Cotton and polyester cloth.n. Hourly wages of sewing machine operators.Variable Cost:Variable cost is the cost incurred in the production of goods or services and changes with the level of output produced by an organization. It includes costs directly linked with production, like material cost.
Following are the answers to all the asked questions for classifying types of cost.
a. Electricity costs of $0.025 per kilowatt-hour - variable cost
b. Warehouse rent of $6,000 per month plus $0.50 per square foot of storage used - mixed cost
c. Thread - variable cost
d. Lubricants used to oil machinery - variable cost
e. Janitorial costs of $2,000 per month - fixed cost
f. Advertising costs of $10,000 per month - fixed cost
g. Sales salaries - fixed cost
h. Color dyes for producing different colors of sweatshirts - variable cost
i. Salary of the production supervisor - fixed cost
j. Patterns for different designs. Patterns typically last many years before being replaced - fixed cost
k. Maintenance cost with sewing machine company. The cost is $2,000 per year plus $0.001 for each machine hour of use - mixed cost
l. Property taxes on factory, building, and equipment - fixed cost
m. Cotton and polyester cloth - variable cost
n. Hourly wages of sewing machine operators - variable cost
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When calculating simple interest, what must you do if you want to invest for months or weeks instead of years? a. Simple interest only pays yearly, so you should find another investment. b. The simple interest calculation does not use those units, so you should just calculate as normal. c. The units have changed, so the interest formula switches to I·t=P·r. d. Interest rates are annual, so you must convert the units into years. Please select the best answer from the choices provided A B C D
When calculating simple interest, if we want to invest for months or weeks instead of years we must convert the interest rate units into years.
Option D is the correct answer
What is simple interest?It is the interest calculated on the principal at a given rate for a time period.
We have,
The formula for simple interest.
SI = PRT/100
P = principa
R = rate
T = time
Now,
Time is in years when we are calculating the simple interest.
Example:
If time is in months.
T = 4 months
T = 4/12 = 1/3 years.
Thus,
We must convert the units into years.
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Una granja tiene q
cerdos, 2 caballos.
y 4 vacas. ¿Cuantos
animales hay ahi?
animales
Answer: letra “q” no es un número
Step-by-step explanation:
Debt payments of $2,100 and $1,950 are due in six months and nine months, respectively. What single payment is required to settle both debts in one month? Assume a simple interest rate of 6.30% p.a. and use one month from now as the focal date.
The single payment that is required to settle both debts is given as follows:
$3,897.9.
How to obtain the balance?The balance of an account after t years, using simple interest, that is, a single compounding per year, is given by the equation presented as follows:
A(t) = P(1 + rt).
In which:
P is the principal.r is the rate, as a decimal.t is the time, in years.For the $2,100 payment in 6 months = 0.5 years, the principal is obtained as follows:
(1 + 0.5 x 0.063) P = 2100
P = 2100/(1 + 0.5 x 0.063)
P = $2,035.88.
For the $1,950 payment in 9 months = 0.75 years, the principal is obtained as follows:
(1 + 0.75 x 0.063) P = 1950
P = 1950/(1 + 0.75 x 0.063)
P = $1,862.02.
Hence the single payment is obtained as follows:
2035.88 + 1862.02 = $3,897.9.
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what size bands would result from cutting the following plasmid using hind iii, eagi and smai? the plasmid has a total of 5541 base pairs
The size of the bands that would result from cutting the plasmid with Hind III, Eco RI and Sma I would be 5530
The size of bands resulting from cutting a plasmid with Hind III, EagI and SmaI can be determined using the following formula:
Size of Band = Plasmid Size - (Hind III + EagI + SmaI)
The size of the bands that would result from cutting the plasmid with Hind III, Eco RI and Sma I can be calculated using the following formula:
For this example, the plasmid has a size of 5541 base pairs. After cutting with Hind III, EagI and SmaI, the size of the bands will be:
Size of Band = 5541 - (6 + 4 + 4)
Size of Band = 5527 base pairs
Band size = Total base pairs - [size of sites (Hind III + Eco RI + Sma I)] + 1
In this case, the band size would be calculated as follows:
Band size = 5541 - (6 + 6 + 4) + 1
Band size = 5530 bp
Therefore, the size of the bands that would result from cutting the plasmid with Hind III, Eco RI and Sma I would be 5530
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Complete question:
What size bands would result from cutting the following plasmid using Hind III, EcoRI and SmaI? The plasmid has a total of 5541 base pairs.
