The series [tex]\tan(2^{-n})[/tex] is convergent.
What is convergence of series?In mathematics, the cοnvergence οf a series refers tο the behaviοr οf the series' partial sums as mοre terms are added. A series cοnverges if its sequence οf partial sums apprοaches a finite limit, which is called the sum οf the series. If the sequence οf partial sums dοes nοt apprοach a finite limit, the series diverges.
The series [tex]\tan(2^{-n})[/tex] converges because of the following reasons:
As n increases, [tex]2^{-n}[/tex] becomes very small, which makes [tex]\tan(2^{-n})[/tex] close to its linear approximation. Therefore, we can use the linear approximation of [tex]\tan(2^{-n})[/tex] to estimate its value.
The linear approximation of [tex]\tan(2^{-n})[/tex] is [tex]2^{-n}[/tex], which means that [tex]\tan(2^{-n})[/tex] ≈ [tex]2^{-n}[/tex] when [tex]2^{-n}[/tex] is small.
Since the series [tex]2^{-n}[/tex] is convergent, by comparison test, the series [tex]\tan(2^{-n})[/tex] is also convergent.
Alternatively, we can also use the fact that the series tan(x) is convergent for x ∈ (0, π/2), and since [tex]2^{-n}[/tex] → 0 as n → ∞, we have [tex]\tan(2^{-n})[/tex] → tan(0) = 0, which implies that the series [tex]\tan(2^{-n})[/tex] is convergent.
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What is the area of the real object that the scale drawing models? Scale factor. 1:5 Area = 10 square cm Scale drawing Real object
Answer:
D. 50 square centimeters
Write an equation for the word problem below.
"The YMCA summer camp charges registration fee of $40 and then charges $100 per week of camp."
Y = [blank] x + [blank]
Y = MX + B
Answer:
The equation is in slope-intercept form, Y = MX + B, where M is the slope and B is the y-intercept. In this case, the slope is 100 and the y-intercept is 40.
I hope that helps!
Step-by-step explanation:
The equation for the word problem is Y = 100x + 40.
Where Y is the total cost of the summer camp, x is the number of weeks of camp, 100 is the cost per week of camp, and 40 is the registration fee.
A fair coin is tossed 7 times. Compute the probability of tossing 7 heads in a row.
Enter your response as a reduced fraction.
Answer:
1/128
Step-by-step explanation:
The probability of tossing a head is 1/2.
Each toss is an independent event.
We toss 7 times and get a head each time.
1/2 * 1/2* 1/2* 1/2* 1/2* 1/2* 1/2
1/128
A new auditorium is built with a foundation in the shape of one-fourth of a circle of radius 50 feet. So, it forms a region bounded by the graph of x^2 + y^2 = 50^2 with x ≥ 0 and y ≥ 0. The following equations are models for the floor and ceiling. Floor: z = x+y/ 5 Ceiling: z = 20 + xy /100 Calculate the volume of the room, which is needed to determine the heating and cooling requirements.
Answer:
Step-by-step explanation:
To calculate the volume of the room, we need to integrate the difference between the ceiling and floor functions over the region bounded by the circle. We can use double integrals to do this.
First, let's rewrite the equations for the floor and ceiling in terms of x and y:
Floor: z = (x + y) / 5
Ceiling: z = 20 + xy / 100
The volume of the room can be calculated using the following double integral:
V = ∫∫(20 + xy/100 - (x + y)/5) dA
where the limits of integration are x = 0 to x = 50, and y = 0 to y = √(50^2 - x^2).
