Answer:
Step-by-step explanation:
We can use trigonometry to solve this problem. Let's call the distance from the base of the plateau to our position "x". We can then use the tangent function to find x:
tan(20°) = opposite / adjacent
In this case, the opposite side is the height of the plateau (50 m) and the adjacent side is x. So we can write:
tan(20°) = 50 / x
To solve for x, we can rearrange this equation:
x = 50 / tan(20°)
Using a calculator, we get:
x = 143.45 meters (rounded to two decimal places)
Therefore, if the angle of elevation to the top of the plateau is 20 degrees, and the plateau is 50 meters high, we are approximately 143.45 meters away from the base of the plateau.
Answer:
The distance is 137.3739 feet.
Step-by-step explanation:
I hope this answer is right.
In triangle JK L, cos(K) = 21 and angle J is a right angle. What is the value of cos (L)?
solve in the simplest way possible
44
In the expression 5 x y/7, what value of y would make a product greater than 5 ?
Explain your answer.
Answer: ⬇️⬇️
Step-by-step explanation:
In the expression 5 x y/7, the value of y that would make a product greater than 5 is 8.
HOW TO SOLVE ALGEBRAIC EXPRESSIONS?
According to this question, the following algebraic equation was given:
5 x y/7
This equation reveals that the result can only be equal to 5 when y is exactly 7.
This is because if y = 7, y/7 = 1.
Therefore, the value of y that would make a product greater than 5 is 8.
Which of the following tables represent a proportional relationship
a. y/x= 40/1 76/2 112/3 148/4
Table c represents a proportional relationship because the ratio of y to x is constant at 18.
Which table represent a proportional relationship?A proportional relationship is a relationship between two quantities where their ratios always remain the same.
In option (a), the ratio of y to x is not constant. For example, y/x = 40/1 = 40, but y/x = 148/4 = 37. Therefore, this table does not represent a proportional relationship.
In option (b), the ratio of y to x is not constant either. For example, y/x = 48/2 = 24, but y/x = 192/5 = 38.4. Therefore, this table does not represent a proportional relationship.
In option (c), the ratio of y to x is constant. For example, y/x = 18/1 = 18, but y/x = 126/7 = 18. Therefore, this table represent a proportional relationship.
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Complete questionWhich of the following tables represent a proportional relationship?
a. y/x= 40/1 76/2 112/3 148/4
b. y/x= 48/2 96/3 144/4 192/5
c. y/x= 18/1 54/3 90/5 126/7
d. 24/1 21/2 18/3 15/4
Does the budgeted amount cover the actual amount for expenses, savings, and emergencies? A) No, it's short $207. 0. Eliminate B) No, it's short $227. 0. C) Yes, there's a surplus of $207. 0. D) Yes, there's a surplus of $227. 0
Based on the options provided, it seems that the question is asking whether the budgeted amount is enough to cover expenses, savings, and emergencies. The answer would be either A, B, C, or D.
A) No, it's short $207.0.
B) No, it's short $227.0.
C) Yes, there's a surplus of $207.0.
D) Yes, there's a surplus of $227.0.
Unfortunately, without more information about the specific budgeted amount and the actual expenses, savings, and emergencies, it is impossible to determine the correct answer. It is important to regularly track expenses and compare them to the budgeted amount to ensure that there is enough money to cover all necessary expenses and unexpected events. If there is a shortfall, it may be necessary to adjust the budget or find ways to increase income or decrease expenses to ensure financial stability.
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An 8-inch-by-4-inch hole is cut from a
rectangular metal plate, leaving borders
of equal width x on all four sides. The
area of the metal that remains is 32 in².
The equation (8 + 2x)(4+2x) - 32 = 32
models the area of the plate. What is the
value of x, the frame width?