a 5 ich photo was enlarged to make a poster as shown. if the dimensions of the poster are (x^2 - 6) inches by (x^ + 12) inches, what is the area of the poster
The dimensions of the poster are given as (x^2 - 6) inches by (x^2 + 12) inches.
The area of the poster is given by the product of these two dimensions:
Area = (x^2 - 6) inches * (x^2 + 12) inches
Area = x^4 - 6x^2 + 12x^2 - 72
Area = x^4 + 6x^2 - 72 inches^2
So, the area of the poster is x^4 + 6x^2 - 72 inches^2.
Consider the graph of the function f(x) shown above. It consists of two line segments, an arc of a semi-circle centered at (-4,0) with radius 3, and an arc of a quarter-circle centered at (1,0) with radius 2. Find the value of [tex]\int\limits^4_ {-7} \, f(x) dx[/tex]
The value of the integral is given as follows:
C. -7π/2.
How to obtain the result of the integral?The result of the integral is given by the total area of the figure.
From x = -7 to x = -1, the figure is a semi-circle of radius 3, below the y-axis, hence it's area is given as follows:
A1 = -0.5π x 3²
A1 = -4.5π.
From x = -1 to x = 1, with the line, the figure is a right triangle with sides 2 and 2, hence the area is given as follows:
A2 = 0.5 x 2 x 2
A2 = 2.
From x = 1 to x = 3, the figure is quarter circle with radius 2, hence the area is given as follows:
A3 = 0.25π x 2²
A3 = π.
From x = 3 to x = 4, the figure is a right triangle with sides 1 and 4, below the y-axis, hence the area is given as follows:
A4 = 0.5 x 1 x 4
A4 = -2.
Hence the total area, representing the integral, is given as follows:
A = A1 + A2 + A3 + A4
A = -4.5π + 2 + π - 2
A = -3.5π
A = -7π/2.
(the result of the integral is negative but the area is positive).
Meaning that the correct option is given by option C.
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1. A community garden has an area of 10 square yards. The community wants to divide it into plots that each
have an area of 1/2 square yard. How many plots can the garden be split into?
The garden can split into 20 plots.
What is division?The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
A community garden has an area of 10 square yards.
The community wants to divide it into plots that each have an area of 1/2 square yard.
The number of community garden,
= 10/(1/2)
= 20
Therefore, the number of required plots is 20.
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Describe the possible lengths of the third side of the triangle given that the lengths of the other two sides are 5 inches and 12 inches.
The possible length of the third side of a triangle, say x, with the other two sides are 5 inches and 12 inches. is
7 ≥ x ≥ 17
How to determine the possible length of third sideThe length of the third side is within the range of
sum of the two sidesdifference of the two sidesThe sum of the two ides is the highest possible length of the third side
= 12 inches + 5 inches
= 17 inches
The difference of the two ides is the lowest possible length of the third side
= 12 inches - 5 inches
= 7 inches
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Consider the numerical expression: 3 + 6 × (5 + 4) ÷ 3 − 7.
Answer:
14
Step-by-step explanation:
When solving something like this use Pemdas
hope this help in some way
How to solve (2n+7)=(3n+3)
Answer:
n = 4
Step-by-step explanation:
Let's put all of our variables on one side, and all of our non-variable numbers on the other. (I like to do this so that the variable isn't negative. You'll still get the same answer.)
So we can minus 2n on both sides, giving us 7 = n + 3.
Then, let's do our non-variables and subtract 3 from each side. This gives us 4 = n or n = 4.
Therefore, n = 4.
