The area of the region cleaned by the 7-in wiper blade with the angle subtended being 92⁰ as described in the task content is; 39.32in².
What is the area cleaned by the 7-in wiper blade?According to the task content, it follows that the wiper blade is 7-in long and subtends an angle of 92°.
Consequently, the area swept by the blade in discuss can be determined by the Area of sector formula and hence, is;
Area = (92/360) × (22/7) × 7².
Area = 39.32 in².
Ultimately, it follows from the solving steps above that the area of the region cleaned by the 7-in wiper blade with the angle subtended being, 92⁰ is; Area = 39.32 in².
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PLEASE HELP IM STUCK PLS
Step-by-step explanation:
the general slope-intercept form is
y = ax + b
a is the slope, b is the y-intercept.
we need to transform x + 2y = 6, so that we end up with y being all alone on one side, and the rest on the other side.
x + 2y = 6
2y = -x + 6
y = -x/2 + 6
Pls help me,this is Canadian MO 1998 question,i'm very confuse with this
The solution to the equation is x = 1.62
How to solve the equation?The equation is given as:
[tex]x =\sqrt{x\ -\ \frac{1}{x}}+\sqrt{1-\ \frac{1}{x}}[/tex]
Next, we split the equations as follows:
y = x
[tex]y =\sqrt{x\ -\ \frac{1}{x}}+\sqrt{1-\ \frac{1}{x}}[/tex]
Next, we graph the equations (see attachment)
From the attached graph, the equations intersect at
(x, y)= (1.62, 1.62)
Remove the y value
x = 1.62
Hence, the solution to the equation is x = 1.62
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Rationalize the right side first. Expand and simplify.
[tex]x = \sqrt{x - \dfrac1x} + \sqrt{1 - \dfrac1x}[/tex]
[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = \left(\sqrt{x - \dfrac1x} + \sqrt{1 - \dfrac1x}\right) \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right)[/tex]
[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = \left(\sqrt{x - \dfrac1x}\right)^2 - \left(\sqrt{1 - \dfrac1x}\right)^2[/tex]
[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = \left(x - \dfrac1x\right) - \left(1 - \dfrac1x\right)[/tex]
[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = x - 1[/tex]
Rearrange terms as
[tex]\dfrac{x - 1}x = \sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}[/tex]
[tex]\dfrac{x - 1}x + \sqrt{\dfrac{x - 1}x} = \sqrt{x - \dfrac1x}[/tex]
Substitute
[tex]y = \sqrt{\dfrac{x-1}x} = \sqrt{1 - \dfrac1x} \\\\ \implies y^2 = 1 - \dfrac1x \implies x = \dfrac1{1-y^2}[/tex]
(noting that [tex]y>0[/tex]) so that
[tex]x - \dfrac1x = \dfrac1{1-y^2} - (1-y^2) = \dfrac{2y^2 - y^4}{1-y^2}[/tex]
and the equation transforms to
[tex]y^2 + y = \sqrt{\dfrac{2y^2 - y^4}{1-y^2}}[/tex]
[tex]y + 1 = \sqrt{\dfrac{2 - y^2}{1-y^2}}[/tex]
Solve for [tex]y[/tex].
[tex](y+1)^2 = \left(\sqrt{\dfrac{2-y^2}{1-y^2}}\right)^2[/tex]
[tex]y^2 + 2y + 1 = \dfrac{2 - y^2}{1 - y^2}[/tex]
[tex]-y^4 - 2y^3 + 2y + 1 = 2 - y^2[/tex]
[tex]y^4 + 2y^3 - y^2 - 2y + 1 = 0[/tex]
[tex]\left(y^4 + 2y^3 + y^2\right) - 2y^2 - 2y + 1 = 0[/tex]
[tex](y^2 + y)^2 - 2(y^2 + y) + 1 = 0[/tex]
[tex]\left(y^2 + y - 1\right)^2 = 0[/tex]
[tex]y^2 + y - 1 = 0[/tex]
[tex]y = \dfrac{-1 \pm \sqrt5}2[/tex]
Since [tex]y>0[/tex], we take the positive root. Then
[tex]y = \sqrt{\dfrac{x-1}x} = \dfrac{-1 + \sqrt5}2[/tex]
Solve for [tex]x[/tex].