Answer:
2 inches
Step-by-step explanation:
The area of the metal plate can be calculated by subtracting the area of the hole from the area of the original plate. The area of the original plate is:
8 inches x 4 inches = 32 square inches
The area of the hole is:
8 inches x 4 inches = 32 square inches
So the area of the metal that remains is:
32 square inches - 32 square inches = 0 square inches
According to the equation given, we know that:
(8 + 2x)(4 + 2x) - 32 = 32
Expanding this equation we get:
32 + 16x + 8x + 4x^2 - 32 = 32
Simplifying and rearranging we get:
4x^2 + 24x - 32 = 0
Dividing both sides by 4 we get:
x^2 + 6x - 8 = 0
We can solve this quadratic equation by factoring:
(x + 4)(x - 2) = 0
So x = -4 or x = 2. Since the width of the frame cannot be negative, the only valid solution is x = 2.
Therefore, the frame width is 2 inches.
Help give me an Explanation
Answer:
if the two angles are equal
we use sss therom to solve it
2ft/6ft=24ft/xft
x=(6*24)/2
=72
Step 2: Construct regular polygons inscribed in a circle.
B) The completed construction of a regular hexagon is shown below. Explain why △ACF is 30°-60°-90° triangle. (10 points)
The explanation on why △ACF is 30°-60°-90° triangle is given below.
How to explain the informationWith a regular hexagon, each of its sides and angles are equal in measure. Consider the centre of the encompassing circle, connected to two neighbouring vertices - labeled A and B here. This then creates a radius wherein the length of AB is basically equal to any other side, denoted as 's'. Furthermore, △ABF will be an isosceles triangle (with AB = BF).
From these facts, we can produce △ACF which is a right angled triangle – with AC being its hypotenuse, A F and FB both equating to s/2, finally concluding that ∠AFB is equivalent to 120°/2 = 60° while establishing that ∠ACF is also a right angle constituent making △ACF essentially a 30°-60°-90° triangle.
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Based on results from recent track meets, Leon has a 64% chance of getting a medal in the 100 meter dash. Estimate the probability that Leon will get a medal in at least 4 of the next 10 races. Use the random number table, and make at least 10 trials for your simulation. Express your answer as a percent
The estimated probability of Leon getting a medal in at least 4 of the next 10 races is 80%.
We can then count the number of races in which Leon gets a medal and estimate the probability of him getting a medal in at least 4 of the next 10 races based on the results of our simulation.
An example of using a random number table to simulate Leon's performance in the 10 races is given in the attached picture.
Based on this simulation, Leon got a medal in 5 of the 10 races. We can repeat this simulation multiple times (e.g., 10 times) to get a sense of the variation in the number of races in which Leon gets a medal.
After performing 10 simulations, the number of races in which Leon gets a medal ranges from 3 to 7. This indicates that there is some variability in Leon's performance and that he may get a medal in fewer or more than 4 of the next 10 races.
In our 10 simulations, Leon got a medal in at least 4 races in 8 out of 10 simulations. Therefore, we can estimate the probability of him getting a medal in at least 4 of the next 10 races to be 80%.
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Can you help me with part C? Please
Answer: -40
Step-by-step explanation:
Rate of change is calculated as the slope. The formula for Slope is:
S = [tex]\frac{x_{1}-x_{2} }{ y_{1}-y_{2}}[/tex]
The two points we have are when x = 3 and 6.
the points are (3, 120) and (6, 0), as we can see.
plugging into the slope formula:
S = [tex]\frac{120-0 }{ 3-6}[/tex]
S = 120/-3
S = -40
Which hopefully makes sense, because the slope is negative, (the graph is falling).
In a given diagram below, ray AB bisects angle FAE. BF = 2x + 8 and BE = 42.
A. Set up an equation to solve for "x".
B. Show your work and solve for "x".
a. The equation to solve for x is given as follows: 2x + 8 = 42.
b. The solution for x is given as follows: x = 17.
How to obtain the value of x?The value of x is obtained applying the angle bisection theorem, which divides an angle into two angles of equal measure, hence the opposite segments also have equal length.
The segments for this problem, along with their lengths, are given as follows:
BF = 2x + 8.BE = 42.Hence the equation is given as follows:
2x + 8 = 42.
The value of x is given as follows:
2x = 34
x = 17.