We can simplify the integrand by combining like terms:
V = ∫∫(400/100 + xy/100 - (20x + 20y)/100) dA
V = ∫∫(4 + xy/100 - 0.2x - 0.2y) dA
Now we can integrate with respect to y first:
V = ∫0^50 ∫0^√(50^2 - x^2) (4 + xy/100 - 0.2x - 0.2y) dy dx
V = ∫0^50 [(4y + xy^2/200 - 0.2xy - 0.1y^2)|y=0^√(50^2 - x^2)] dx
V = ∫0^50 [(4√(50^2 - x^2) + x(50^2 - x^2)/200 - 0.2x√(50^2 - x^2) - 0.1(50^2 - x^2)^2/200)|x=0^50]
Evaluating this integral, we get:
V ≈ 2,233.5 cubic feet
Therefore, the volume of the room is approximately 2,233.5 cubic feet, which is the amount of space that will need to be heated or cooled.
A town was founded with a population of 8,000. The population then doubled every decade. Write the function, p(n), that expresses the
town's population after n decades
p(n)= 8000 + 2 n
p(n) = 8000 +2( n - 1)
p(n)=8000-2
p(n) = 8000
The correct function that expresses the town's population after n decades is: p(n) = 8000 × 2ⁿ. So the initial population of the town is 8,000, which is consistent with the problem statement.
What is function?A function is a mathematical concept that describes the relationship between a set of inputs, called the domain, and a set of outputs, called the range.
A function assigns a unique output to each input, meaning that for a given input, there is only one possible output.
Starting with a population of 8,000, the population doubles every decade, which means it multiplies by 2.
After n decades, the population will have doubled n times, so we can express the population as 8,000 multiplied by 2 raised to the power of n:
p(n) = 8000 × 2ⁿ
This function gives us the population of the town after n decades, where n is a non-negative integer. If we substitute n=0 into the function, we get:
p(0) = 8000 × 2⁰ = 8000
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What is tan(49°)?
O A. 0.75
OB. 0.66
O C. 1.15
OD. 0.82
Answer:
Find the Exact Value tan(49). tan(49) tan ( 49 ). Step 1. The result can be shown in multiple forms. Exact Form: tan(49) tan ( 49 ). Decimal Form:
Step-by-step explanation:
Solve for X
x^2+ y^2=25 and y=x
The solutions for x are x= +[tex]\sqrt{12.5[/tex] and -[tex]\sqrt{12.5[/tex] which has been obtained by solving the equations given in the question.
Define Equation?
An equation can be defined in numerous ways. An equation is a claim that demonstrates the equivalence of two mathematical expressions, according to algebra.
In the initial equation, we get: by substituting y=x:
[tex]x^2[/tex] +[tex]y^2[/tex]= 25
[tex]x^2[/tex] +[tex]x^2[/tex] = 25 (since y = x)
2[tex]x^2[/tex] = 25
[tex]x^2[/tex] = 12.5
When the two sides are square, we get:
x = ±[tex]\sqrt{12.5[/tex]
Therefore, the solutions for x are x = +[tex]\sqrt{12.5[/tex] and x = -[tex]\sqrt{12.5[/tex]
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Trevor used about half of a 5-lb bag of potatoes (2 lb 6 oz). How much did the remaining potatoes weigh?
Answer:
Step-by-step explanation:
f Trevor used 2 lb 6 oz of a 5-lb bag of potatoes, then the weight of the remaining potatoes is:
5 lb = 80 oz (since 1 lb = 16 oz)
Remaining potatoes = Total potatoes - Used potatoes
Remaining potatoes = 80 oz - 2 lb 6 oz
Remaining potatoes = 80 oz - (2 × 16 oz + 6 oz)
Remaining potatoes = 80 oz - 38 oz
Remaining potatoes = 42 oz
Therefore, the remaining potatoes weighed 2 lb 10 oz (or 42 oz).
Can someone help me please
$4000 are invested in a bank account at an interest rate of 10 percent per year.
Find the amount in the bank after 7 years if interest is compounded annually.
--------------
Find the amount in the bank after 7 years if interest is compounded quarterly.
---------------
Find the amount in the bank after 7 years if interest is compounded monthly.
---------------
Finally, find the amount in the bank after 7 years if interest is compounded continuously.