Sine/Cosine of Complementary Angles Solve for x
In a right triangle, sin (8x-9) = cos (x+6) Solve for X Round your answer to the nearest tenth
Using complementary angles, the value of x = 10.3
What are complementary angles?Complementary angles are angles in which the sum add up to 90°.
How to find the value of x in the given right triangle?Since in a right triangle, sin(8x - 9) = cos(x + 6). To solve the problem, we proceed as follows.
sin(8x - 9) = cos(x + 6).
Given that cosФ = sin(90 - Ф), we have that
cos(x + 6) = sin[90 - (x + 6)]
So, susbstituting this into the equation, we have that
sin(8x - 9) = cos(x + 6)
sin(8x - 9) = sin[90 - (x + 6)]
This implies that
8x - 9 = 90 - (x + 6)
8x - 9 = 90 - x - 6
8x - 9 = 84 - x
Adding x to both sides of the equation, we have
8x + x - 9 = 84 - x + x
9x - 9 = 84 + 0
9x - 9 = 84
Adding 9 to both sides of the equation, we have that
9x - 9 + 9 = 84 + 9
9x + 0 = 93
9x = 93
x = 93/9
x = 10.33
x ≅ 10.3
So, the value of x = 10.3
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A study found that a student's GPA, g, is related to the number of hours worked each week, h, by the equation g=−0.0007h2+0.018h+3.02. Estimate the number of hours worked each week for a student with a GPA of 2.72.
After solving the given equation we know that students worked for 17 hours a week.
What are equations?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
So, the given equation is:
g=−0.0007h+0.018h+3.02
Now, we know that g is 2.72, so we can calculate for h as follows:
2.72=−0.0007h+0.018h+3.02
2.72 = -0.0173h + 3.02
- 0.3 = -0.0173h
h = -0.3/-0.0173
h = 17.34
Rounding off: 17 hours
Therefore, after solving the given equation we know that students worked for 17 hours a week.
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Suppose we want to choose letters, without replacement, from distinct letters. (a) If the order of the choices does not matter, how many ways can this be done? (b) If the order of the choices matters, how many ways can this be done?
The number of ways to choose 6 letters from a set of 13 is given as follows:
a) Order does not matter: 1716.
b) Order matters: 1,235,520.
When to use permutation and when to use combination?A combination is used when the order does not matter, hence, for item a, the number of ways is obtained as follows:
C(13,6) = 13!/(6!7!) = 1716.
A permutation is used when the order matters, hence, for item b, the number of ways is obtained as follows:
P(13,6) = 13!/7! = 1,235,520.
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Find all the solutions, including imaginary solutions if they exist, of the following equation. x^4+17x^2+16=0
The solution to the equation is i , -i , 4i , -4i
What is Quadratic Equation?A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.
The roots of the quadratic equations are
x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
where ( b² - 4ac ) is the discriminant
when ( b² - 4ac ) is positive, we get two real solutions
when discriminant is zero we get just one real solution (both answers are the same)
when discriminant is negative we get a pair of complex solutions
Given data ,
Let the equation be represented as A
Now , the value of A is
x⁴ + 17x² + 16 = 0 be equation (1)
On simplifying the equation , we get
Substitute the value of x² = u
u² + 17u + 16 = 0
The roots of the quadratic equations are
u = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
u = [ -17 ± √ ( 289 - 4 ( 1 ) ( 16 ) ) ] / 2
On further simplification , we get
u = [ -17 ± √ ( 289 - 64 ) ] / 2
u = ( -17 ± √ 225 ) / 2
u = ( -17 ± 15 ) / 2
The two values of u are
u = ( -17 - 15 ) / 2
u = -16
And , u = ( -17 + 15 ) / 2
u = -1
Substitute the value of x² = u , we get
x² = -16
Taking square roots on both sides of the equation , we get
x = ± √ ( -16 )
x = ± 4i
And ,
x² = -1
x = ± i
So , the values of x are i , -i , -4i and 4i
Hence , the equations are solved
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