The given series are in conditionally convergent
To determine whether the given series is absolutely convergent, conditionally convergent, or divergent, we will use the Comparison Test.
Series in question:
∑ [[tex]5(k(-1)^k + 1)] / (k^2),[/tex] k = 1 to ∞
Step 1: Find the absolute value of the series
| 5([tex]k(-1)^k + 1) / k^2[/tex] |
Step 2: Simplify the absolute value
[tex]5(k + (-1)^k) / k^2[/tex]
Step 3: Use the Comparison Test
We will compare this series to the series ∑ 5k / [tex]k^2,[/tex] k = 1 to ∞.
Since [tex](-1)^k[/tex] is always either 1 or -1, we know that [tex]5(k + (-1)^k) / k^2 \leq 5k / k^2.[/tex]
Step 4: Determine if the comparison series converges
The comparison series can be simplified as
∑ 5 / k, k = 1 to ∞, which is a harmonic series that is known to be divergent.
Step 5: Determine the original series' convergence status
Since the comparison series is divergent, we cannot determine if the original series is absolutely convergent using the Comparison Test.
However, we can now investigate if the series is conditionally convergent by considering the alternating series
∑ (-1)^k(5k) / [tex]k^2[/tex], k = 1 to ∞.
Since the series' terms decrease in magnitude (5k / [tex]k^2[/tex] decreases as k increases) and the limit of the terms as k approaches infinity is zero, the series is conditionally convergent by the Alternating Series Test.
In conclusion, the given series is conditionally convergent.
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the measure of an angle is 156°. what is the measure of its supplementary angle
Answer:
24°
Step-by-step explanation:
Supplementary angles: 2 angles that add up to 180°.
We are given that one angle is 156°, so we can write an equation:
180=156+x
subtract both sides by 156
24=x
So, the measure of the supplementary angle is 24°.
Hope this helps!
Which function is increasing and has a domain of (1,infiniti)?
A. F(x) = log(x - 1) + 2
B. F(x) = -log(x - 2) + 1
C. F(x) = -log(x - 1) + 2
D. F(x) = log(x - 2) + 1
The function that is increasing and has a domain of (1, infinity) is A. F(x) = log(x - 1) + 2.
To determine the increasing function with the specified domain, let's analyze each option:
A. F(x) = log(x - 1) + 2
This function is increasing because the logarithm of a positive number is always increasing. The domain is (1, infinity), which matches the requirement.
B. F(x) = -log(x - 2) + 1
This function is decreasing because the negative sign in front of the logarithm inverts the increase. The domain is (2, infinity), which does not match the requirement.
C. F(x) = -log(x - 1) + 2
This function is also decreasing because of the negative sign in front of the logarithm. The domain is (1, infinity), which matches the requirement, but the function is not increasing.
D. F(x) = log(x - 2) + 1
This function is increasing because the logarithm of a positive number is always increasing. However, the domain is (2, infinity), which does not match the requirement.
The function that is increasing and has a domain of (1, infinity) is A. F(x) = log(x - 1) + 2.
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MarÃa is shopping for party supplies. She finds that paper plates come in packages of 8, napkins come in packages of 10, and paper cups come in packages of 12. What is the least number of packages of plates, napkins, and cups she has to buy so that she has the same number of each item for her party?
The least number of packages of plates, napkins, and cups Maria has to buy is 15 packages .
To find the least number of packages, we need to find the least common multiple (LCM) of the three package sizes: 8 (plates), 10 (napkins), and 12 (cups).
The prime factors of 8 are 2x2x2, of 10 are 2x5, and of 12 are 2x2x3. To find the LCM, we multiply the highest powers of each prime factor: 2³x3x5 = 120.
This means Maria needs 120 of each item. She will buy 120/8 = 15 packages of plates, 120/10 = 12 packages of napkins, and 120/12 = 10 packages of cups.
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Newton’s Method!!!!!!