[tex]\left(\sqrt{\dfrac{x-1}x}\right)^2 = \left(\dfrac{-1 + \sqrt5}2\right)^2[/tex]
[tex]\dfrac{x-1}x = \dfrac{6 - 2\sqrt5}4 = \dfrac{3 - \sqrt5}2[/tex]
[tex]x - 1 = \dfrac{3-\sqrt5}2x[/tex]
[tex]\dfrac{-1+\sqrt5}2 x = 1[/tex]
[tex]x = \dfrac2{-1+\sqrt5}[/tex]
Rationalize to clean up the answer.
[tex]x = \dfrac2{-1+\sqrt5}\cdot\dfrac{-1-\sqrt5}{-1-\sqrt5} = \boxed{\dfrac{1+\sqrt5}2} \approx 1.6108[/tex]
i.e. the golden ratio constant.
Alex travels 46
miles per hour for
3.2 hours. How far
has he gone?
Answer: 147.2 miles
Step-by-step explanation:
We can answer this by knowing the formula [tex]speed=\frac{distance}{time}[/tex]. Here, the speed is 46 miles/hour and the time is 3.2 hours. Let's put these values into the formula and solve for distance.
[tex]46=\frac{distance}{3.2}\\46*3.2=\frac{distance}{3.2}*3.2\\d=147.2[/tex]
Alex traveled 147.2 miles.
50 POINTS !! Charlotte has a map of the village. The scale on the map is such that 1cm represents 25m.
a) On the map, the church is 7cm from the village store. What is the real distance from the church to the village store?
b) Charlotte calculates her car is parked approximately half a kilometer from the church. On the map, how many cm would represent this distance?
Answer:
a) 175 m
b) 20 cm
Explanation:
scaled size → actual size
1 cm represents 25 m
a)
In the scaled size on the map, the distance from church to village is 7 cm.
Then actual size,
1 cm → 25 m
7 cm → (25 × 7) m
7 cm → 175 m
b)
half a kilometer = 0.5 km = 500 m
Then scaled size,
25 meter → 1 cm
500 meter → 500/25 cm = 20 cm
As of 2018, approximately what percentage of americans rely upon television broadcasts for their news?
More than 50% of Americans rely upon television broadcasts for their news.
News, current occasions and ancient programming can help make younger humans extra aware of other cultures and those. Documentaries can help increase vital thinking about society and the arena. television can help introduce teens to classic Hollywood movies and overseas movies that they might not otherwise see.
It broadens the know-how of various cultures, and promotes tolerance and international information of worldwide issues. Through contemporary affairs, discovery, lifestyle, cooking indicates and children's programs, television encourages medical and cultural interest.
Agree with and effect: television is the most relied on a form of marketing and stays maximumly probable to make consumers chortle, circulate them to tears or cause feelings. For example, television is with the aid of a long way the maximum relied on shape of video advertising in Canada. 70% of 18-34-year-olds named tv the most truthful compared to 12% for online video.
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Mrs. kelley asked her students to plot the number of books they read over the summer. a dot blot titled books read over the summer goes from 0 to 7. 0 has 1 dot, 1 has 3 dots, 2 has 6 dots, 3 has 2 dots, 4 has 3 dots, 5 has 1 dot, 6 has 2 dots, and 7 has 0 dots. using the dot plot, what was the total number of students that plotted the number of books they read? 3 6 17 18
The total number of students that plotted the number of books they read was 18, making the 4th option, a correct choice, using the dot plot.