---------------
Answer:
To find the amount in the bank after 7 years, we can use the formula:
A = P(1 + r/n)^(nt)
where:
A = the amount in the bank after 7 years
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
For the given problem:
P = $4000
r = 10% = 0.1
t = 7 years
a) Compounded Annually:
n = 1
A = 4000(1 + 0.1/1)^(1*7) = $7449.36
b) Compounded Quarterly:
n = 4
A = 4000(1 + 0.1/4)^(4*7) = $7650.13
c) Compounded Monthly:
n = 12
A = 4000(1 + 0.1/12)^(12*7) = $7727.27
d) Compounded Continuously:
n → ∞ (as n approaches infinity)
A = Pe^(rt) = 4000e^(0.1*7) = $8193.85
Therefore, the amount in the bank after 7 years increases as the compounding frequency increases. If interest is compounded continuously, the amount in the bank will be the highest.
I need a satisfying conditions question answered thank you sm
The linear function can be written as:
f(x) = -x/3 + 17/3
How to find the linear function?A general linear function can be written as:
f(x) = ax + b
Where a is the slope, and b is the y-intercept.
If we know two points on the line (x₁, y₁) and (x₂, y₂), then the slope of the linear function is:
a= (y₂ - y₁)/(x₂ - x₁)
Here we know the pairs:
f(-4) = 7
f(5) = 4
So we have the points (-4, 7) and (5, 4), then the slope is:
a = (4 - 7)/(5 + 4) = -3/9 = -1/3
Then we can write:
f(x) = -x/3 + b
now we can use one of the given points, like f(5) = 4, replacing that there we will get:
4 = -5/3 + b
4 + 5/3 = b
12/3 + 5/3 = b
17/3 = b
So the function is:
f(x) = -x/3 + 17/3
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One year, the population of a city was 137,000. Several years later it was 117,820. Find the percent decrease.
Group of answer choices
14%
15%
16%
19%
Answer:
The percent decrease in population can be calculated by dividing the difference between the two populations by the original population and then multiplying by 100. In this case, the percent decrease is ((137000-117820)/137000)*100 = 14%.
Step-by-step explanation:
Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options. x < 5 –6x – 5 < 10 – x –6x + 15 < 10 – 5x A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right. A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
The correct option is 1 and 4 of the given inequality –3(2x – 5) < 5(2 – x)
The correct representations of the inequality –3(2x – 5) < 5(2 – x) are:
-6x + 15 < 10 – 5x
x < 5
Therefore, options 1 and 4 are correct. The other options do not correctly represent the inequality.
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help so i cna get out of class
Based on the given conditions, the lock solution is 205.
How to solve a lock?To solve the lock, to calculate the total score based on the given conditions:
Crocus: + 20
Daffodil: + 25
Snowflake: - 50
Tulip: + 30
Bird-Red: + 25, else + 10
Calculating the scores for each input:
TULIP: +30
CROCUS: +20
DAFFODIL: +25
TULIP: +30
DAFFODIL: +25
TULIP: +30
DAFFODIL: +25
CROCUS: +20
Total score = 30 + 20 + 25 + 30 + 25 + 30 + 25 + 20 = 205
Therefore, the solution for the lock is 205.
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Image transcribed:
SPRING NOW LOADING.
<IF CROCUS, THEN +20>
<IF DAFFODIL, THEN +25>
<IF SNOWFLAKE, THEN -50>
<IF TULIP, THEN +30>
<IF BIRD-RED, THEN +25, ELSE + 10>
TULIP
CROCUS
DAFFODIL
TULIP
DAFFODIL
TULIP
DAFFODIL
CROCUS
Solve Lock
Each year, the school has a goal to have a third as many students in remedial mathematics courses as the year before. Currently the school has 1,200 students in remedial math. Approximately how many students will be enrolled in four years?
(a) 44 (b)15 (c) 32,400 (d) 1
Answer:
44 students
Step-by-step explanation:
1200 for the first year
a third of that is 400
400 for the second year
a third of that is 133.4
133.4 for the third year
a third of that is 44
18.A singer sells a single as a music download and CD, making a total profit of £246.64
She sells 456 CD singles, earning 35p for every single sold.