The approximate value of x using the newton method is 0.7
Calculating the value of x using the newton methodFrom the question, we have the following expression that can be used in our computation:
[tex]\frac{x}{x^2+1}-\sqrt{1-x}[/tex]
Also, we have the function f(x) to be
[tex]f(x) = x(x^2+1)^{-1} -\sqrt{1-x}[/tex]
And we have the differentiated function to be
[tex]f'(x) = \frac{1}{x^2+1} - \frac{2x^2}{(x^2 + 1)^2} + \frac{1}{2\sqrt{1-x}}[/tex]
The value of x using the newton method is given as
[tex]x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}[/tex]
Set [tex]x_{n-1}[/tex] = 0
So, we have
x₁ = 0 - -1/1.5 = 0.67
x₂ = 0.67 - undefined = undefined
So, we have
x₁ = 0.67
When approximated, we have
x = 0.7
This means that the value of x using the newton method is 0.7
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Can someone explain ._.
The mark-up value percentage is 25 %
Given data ,
The markup amount is the selling price minus the cost price, so:
Markup = $8630 - $6900 = $1730
The markup percentage is the markup amount divided by the cost price, expressed as a percentage:
Markup percentage = (Markup / Cost price) x 100%
Markup percentage = ($1730 / $6900) x 100%
Markup percentage = 0.25 x 100%
Markup percentage = 25%
Hence , the markup percentage is 25%
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Which of the data sets below have striking deviations? Select all that apply.
A) 65, 68, 61, 63, 71
B) 99, 95, 93, 97, 98
C) 22, 83, 85, 88, 91
D) 45, 47, 43, 45, 97
E) 72, 78, 71, 67, 35
What is the first quartile (Q1) of the data set? 51, 42, 46, 53, 66, 70, 90, 79
Answer:47.25
Step-by-step explanation:
Answer:
48.5
Step-by-step explanation:
To find the first quartile (Q1) of the data set, we need to arrange the numbers in ascending order:
42, 46, 51, 53, 66, 70, 79, 90
Q1 is the median of the lower half of the data set. Since we have 8 data points, the lower half will be the first four numbers.
42, 46, 51, 53
To find the median of these numbers, we take the average of the two middle numbers:
(Q1) = (46 + 51) / 2 = 48.5
Therefore, the first quartile (Q1) of the data set is 48.5.
For what primary reason have many present-day African nations struggled to unite their people?
F.
Because many African peoples have strong tribal ties
G.
Because much of the African population remains loyal to former colonial powers
'H.
Because the African people refuse to accept centralized authority
J. Because many Africans seek to migrate from the continent
The primary reason many present-day African nations struggled to unite their people is F. Because many African peoples have strong tribal ties.
The strong tribal identities and loyalties that exist among various ethnic groups inside African states today are one of the main causes of the effort to bring people together. Through colonisation, many African nations were developed, and many of them contained numerous groups with various cultural, linguistic, and ethnic origins.
Therefore, as a result, it has been challenging to forge a strong sense of national identity among many individuals who still identify primarily with their own tribe or ethnic group. This has occasionally resulted in disputes and hostilities amongst various diverse groups, impeding efforts to create a community that is more unified.
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!!I need help seriouslyyy!!
The average cost per day for the four service is $3.23 per day
What is average cost?Average cost refers to the per-unit cost of production, which is calculated by dividing the total cost of production by the total number of units produced.
Therefore average cost = total cost/ number of unit
total cost = $108
average cost for the three services = $108/3
= $36
total average cost = $36+$54.30
= $90.30
therefore average cost for a day will be average cost for a month over 28day i.e 7days ×4
= 90.30/28
= $3.23 per day
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n^2 - 5n + 6, n^2 - 4n+ 4 I need help, i know the answer may possibly be (n-3)(n-2)^2 factor it, an di need steps.
The simplified expression is (n - 3) / (n - 2).