Any data that may be shown as dots or tiny circles is called a dot plot. Given that the height of the bar created by the dots indicates the numerical value of each variable, it is comparable to a bar graph or a simple histogram. Small amounts of data are shown using dot plots.
In the question, we are informed that Mrs. Kelley asked her students to plot the number of books they read over the summer. a dot plot titled books read over the summer goes from 0 to 7. 0 has 1 dot, 1 has 3 dots, 2 has 6 dots, 3 has 2 dots, 4 has 3 dots, 5 has 1 dot, 6 has 2 dots, and 7 has 0 dots.
We are asked to find the total number of students that plotted the number of books they read using the dot plot.
The dot plot is attached to the solution.
The total number of students = The total number of dots = 1 + 3 + 6 + 2 + 3 + 1 + 2 + 0 = 18.
Thus, the total number of students that plotted the number of books they read was 18, making the 4th option, a correct choice, using the dot plot.
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The population of a country was increased in the past 4 years. The annual presentage increases are 20%,25%,30%,10% respectively.What the overall percentage increase in these 4years?
The answer is 114.5%.
Let the original population be denoted by x.
Now, let's go through the percentage increase per year.
Year 1
x (1 + 20%)x (1.2)1.2xYear 2
1.2x (1 + 25%)1.2x (1.25)1.5xYear 3
1.5x (1 + 30%)1.5x (1.3)1.95xYear 4
1.95x (1 + 10%)1.95x (1.1)2.145xOverall increase : 214.5% - 100% = 114.5%
Hence, the overall percentage increase in these 4 years is 114.5%.
The seventh grade class at a Christian school had 24 students enrolled at the beginning of the school year. At the end of the school year, there were 27 students enrolled in the seventh grade. What is the percent increase in the enrollment of the seventh grade?
Answer:
12.5%
Step-by-step explanation:
27-24=3
3/24=1/8
1/8=0.125
0.125 in percent is 12.5%
Select the correct answer.
What is the approximate measure of exterior angle B in the polygon?
Answer:
C
Step-by-step explanation:
the sum of the exterior angles of a polygon = 360°
sum the 4 exterior angles and equate to 360
8x + 9x + 5 + 5x + 4 + 9x + 3 = 360 , that is
31x + 12 = 360 ( subtract 12 from both sides )
31x = 148 ( divide both sides by 31 )
x ≈ 11.23 ( to 2 dec. places )
then exterior angle at B is
9x + 3 = 9(11.23) + 3 ≈ 104° → C
The approximate measure of exterior angle B in the polygon is 104° option (C) is correct.
What is a regular polygon?A polygon is a geometric figure with a finite number of sides in two dimensions. On the sides or edges of a polygon, straight-line segments are joined end to end to form a closed shape. The vertices, also known as corners, are the points where two line segments meet and form an angle.
[tex]\rm Area = |\dfrac{(x_1y_2-y_1x_2)+(x_2y_3-y_2x_3)....+(x_ny_1-y_nx_1)}{2}|[/tex]
It is given that:
A polygon is given in the picture with angle measures.
As we know,
The sum of the exterior angles of a polygon = 360°
8x + 9x + 5 + 5x + 4 + 9x + 3 = 360
31x + 12 = 360
31x = 148
x ≈ 11.23
= 9x + 3 = 9(11.23) + 3
≈ 104°
Thus, the approximate measure of exterior angle B in the polygon is 104° option (C) is correct.
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need answer
solve for w
w + 7.1
———- = 2.87
3
Answer: 1.51
Step-by-step explanation:
[tex]\frac{w + 7.1}{3} =2.87[/tex]
[tex]3(\frac{w+7.1}{3})=3(2.87)[/tex]
w + 7.1 = 8.61
w + 7.1 - 7.1 = 8.61 - 7.1
w = 1.51
Miguel created the ratio table below to show how he uses the money he earns at work.
Miguel’s Money
Money Earned
(dollars)
Money Saved
(dollars)
10
8
50
40
120
?