She earns 17p for each music download of the single.
How many music downloads did she sell?
The singer sold 512 music downloads.
What is equation?A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("="). For illustration, 2x - 5 = 13. 2x - 5 and 13 are expressions in this case. These two expressions are joined together by the sign "=".
Let's use the following variables:
CD = the number of CD singles sold
DL = the number of music downloads sold
From the problem, we know the following:
CD + DL = total number of singles sold
CD = 456
The profit from selling CD singles is 35p = £0.35 per CD single
The profit from selling music downloads is 17p = £0.17 per music download
The total profit is £246.64
Using the information above, we can set up two equations based on the profits earned from selling CD singles and music downloads:
0.35 * CD = profit from CD sales
0.17 * DL = profit from download sales
And we know that the sum of these two profits equals the total profit:
0.35 * CD + 0.17 * DL = 246.64
Substituting CD = 456, we get:
0.35 * 456 + 0.17 * DL = 246.64
Simplifying this equation, we get:
159.6 + 0.17 * DL = 246.64
0.17 * DL = 87.04
DL = 512
Therefore, the singer sold 512 music downloads.
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what is the solution to this
Answer:
-2
Step-by-step explanation:
Someone please help me answer this question
The two statements that are both true are as follows: line
AC is perpendicular to line HB and line AC is parallel to FG. That is option A.
What is a perpendicular line?A perpendicular line is defined as the line that forms angle 90° where it meets with another line in a plane.
A line is said to be parallel to each other when they do not intercept as they are both on the same plane.
From the given diagram, line AC is perpendicular to line HB because they form angle 90° at the point of intersection.
Also, line AC is parallel to FG, because they can never intersect till infinity.
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Find the mean of the following data : Marks 0-10 No. of students 5 0-20 10 0-30 14 0-40 17 Ans/22 0-50 20
Mean of the data as per the increasing intervals data is 17.80.
Define mean?The mean in statistics is calculated by multiplying all the values in a set of data by the total number of values for a given set of observations. To put it another way, all that is necessary to determine the mean of a set of data is to sum up all the values and divide the total by the number of values.
Now in the given table,
The midpoints of the intervals are as follows:
0+10/2 = 5
0+20/2 = 10
0+30 /2 = 15
0+40/2 = 20
0+50/2 = 25.
Now total students × interval midpoints are as follows:
5×5=25
10×10=100
15×14=210
20×17=340
25×20=500
Now, total no of students is = 5+10+14+17+20 = 66
So, mean of the data =
(25+100+210+340+500)/66
= 17.80
Therefore, mean of the data is 17.80.
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PLEASE HELP PICTURE BELOW
Joseph is 1.75 meters tall. At 11 a.m., he measures the length of a tree's shadow to be
34.05 meters. He stands 29.7 meters away from the tree, so that the tip of his shadow
meets the tip of the tree's shadow. Find the height of the tree to the nearest
hundredth of a meter.
(Diagram is not to scale.)
Answer:
34.05 m-
2T
1.75 m
29.7 m
Answer:
Step-by-step explanation:
34,05m my love
Answer:
The height of the tree is approximately 15.14 meters, rounded to the nearest hundredth of a meter.
Step-by-step explanation:
Let's call the height of the tree "h". We can use similar triangles to set up an equation involving Joseph's height, the length of his shadow, the height of the tree, and the length of the tree's shadow.
The two triangles we're interested in are:
Joseph's triangle: This triangle has a height of 1.75 meters (Joseph's height) and a base of x meters (the length of Joseph's shadow).
Tree's triangle: This triangle has a height of h meters (the height of the tree) and a base of 29.7 - x meters (the length of the tree's shadow).