What is the simplification of the expression?The expression is simplified as follows;
(n² - 5n + 6) / (n² - 4n + 4)
n² - 5n + 6 can be factored as (n - 2) (n - 3)
n² - 4n + 4 can be factored as (n - 2) (n - 2)
Therefore, the expression becomes:
[(n - 2) (n - 3)] / [(n - 2) (n - 2)]
We can cancel the (n - 2) factor in the numerator and denominator, leaving us with:
(n - 3) / (n - 2)
So the simplified expression is (n - 3) / (n - 2).
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list the elements of U
Find the perimeter of a square that has a side length of 4.3x + 2 inches.
The calculated perimeter of the square from the side length is 17.2x + 8
Finding the perimeter of a square from the side lengthFrom the question, we have the following parameters that can be used in our computation:
A square that has a side length of 4.3x + 2 inches.
Using the above as a guide, we have the following:
Perimeter = 4 * side length
Substitute the known values in the above equation, so, we have the following representation
Perimeter = 4 * (4.3x + 2)
When teh brackets are opened, we have
Perimeter = 17.2x + 8
Hence, the perimeter is 17.2x + 8
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39 A sample of a substance with an initial mass of 963 grams is decaying
at a rate of 27% per hour.
Create a function that can be used to find y, the mass of the
substance in grams remaining after x hours.
Record your answer in the space provided.
A function that can be used to find y, the mass of the substance in grams remaining after x hours is [tex]P(x) = 963(0.63)^x[/tex]
How to create a function that can be used to find the mass of the substance?In Mathematics and Statistics, a population or substance that decreases at a specific period of time represent an exponential decay. This ultimately implies that, a mathematical model for any population or substance that decreases by r percent per unit of time is an exponential equation of this form:
[tex]P(x) = I(1 - r)^x[/tex]
Where:
P(x) represents the total mass or population.x represents the time or number of years.I represents the initial value of the substance.r represents the decay rate.By substituting given parameters into the , we have the following:
[tex]P(x) = I(1 - r)^x\\\\P(x) = 963(1 - 0.27)^x\\\\P(x) = 963(0.63)^x[/tex]
In conclusion, we can reasonably infer and logically deduce that the decay rate is equal to 63%.
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If k= ∫ from zero to π/2 of sec²(x/k) dx, find k where k>0.
The value of k = 2
If k= ∫ from zero to π/2 of sec²(x/k) dx, what is value of k?Let u = x/k, then du/dx = 1/k and dx = k du.Substituting into the integral:
k ∫₀^(π/2k) sec²(u) du
= k [tan(u)]₀^(π/2k)
= k [tan(π/2k) - tan(0)]
= k [∞ - 0]
= ∞
This means that the integral diverges unless k = 0.
However, if we instead use the identity sec²(x) = 1 + tan²(x), we can rewrite the integral as:∫₀^(π/2k) sec²(x/k) dx
= ∫₀^(π/2k) (1 + tan²(x/k)) dx
= [x + k tan(x/k)]₀^(π/2k)
= π/2
So we have:
π/2 = [π/2k + k tan(π/2k)] - [0 + k tan(0)]
= π/2k + k tan(π/2k)
Multiplying through by k:
π/2 = π/2 + k² tan(π/2k)
Subtracting π/2 from both sides:
0 = k² tan(π/2k)
The only way for this equation to hold for k > 0 is if tan(π/2k) = 0. This occurs when π/2k is an integer multiple of π/2, i.e., when k is an even integer.
Therefore, the value of k that satisfies the original integral is k = 2.
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Select the correct answer from each drop-down menu.
The three vertices of a triangle drawn on a complex plane are represented by 0 + 0i, 4 + 0i, and 0+ 3i.
The length of the hypotenuse is
units, and the area of the triangle is
square units. (Hint: Use the Pythagorean theorem.)
The area of the triangle is (a-6, b-12) square units 6 sq unit.
What is the triangle?A triangle is described as a polygon with three sides having three vertices. The angle formed inside the triangle is equal to 180 degrees.
This means that the sum of the interior angles of a triangle is equal to 180°
Now that we have the points they make a 3-4-5 triangle.
The two legs are 3 and 4, so the hypotenuse has to be 5.