150
120
How much money will he save if he earns 120 dollars?
The amount of money that Miguel would save when he earns $120 is $96.
How much would Miguel save?Ratio expresses the relationship between two or more numbers. It shows the frequency of the number of times that one value is contained within other value(s).
A ratio table can be described as table that shows equivalent values between two or more numbers and helps us to understand the relationship between the numbers.
The first step is to determine the ratio between the money earned and the money saved.
Ratio = money saved / money earned
8/10 = 0.8
Now, multiply the ratio gotten in the previous step by the amount earned to determine the amount saved.
Amount saved = amount earned x ratio
0.8 x 120 = $96
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Answer:
96
Step-by-step explanation:
I took the quiz edge 2022
A written contract is not legally binding until _____. a. money is exchanged b. it is signed by all parties involved c. all requirements of the contract have been fulfilled d. it is signed by all parties involved and notarized by an authorized notary official please select the best answer from the choices provided a b c d
Answer: Answer is B, Mark me brainliest please
Step-by-step explanation:
How to Simplify using suitable property: (-3/7) * 6/5 + (3/2) - (6/5) * 1/14
I will mark the first answerer as brainliest
The value of the given expression is 9/10. The properties of real numbers, such as commutative and distributive are used for solving this expression.
What are the required properties for solving an expression?The properties are:
Associative property: (a + b) + c = a + (b + c) Commutative property: a + b = b + c or a × b = b × aDistributive property: a × (b + c) = a × b + b × cCalculation:The given expression is
(-3/7) × 6/5 + (3/2) - (6/5) × 1/14
applying commutative law (a + b = b + c)
⇒ (-3/7) × 6/5 - (6/5) × 1/14 + (3/2)
again applying commutative law (a × b = b × a)
⇒ (-3/7) × 6/5 - 1/14 × (6/5)+ (3/2)
applying distributive law (a × (b + c) = a × b + b × c) in reverse order
⇒ (-3/7 - 1/14) × (6/5)+ (3/2)
⇒ (-7/14) × (6/5)+ (3/2)
⇒ (-1/2) × (6/5)+ (3/2)
⇒ (-6/10) + (3/2)
⇒ (-6 + 15)/10
⇒ 9/10
Thus, the required value is 9/10 and the suitable properties are commutative and distributive.
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The number 20 is multiplied either by 4 or by 5, then the product is increased either by 3 or by 5, and then the sum is divided either by 6 or by 7. What is the final quotient, if it is a natural number?
PLEASE HELP ITS MATH PLEASE
Answer:
y=[4]x + [12]
Step-by-step explanation:
Remember that parallel lines have identical slopes; and to make a line which passes through a point we will use point-slope form.
Familiarize yourself with point-slope form:
[tex]y-y1=m(x-x1)[/tex]
If the line is parallel to [tex]y=4x+11[/tex] the slope must be 4.
[tex]y-y1=4(x-x1)[/tex]
And we know the point it must pass through is (-2,4), so we can fill those values in too.
[tex]y-4=4(x+2)[/tex]
Now all we have to do is solve for y:
[tex]y-4=4(x+2)\\y-4=4x+8\\y=4x+12[/tex]
Answer:
[tex]y = 4x + 12[/tex]
Step-by-step explanation:
The solution is in the attached image
Match the key features of functions terms to their corresponding definitions.
The key features of the above given functions are correctly matched to their corresponding definition.
Definition of termsNegative sections of the graph: They are the parts where the graph is below the x-axis. That is option C.End behaviour: This is what happens to the graph on the far left or far right. That is option E.Positive sections of the graph: This is the parts where the graph is above the x-axis. That is option D.Intercepts: This is the points where the graph crosses an axis. That is option BRelative extrema: This is the points of relative minimum or maximum in a graph. That is option A.Learn more about graphs here:
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The chart below shows conversion between kilometers and miles.