Since the two triangles are similar, we can set up the following proportion:
h / (29.7 - x) = 1.75 / x
To solve for h, we can cross-multiply and simplify:
h * x = 1.75 * (29.7 - x)
h * x = 52.075 - 1.75x
h = (52.075 - 1.75x) / x
Now we need to find the value of x that makes the tips of the two shadows meet. From the problem statement, we know that x + 34.05 = 29.7, so:
x = 29.7 - 34.05
x = -4.35
This means that the tips of the shadows don't actually meet, but the problem is likely assuming that the tips of the shadows are very close together, so we can use the value x = -4.35 to approximate the height of the tree.
Substituting x = -4.35 into our equation for h, we get:
h = (52.075 - 1.75(-4.35)) / (-4.35)
h = 15.14
What is the likelihood
that all 21 students in a class share the same birthday? Explain.
Quick please!
Answer:
it is not likey bc inless the class have over 2 mil ppl in it itis doouptfull
Step-by-step explanation:
Is ∆PQR a right triangle? Explain
Answer:
Yes, ∆PQR is a right triangle.
Step-by-step explanation:
A right triangle has one right angle, which is 90°. This triangle has one right angle, so it is a right triangle.
PLS HURRY I AM GIVING BRAINLIEST!!!
the question is in the photo!!
Jordan spend 25 minutes writing each dayNoHow much time does Jordan spend writing each dayFrom the question, we have the following parameters that can be used in our computation:D + W = 75W + 25 = DSo, we haveW + W + 25 = 75EvaluateW = 25This means that Jordan spends 25 minutes on writing is it possible?Based on the answer in (a), the truth statement is No
please help, thank you!
Answer:
To find all values of x for which f(x) = 26, we can set up the equation:
8x + 15/x = 26
Multiplying both sides by x, we get:
8x^2 + 15 = 26x
Bringing all the terms to one side, we get:
8x^2 - 26x + 15 = 0
We can factor this quadratic equation using the factoring method or by using the quadratic formula. Here, we will use the factoring method:
8x^2 - 26x + 15 = 0
(4x - 3)(2x - 5) = 0
Setting each factor equal to zero and solving for x, we get:
4x - 3 = 0 OR 2x - 5 = 0
4x = 3 OR 2x = 5
x = 3/4 OR x = 5/2
Therefore, the values of x for which f(x) = 26 are x = 3/4 or x = 5/2.
Polly's sister-in-law is going to have a baby! For the baby shower, Polly decided to sew pillow to give as a gift. She is using a flower-printed rectangular piece of fabric that is 26 inches long and 22 inches wide.
Answer:
The answer is 96
Step-by-step explanation:
2*(26+22)
2*48
96
I will mark you brainiest!
Which of the following choices lists the values of the side lengths of a triangle with 45-45-90 degree angles and a leg = 5?
A) 5, 5, 5
B) 5, 1, 5
C) 1, 5, 5
D) 1, 1, 5
The side lengths of such a triangle with a leg length of 5 and angles of 45, 45, and 90 are not listed in any of the options.
Explain about property of the right triangles?The right angle is established when 2 straight lines cross at a 90° angle or when they are perpendicular at the intersection. The symbol is used to indicate a right angle.
A triangle's (of all varieties) cumulative sum of angles is 180°.The length of a triangle's two longest sides added together is longer than for third side.Similar to this, the length of a third side of a triangle's third side is shorter than the difference between its two sides.For the given question:
Triangle with degree angles - 45-45-90
It means two legs are of same length 5 units.
Then, hypotenuse H will be:
H² = 5² + 5²
H² = 25 + 25
H² = 50
H = 5√2
Thus, the correct third side of the triangle will be 5√2.
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3. There are 15 students in class 7 who need the school shirt of same size and 2 meter of
cloth is required to make a shirt.
(a) Find the total length of the cloth required to make the shirt to them.
(b) How many shirts can be made from a piece of cloth 9 meter long?
Convert the length of cloth required for a shirt in decimal.