We could also use the Pythagorean theorem a² + b² = c² 3² + 4² = c² 25 = c² c = 5
We then calculate the area
Area =1/2 b*h
Area = 1/2(3*4)
Area = 6 sq unit
The area of the triangle is (a-6, b-12) square units 6 sq unit.
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Find the slope from the data below
Answer:
m = -4
Step-by-step explanation:
Slope = rise/run or (y2 - y1) / (x2 - x1)
Pick 2 points (-3,12) (-2,8)
We see the y decrease by -4, and the x increase by 1, so the slope is
m = -4
So, the slope of the line representing by the data is -4.
Analyzing Solution Sets to Linear Equations with the Variable on Both Sides
2x + 5 = 3 + 2(x + 1)
Answer: it will need to be rewritten so that the variable is only on one side of the equation
Step-by-step explanation:f the equation contains fractions, you may elect to multiply both sides of the equation by the least common denominator
A particle moves along the x-axis so that its position at time t>0 is given by
X(t) = (t^2 - 9)/(3t^2 + 8)
A) show that the velocity of the particle at time this given by v(t) = 70t/(3t^2 + 8)^2
B) is the particle moving toward the origin or away from the originator time t = 2? Give a reason for your answer.
C) The acceleration of the particle is given by £(t). Write an expression for £(t), and find the value of £(2).
D) What position does the particle approach as t approaches infinity?
The velocity of the particle at time t is given by [tex]v(t) = 70t/(3t^2 + 8)^2.[/tex]
The expression for the acceleration of the particle is a(t) [tex]= (210t^2 + 1120)/(3t^2+8)^3.[/tex]
A) To find the velocity of the particle, we need to take the derivative of its position with respect to time:
[tex]X(t) = (t^2 - 9)/(3t^2 + 8)[/tex]
[tex]v(t) = dX/dt = [(2t)(3t^2+8) - (t^2-9)(6t)]/(3t^2+8)^2[/tex]
[tex]v(t) = (6t^3 + 16t - 6t^3 + 54t)/(3t^2 + 8)^2[/tex]
[tex]v(t) = 70t/(3t^2 + 8)^2[/tex]
Therefore, the velocity of the particle at time t is given by[tex]v(t) = 70t/(3t^2 + 8)^2.[/tex]
B) To determine whether the particle is moving toward or away from the origin at time t = 2, we need to examine the sign of the velocity v(2). Plugging t = 2 into the expression for v(t), we get:
[tex]v(2) = 70(2)/((3(2)^2 + 8)^2) = 280/169[/tex]
Since v(2) is positive, the particle is moving away from the origin at time t = 2.
C) The acceleration of the particle is given by the derivative of its velocity with respect to time:
[tex]v(t) = 70t/(3t^2 + 8)^2[/tex]
[tex]a(t) = dv/dt = (70(3t^2+8)^2 - 2(70t)(2t)(3t^2+8))/(3t^2+8)^4[/tex]
[tex]a(t) = (210t^2 + 1120)/(3t^2+8)^3[/tex]
Therefore, the expression for the acceleration of the particle is [tex]a(t) = (210t^2 + 1120)/(3t^2+8)^3[/tex]. To find the value of a(2), we plug in [tex]t = 2:a(2) = (210(2)^2 + 1120)/(3(2)^2+8)^3 = 175/677[/tex]
D) To find the position that the particle approaches as t approaches infinity, we examine the behavior of X(t) as t gets very large. We can do this by looking at the leading term of the numerator and denominator of X(t) as t approaches infinity:
[tex]X(t) = (t^2 - 9)/(3t^2 + 8)[/tex]
As t approaches infinity, the numerator is dominated by the t^2 term, and the denominator is dominated by the 3t^2 term. Therefore, as t approaches infinity, X(t) approaches:
[tex]X(infinity) = t^2/3t^2 = 1/3[/tex]
So the particle approaches the point x = 1/3 as t approaches infinity.
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For 50 points can you help me with this I really need help.