Conversion Chart
Kilometer
Miles
2
1.2
7
4.2
20
?
30
18
What is the missing value in the table?
Answer:Answer:
12 (D)
Step-by-step explanation:
The first one is 2:1.2 so i did x10 on it to get 20:12.
Step-by-step explanation:Oh and hey if it's right u have to make me brainliest i need award
Apyrotechnician plans for two fireworks to explode together at the same
height in the air. They travel at speeds shown below. Firework B is launched
0.25 s before Firework A. How many seconds after Firework B launches will
both fireworks explode?
Firework A Firework B
320 fts
240 fuis
Both fireworks will explode _?seconds after Firework B launches.
How many solutions does the equation a+b+c+d+e+f=2006. They are positive integers. Your FINAL answer should be in the form x!/x!•x!, where x is a placeholder
The number of solutions of a+b+c+d+e+f = 2006 is 9.12 * 10^16
How to determine the number of solutions?The equation is given as:
a+b+c+d+e+f = 2006
In the above equation, we have:
Result = 2006
Variables = 6
This means that
n = 2006
r = 6
The number of solutions is then calculated as:
(n + r - 1)Cr
This gives
(2006 + 6 - 1)C6
Evaluate the sum and difference
2011C6
Apply the combination formula:
2011C6 = 2011!/((2011-6)! * 6!)
Evaluate the difference
2011C6 = 2011!/(2005! * 6!)
Expand the expression
2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006 * 2005!/(2005! * 6!)
Cancel out the common factors
2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006/6!
Expand the denominator
2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006/720
Evaluate the quotient
2011C6 = 9.12 * 10^16
Hence, the number of solutions of a+b+c+d+e+f = 2006 is 9.12 * 10^16
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Answer:
210 = 6!/1!•1!•1!•1!•1!•1!
Step-by-step explanation:
We can use the stars and bars method to solve this problem. Imagine we have 2006 stars and we want to distribute them among 6 bins (a, b, c, d, e, and f). We can represent the stars as follows:
... * (a stars)
| * * * ... * (b stars)
| | * * * ... * (c stars)
| | | * * * ... * (d stars)
| | | | * * * ... * (e stars)
| | | | | * * * ... * (f stars)
The bars divide the stars into 6 bins, and the number of stars in each bin represents the value of the corresponding variable (a, b, c, d, e, or f).
To ensure that each variable is a positive integer, we can add 1 to each variable and distribute the remaining stars. For example, if we add 1 to a, b, c, d, e, and f, the equation becomes:
(a+1) + (b+1) + (c+1) + (d+1) + (e+1) + (f+1) = 2012
Now we have 6 stars and 5 bars, and we can use the stars and bars formula to find the number of solutions:
Number of solutions = (6+5-1) choose (5-1) = 10 choose 4 = 210
Therefore, the equation a+b+c+d+e+f=2006 has 210 positive integer solutions.
Expressing the answer in the form x!/x!•x!, we have:
210 = 6!/1!•1!•1!•1!•1!•1!
The stars and bars formula:The stars and bars formula is a combinatorial formula that allows us to count the number of ways to distribute identical objects into distinct groups.
Suppose we have n identical objects and k distinct groups. We can represent the objects as stars and the groups as bars. For example, if we have 7 objects and 3 groups, we can represent them as:
| | |
The bars divide the 7 stars into 3 groups, and the number of stars in each group represents the number of objects in that group.
The stars and bars formula tells us that the number of ways to distribute n identical objects into k distinct groups is:
(n+k-1) choose (k-1)
where "choose" is the binomial coefficient. This formula can be derived using a technique called "balls in urns" or by using generating functions.
In the example above, we have n = 7 objects and k = 3 groups, so the number of ways to distribute the objects is:
(7+3-1) choose (3-1) = 9 choose 2 = 36/2 = 18
Therefore, there are 18 ways to distribute 7 identical objects into 3 distinct groups.