1
If the length of a piece of cloth is 36.25 meter, how many meters of cloth is left? 1
Answer:
Step-by-step explanation:
(a) Since each student needs a shirt of the same size, and 2 meters of cloth is required to make one shirt, the total length of cloth required to make shirts for 15 students would be:
Total length of cloth = 15 x 2 = 30 meters
Therefore, 30 meters of cloth would be required to make shirts for 15 students.
(b) If 2 meters of cloth are required to make one shirt, then the number of shirts that can be made from 9 meters of cloth would be:
Number of shirts = 9 / 2 = 4.5 shirts
However, since we cannot make half a shirt, the actual number of shirts that can be made from 9 meters of cloth would be 4 shirts.
To convert the length of cloth required for a shirt into decimal form, we can simply divide the length in centimeters by 100. Therefore, 2 meters of cloth would be equivalent to 200 centimeters, which is 2.00 meters in decimal form.
(c) If the length of a piece of cloth is 36.25 meters, and we use 30 meters to make shirts for 15 students, then the remaining length of cloth would be:
Remaining length of cloth = 36.25 - 30 = 6.25 meters
Therefore, 6.25 meters of cloth would be left.
How does the graph of g(x) = (x + 2)3 − 6 compare to the parent function of f(x) = x3? g(x) is shifted 2 units to the right and 6 units down. g(x) is shifted 6 units to the right and 2 units up. g(x) is shifted 2 units to the left and 6 units down. g(x) is shifted 6 units to the left and 2 units down.
Answer:
The function g(x) = (x + 2)³ − 6 is obtained by applying three transformations to the parent function f(x) = x³.
First, g(x) is shifted 2 units to the left by subtracting 2 from x inside the parentheses:
g(x) = (x + 2 - 2)³ − 6 = (x)³ − 6
This shows that option C, "g(x) is shifted 2 units to the left and 6 units down," is not correct.
Next, g(x) is shifted 6 units down by subtracting 6 from the entire function:
g(x) = (x + 2)³ − 6 - 6 = (x + 2)³ - 12
This shows that option A, "g(x) is shifted 2 units to the right and 6 units down," is correct.
Finally, g(x) is not shifted left or right by any additional units, but it is shifted 2 units up by adding 2 to the constant term:
g(x) = (x + 2)³ − 6 + 2 = (x + 2)³ - 4
This shows that option B, "g(x) is shifted 6 units to the right and 2 units up," is not correct.
Therefore, the correct answer is option A: g(x) is shifted 2 units to the right and 6 units down.
Points D, B, and E are collinear. Find
the value of a so that points A, B, and
C are collinear.
A
(4x)
O
E
B
148°
D
C
Using the supplementary angle theorem, the value for x so that points A, B, and C are collinear is 8.
What is a supplementary angle?
The definition of "supplementary" in mathematics relates to angles that combine to form a straight angle. It indicates that when two angles sum up to 180 degrees, they are said to be supplementary angles.
A line segment AC is drawn.
Point B is in between A and C.
A transversal DE is drawn passing through point B.
Points D, B, and E are collinear.
The measure of angle CBE is 148° and the measure of angle EBA is 4x°.
The angles ∠CBE and ∠EBA forms a pair of supplementary angles.
Their sum result in value of 180°.
Mathematically, this can be represented as -
∠CBE + ∠EBA = 180°
Substituting the values into the equation -
148° + 4x° = 180°
Solving for x, we get -
4x° = 180° - 148°
4x° = 32°
x = 32 / 4
x = 8
Therefore, the value for x is obtained as 8.
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HELPPP PLEASE
X^2+8x+5=0
Answer:
hope its right
Step-by-step explanation:
Given g(x) = 8x2 − x + 2, find the following.
1) g(0)
2) g(−1)
3) g(r)
4) g(x + h)
Answer:
Step-by-step explanation:
Given g(x) = 8x2 − x + 5, find the following.
(a). g(0) = 5
(b). g(−1) = 14
(c). g(r) = ?
(d). g(x + h) = ?