The coordinates of the points in the table indicates that the coordinates of the image following the reflections, can be presented on the coordinate plane as shown in the attached graph created with MS Excel.
What is a reflection transformation?A reflection transformation is one in which the mirror image of the preimage is created across a line of reflection.
The coordinate points of the image after the specified reflections can be presented as follows;
Original [tex]{}[/tex]Reflection across the x-axis Reflection across the y-axis y = -x
(0, 15) [tex]{}[/tex] (0, -15) (0, 15) (-15, 0)
(1, 15) [tex]{}[/tex] (1, -15) (-1, 15) (-15, -1)
(1, 13) [tex]{}[/tex] (1, -13) (-1, 13) (-13, -1)
(3, 15) [tex]{}[/tex] (3, -15) (-3, 15) (-15, -3)
(3, 12) [tex]{}[/tex] (3, -12) (-3, 12) (-12, -3)
(1, 10) [tex]{}[/tex] (1, -10) (-1, 10) (-10, -1)
(1, 8) [tex]{}[/tex] (1, -8) (-1, 8) (-8, -1)
(3, 10) [tex]{}[/tex](3, -10) (-3, 10) (-10, -3)
(3, 7) [tex]{}[/tex] (3, -7) (-3, 7) (-7, -3)
(1, 5) [tex]{}[/tex] (1, -5) (-1, 5) (-5, -1)
(1, 2) [tex]{}[/tex] (1, -2) (-1, 2) (-2, -1)
(4, 4) [tex]{}[/tex] (4, -4) (-4, 4) (-4, -4)
(5, 7) [tex]{}[/tex] (-5, 7) (5, -7) (-7, -5)
(7, 8) [tex]{}[/tex] (7, -8) (-7, 8) (-8, -7)
(6, 5) [tex]{}[/tex] (6, -5) (-6, 5) (-5, -6)
(8, 6) [tex]{}[/tex] (8, -6) (-8, 6) (-6, -8)
(9, 9) [tex]{}[/tex] (9, -9) (-9, 9) (-9, -9)
(11, 10) [tex]{}[/tex] (11, -10) (-11, 10) (-10, -11)
(10, 7) [tex]{}[/tex] (10, -7) (-10, 7) (-7, -10)
(12, 8) [tex]{}[/tex] (12, -8) (-12, 8) (-8, -12)
(13, 6) [tex]{}[/tex] (13, -6) (-13, 6) (-6, -13)
(11, 5) [tex]{}[/tex] (11, -5) (-11, 5) (-5, -11)
(14, 4) [tex]{}[/tex] (14, -4) [tex]{}[/tex] (-14, 4) (-4, -14)
(12, 3) [tex]{}[/tex] (12, -3) (-12, 3) (-3, -12)
(9, 4) [tex]{}[/tex] (9, -4) (-9, 4) (-4, -9)
(7, 3) [tex]{}[/tex] (7, -3) (-7, 3) (-3, -7)
(10, 2) [tex]{}[/tex] (10, -2) (-10, 2) (-2, -10)
(8, 1) [tex]{}[/tex] (8, -1) (-8, 1) (-1, -8)
(5, 2) [tex]{}[/tex] (5, -2) (-5, 2) (-2, -5)
(2, 0) [tex]{}[/tex] (2, 0) (-2, 0) (0, -2)
The above coordinate points can be used to plot the graphs showing the image of the points following the specified reflections across the x-, y-, and y = -x, axis.
Please find attached the required graph of the coordinate points following the reflection transformations, created with MS Excel.
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Can you guys help me on this algebra work? thankscompare b(x) = 6x^2 to f(x)compare m(x) = -1/3x^2 - 4
b(x) = 6x² and m(x) = -1/3x²- 4 have similarities and differences as quadratic functions with different coefficients and signs.
How do b(x) and m(x) compare?We can look at the similarities and differences to compare b(x) = 6x² and m(x) = -1/3x² - 4
Similarities:Both b(x) and m(x) are quadratic functions, which means they have an x² term.