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Find the area of the irregular figure.
6 in.
13 in.
12 in.
5 in.
4 in.
4 in.
A = [? ]in.² please explain why or how you got the answer for future questions I only have so many points
Answer:
153 in²
Step-by-step explanation:
Area of irregular figure[tex]\sf \boxed{\text{\bf Area of rectangle = length * width}}[/tex]
[tex]\sf \boxed{\text{\bf Area of square = side *side}}[/tex]
Figure I:
length = 12 in
width = 6 in
Area of figure I = 12 * 6
= 72 in²
Figure II:
length = 13 in
width = 5 in
Area of figure II = 13 * 5
= 65 in²
Figure III:
side = 4 in
Area of figure III = 4 * 4
= 16 in²
Area of irregular shape = 72 + 65 + 16
= 153 in²
Explain how numbers are sorted in consecutive numbering, terminal-digit numbering, and middle-digit numbering
The numbers are arrange in order of consecutive numbering, terminal-digit numbering, and middle-digit numbering to give a value.
Methods of arranging numbers:
Numbers that observe every other continuously in the order from smallest to biggest are known as consecutive numbers. as an example: 1, 2, three, 4, 5, 6, and so on are consecutive numbers.consecutive numbering technique. a technique wherein consecutively numbered statistics are arranged in ascending range order - from the lowest range to the very best number. decimal-numeric coding. A numeric method of classifying statistics by using challenge in gadgets of ten and coded for association in numeric order. the middle digits are uded because the finging aid to organize the filing device. It uses the middle two or 3 digits of every wide variety as the number one department underneath which a report is filed.Terminal digit order: A device of submitting the use of a six-digit quantity (or higher) this is divided into 3 parts, wherein the ultimate two digits are considered primary. number one digits: The final two digits to the proper within the range. Secondary digits: The middle two digits within the quantity.Learn more about numbering here:-https://brainly.com/question/1746829
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4. Emily’s house has a chimney. What is the volume of her chimney? 4 in 9 in 12 in 3 in 4 in 6 in
Using the volume for rectangular prisms, the volume of the chimney is: 216 in.³.
What is the Volume of a Rectangular Prism?The volume of a rectangular prism = (length)(width)(height).
How to Find the Volume of Composite Shapes?The Chimney in Emily's house is a composite solid consisting of two rectangular prism. To find the volume, we have to decompose the figure into two rectangular prisms. Then solve for the volume of each before finding the total volume.
Volume of the bigger rectangular prism = (length)(width)(height) = (4)(4)(12)
Volume of the bigger rectangular prism = 192 in.³
Volume of the smaller rectangular prism = (length)(width)(height) = (4)(2)(3)
Volume of the smaller rectangular prism = 24 in.³
The volume of the chimney = volume of the two rectangular prism
The volume of the chimney = 192 + 24
The volume of the chimney = 216 in.³
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Wich of the following is not a way to represent the solution of inequality 2(x-2)<2? A number line with a closed circle on 3 and shading to the right x<3 3>x a number line with a closed circle on 3 and shading to the left
Answer:
The solution is x < 3. To represent that on a number line, you'd have an open (not filled in) circle on 3 and shading to the left.
Step-by-step explanation:
You would only have a closed (solid/filled in) circle if the inequality had a ≤
Which of the graphs below represents the equation 8x-y=-4?
Answer:
Graph X
Step-by-step explanation:
First, convert the equation into slope-intercept form. The first step is the bring the x to the right side by subtracting 8x from both sides: -y=-8x-4. Now multiply both sides by -1: y=8x+4. Now, if you know about the slope-intercept form, you'll know that 4 is the y-intercept and that [tex]\frac{8}{1}[/tex] is the slope. 8 is the rise and 1 is the run, so the answer is graph X.
A cone has a radius of 3 and a height given by the expression 2a. which expression represents the volume of the cone?2aπ units34a2π units36aπ units324aπ units3
The expression that represents the volume of the cone is 6aπ.