They both have a constant term, with b(x) having a constant of 0 and m(x) having a constant of -4.
Differences:The coefficients of the x² term are different: b(x) has a coefficient of 6, while m(x) has a coefficient of -1/3.
The signs of the coefficients are different: b(x) has a positive coefficient, while m(x) has a negative coefficient.
Overall, b(x) and m(x) have some similarities in their form as quadratic functions, but their coefficients and signs are different, which means they will have different shapes and behaviors.
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Helpppppppppppppppppppppppp
Answer:
Point in the original figure: (5,4)
Point in the final figure: (0,-2)
Step-by-step explanation:
hope this works and helps! :)
A 60 foot tall building casts a 20 foot. Shadow. Use the principles of similar triangles to determine the length of a shadow cast by a 5 foot 6 inch student
The length of the shadow cast by the 5 foot 6 inch student is approximately 1.83 feet.
To solve this problem, we'll set up a proportion using the principles of similar triangles. The two triangles in this case are the building and its shadow, and the student and their shadow.
Convert the height of the student to feet. 5 feet 6 inches is equal to 5.5 feet.
Set up the proportion. Let x represent the length of the shadow cast by the student. We have the following proportion:
(height of building) / (length of building's shadow) = (height of student) / (length of student's shadow)
60 / 20 = 5.5 / x
Cross-multiply and solve for x:
60 * x = 20 * 5.5
60x = 110
x = 110 / 60
x = 1.83 (rounded to two decimal places)
The length of the shadow cast by the 5 foot 6 inch student is approximately 1.83 feet.
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Sentence:
7) Jenny bought a pair of boots priced at $85. If the boots were on gole for 15% off
regular price, how much did Jenny pay for the boots?
Let x =
(Remember to subtract sale
Inayah claims that the pool is draining at a rate of 1. 36% per hour
The pool is draining at a rate of 1.36% per hour according to Inayah's claim.
What is the claimed rate at which the pool is draining?According to Inayah's claim, the pool is experiencing a draining at a rate of 1.36% per hour. This means that for every hour that passes, the pool's water level decreases by 1.36% of its total volume.
Understanding the rate at which a pool is draining is essential for monitoring and managing water levels. If the rate of drainage is accurate, it can help estimate how long it would take for the pool to reach a certain level or completely drain. Additionally, it aids in determining the necessary actions to maintain the pool's water balance and prevent potential issues such as overflow or inadequate water supply.
It is crucial to verify the accuracy of the claim by monitoring the pool's water level over a specific period. This can be done by measuring the change in water volume or using other reliable methods to ensure the drainage rate aligns with the claimed percentage.
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The set of values for x that satisfies a quadratic inequality is x < -0.5 or x > 1.5
Write down a possible quadratic inequality. Please help - it’s a gcse maths question
Answer: x-0.5y=1.5
Step-by-step explanation:
Steve bought 10 gallons of gas at the gas station with the highest price. how much more than cole did he pay for gas?
please help, thank you.
Steve paid $10 more than Cole for gas, regardless of the actual prices of gas at the two gas stations.
To answer this question, we need to know the prices of gas at both gas stations. Let's assume that Cole bought 10 gallons of gas at the gas station with the lowest price, and that the price difference between the two gas stations is $x per gallon.
If we denote the price per gallon of gas at the gas station where Cole bought gas as "c", then the total amount Cole paid for gas is:
10c
If we denote the price per gallon of gas at the gas station where Steve bought gas as "s", then the total amount Steve paid for gas is:
10s
The difference between the two amounts is:
10s - 10c
But we know that Steve bought gas at the gas station with the highest price, so we can assume that s = c + x.
Substituting this expression for s into the equation above, we get:
10s - 10c = 10(c + x) - 10c = 10x
So Steve paid $10 more than Cole for gas, regardless of the actual prices of gas at the two gas stations.
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Unit 7: Right Triangles & Trigonometry Homework 4: Trigonometry Ratios & Finding Missing Sides #’s 10&11
The value of the sides are;
x = 20.4
x = 13.84
How to determine the valueThere are six different trigonometric identities. They include;
sinetangentcosinecosecantsecantcotangentGiven that the ratios are;
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
Using the tangent identity, we have;
tan 64 = 42/x
cross multiply the values
x = 42/2. 050
x = 20. 4
Using the sine identity;
sin 70 = 13/x
cross multiply the values
x = 13/0. 939
x = 13. 84
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CAN SOMEONE SHOW ME STEP BY STEP ON HOW TO DO THIS SOMEONE GAVE ME THE WRONG STEPS
A city just opened a new playground for children in the community. An image of the land that the playground is on is shown.
A polygon with a horizontal top side labeled 45 yards. The left vertical side is 20 yards. There is a dashed vertical line segment drawn from the right vertex of the top to the bottom right vertex. There is a dashed horizontal line from the bottom left vertex to the dashed vertical, leaving the length from that intersection to the bottom right vertex as 10 yards. There is another dashed horizontal line that comes from the vertex on the right that intersects the vertical dashed line, and it is labeled 12 yards.
What is the area of the playground?
900 square yards
855 square yards
1,710 square yards
1,305 square yards
The area of the playground include the following: 900 square yards.
How to calculate the area of a regular polygon?In Mathematics and Geometry, the area of a regular polygon can be calculated by using this formula:
Area = (n × s × a)/2
Where:
n is the number of sides.s is the side length.a is the apothem.In Mathematics and Geometry, the area of a parallelogram can be calculated by using the following formula:
Area of a parallelogram, A = base area × height
Area of playground, A = 45 × 20
Area of a playground, A = 900 square yards.
Read more on polygon here: brainly.com/question/16691874
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Find the cost of one dozen exercise books, if 3 similar exercise books cost $2.70.
Answer:
$10.8
Step-by-step explanation:
We know that 3 books are $2.70, so 1 book is 2.70/3 which is $0.9
1 dozen = 12
To find the price of 12 books, we multiply by the cost of 1 book
12 * 0.9 = $10.8
If p and q are real numbers in the function f(x)=x4+5x3+px2+qx+30 and f(1)=0, which equation is true?
p + q + 36 = 0 is the equation that is true. From this equation, we can say that p + q = -36.
If we substitute x = 1 in the given function, we get:
f(1) = [tex](1)^4 + 5(1)^3 + p(1)^2 + q(1) + 30[/tex]
= 1 + 5 + p + q + 30
= p + q + 36
We are given that f(1) = 0. Therefore, we have:
p + q + 36 = 0
This is the equation that is true. From this equation, we can say that p + q = -36.
Now, let's see why this is the case. When we substitute x = 1 in the given function and get f(1) = 0, it means that (1, 0) is a point on the graph of the function. This point lies on the x-axis, which means that the function intersects the x-axis at x = 1. Since the x-axis is the line y = 0, this means that f(x) = 0 when x = 1.
Using this information, we can write an equation in terms of p and q. If we substitute x = 1 in the given function and get f(1) = 0, it means that:
f(x) = [tex](x - 1)(x^3 + 6x^2 + (6+p)x + (30-q))[/tex]
Therefore, we have:
[tex](x - 1)(x^3 + 6x^2 + (6+p)x + (30-q)) = 0[/tex]
Since (1,0) is a point on the graph of the function, we know that x = 1 is a root of this equation. Therefore, we have:
[tex](x - 1)(x^3 + 6x^2 + (6+p)x + (30-q)) = (x - 1)(x^2 + (7+p)x + (36-p-q))[/tex]
We know that (x - 1) is a factor of this equation, so we can use polynomial long division to divide the expression [tex](x^2 + (7+p)x + (36-p-q))[/tex] by (x - 1). The quotient will give us the other two roots of the equation. However, we do not need to find these roots to answer the question. All we need to know is that p + q = -36.
To know more about function, refer to the link below:
https://brainly.com/question/25394520#
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