What is called cone?
A cone is a three-dimensional shape in geometry that narrows smoothly from a flat base (usually circular base) to a point(which forms an axis to the center of base) called the vertex.The volume of a cone is expressed as;V = (1/3)πr²hWhere r is the radius, h is height and π is pieGiven the data in the question;
Radius r = 3
Height h = 2a
Volume of the cone V = ?
We substitute our given values into the expression above.
V = (1/3) × π × r² × h
V = (1/3) × π × (3)² × 2a
V = (1/3) × π × 9 × 2a
V = 18aπ / 3
V = 6aπ
Therefore, the expression that represents the volume of the cone is 6aπ.
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I don't get the minutes part
please answer these questions
Answer:12
Step-by-step explanation:
Answer:
8. -3, 16
9. -3, 4.5
Step-by-step explanation:
See attached worksheet.
Given u=(-12,-5) and v=(3,9) what is proj v u?
[tex]\frac{(-12)(3)+(-5)(9)}{3^2 + 9^2} \langle 3, 9 \rangle \\ \\ =-\frac{9}{10} \langle 3, 9 \rangle \\ \\ =\langle -\frac{27}{10}, -\frac{81}{10} \rangle[/tex]
So, the correct answer is Option 1.
Determine whether the following series converges or diverges using the integral test. Be sure to verify that the integral test can be applied.
By the integral test, the series is convergent.
The sequence [tex]k^3/e^{k^4}[/tex] is clearly positive and decreasing for [tex]k\in\Bbb N[/tex]; then by the integral test,
[tex]\displaystyle \int_1^\infty \frac{x^3}{e^{x^4}} \, dx \le \sum_{k=1}^\infty \frac{k^3}{e^{k^4}}[/tex]
and
[tex]\displaystyle \int_1^\infty \frac{x^3}{e^{x^4}} \, dx = \frac14 \int_1^\infty e^{-u}\, du = \frac1{4e} < \infty[/tex]
By using the integral test, series converges
What is an integral test for convergence and divergence?
⇒ It is used to prove the divergence or convergence of series. This test is called the integral test, which compares an infinite sum to an improper integral. It is important to note that this test can only be applied when we are considering a series whose terms are all positive.
How to know if a series is converging or diverging?
If the limit exists and is a finite number (a number less than infinity), we say the integral converges. If the limit is ±∞ or does not exist, we say the integral diverges.
let aₓ= f (x) is any function
⇒ In order for the integral test to work The function must be positive it has to be continuous and it has to be decreasing when x≥1
If the integral converges it means the series is also converging
If the integral diverges it means the series is also diverging
The sequence [tex]k^{3} /{e^{k^{4} }[/tex] is clearly positive and decreases for k∈N then by the integral test,
[tex]\int\limits^\infty_1 {x^{3} /e^{x^4} dx[/tex] [tex]\leq\displaystyle \sum^{\infty}_{k = 1} {x^{3} /e^{x^{4}[/tex]
and
[tex]\int\limits^\infty_1 {x^{3} /e^{x^4} dx[/tex][tex]=1/4\int\limits^\infty_1 {x^{3} /e^{-u} du[/tex]
⇒ [tex]1/4 < \infty[/tex]
So it comes with a finite value hence the series converges
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Gareth pays $ 60 for 9 m of climbing rope. how much will sophie pay for 15 m at the same store? $
Sophie will pay money equivalent to $ 100 for 15 m at the same store.
The term "money" in mathematics refers to a form of payment, such as bills, coins, and demand deposits, that is used to purchase goods and services. Money is used to pay for the worth or price of an item or service.
A country's monetary system is referred to as its currency.
In the case of Gareth,
Money paid for 9 m of climbing = $ 60
Money paid per m of climbing = $ 60/9
Thus, money paid by Sophie for 15 m of climbing = 15 * 60/9
= $ 100
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