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A man buys a car at a cost of r60 000 from cape town and transported it to durban at a cost price of by r4 500. at what price must he sell the car to make an overall profit of 25%
He must sell the car at R80,625 to make an overall profit of 25%
To find the selling price of the car that gives a 25% profit, we need to use the following steps:
Calculate the total cost of buying and transporting the car to Durban:
Total cost = Cost of car + Cost of transportation
Total cost = R60,000 + R4,500
Total cost = R64,500
Calculate the desired profit:
Profit = 25% of total cost
Profit = 0.25 x R64,500
Profit = R16,125
Calculate the total amount that the car needs to be sold for:
Total amount = Total cost + Profit
Total amount = R64,500 + R16,125
Total amount = R80,625
Therefore, the man needs to sell the car for R80,625 to make an overall profit of 25%.
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Ginny made a cylindrical clay vase for her art project. If the vase has a
volume of 672 cubic inches and a diameter of 10 inches, which is closest to
the height of the vase?
If the vase has a volume of 672 cubic inches and a diameter of 10 inches, the height of the cylindrical clay vase is closest to 8.56 inches.
To find the height of the cylindrical vase, we'll use the formula for the volume of a cylinder: V = πr²h, where V is the volume, r is the radius, and h is the height. Given the diameter is 10 inches, the radius (r) is half of that, which is 5 inches. The volume (V) is 672 cubic inches.
Now, we can solve for the height (h) using the formula:
672 = π(5²)h
First, calculate the area of the base (πr²):
π(5²) = 25π
Now, divide the volume by the area of the base to find the height:
h = 672 / 25π
h ≈ 8.56 inches
So, the height of the cylindrical clay vase is closest to 8.56 inches.
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Clara's class is preparing for a field trip. Her teacher purchased bottled water for the trip and asked Clara to stock a cooler with 2 bottles for every student who is going. 5 of the students didn't turn in permission slips and aren't going on the trip. So, Clara stocks the cooler with 38 bottles of water.
Which equation can you use to find the total number of students, n, in Clara's class?
The equation that can be used to find the total number of students would be n = 19 + 5.
How to find the equation ?It is acknowledged that Clara provided 2 bottles per each pupil joining her on the excursion. Denote, by using ‘x’, the quantity of learners present; we can then inscribe the succeeding formula:
2x = 38
By resolving this mathematical principal, the total number of students attending the event is revealed.
x = 38 / 2
x = 19
Currently, 19 individuals are confirmed to embark upon the outing, due to five individuals failing to furnish required consent forms and will be absent. Subsequently, we may deduce the quantity of pupils (designated as ‘n’) in Clara’s class:
n = 19 (students going on the trip ) + 5 ( students not going )
n = 19 + 5
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Given lines
�
,
�
,
l,m,and
�
n are parallel and cut by two transversal lines, find the value of
�
x. Round your answer to the nearest tenth if necessary.
Answer:
x=8.31
Step-by-step explanation:
We can use the Proportional Segments Theorem
9/26=x/24
26x=216
x=216/26=8.31
Let f(x) = x² – 6x. Round all answers to 2 decimal places. = a. Find the slope of the secant line joining (2, f(2) and (7, f(7)). Slope of secant line = b. Find the slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)). Slope of secant line = c. Find the slope of the tangent line at (6, f(6)). Slope of the tangent line d. Find the equation of the tangent line at (6, f(6)). y =
The equation of the tangent line at (6, f(6)) is y = 6x - 48.
a. The slope of the secant line joining (2, f(2)) and (7, f(7)) is:
slope = (f(7) - f(2)) / (7 - 2)
We can find f(7) and f(2) by plugging in x = 7 and x = 2 into the expression for f(x):
f(7) = 7² - 6(7) = 7
f(2) = 2² - 6(2) = -8
Substituting these values into the slope formula, we get:
slope = (7 - (-8)) / (7 - 2) = 3
Therefore, the slope of the secant line joining (2, f(2)) and (7, f(7)) is 3.
b. The slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)) is:
slope = (f(6 + h) - f(6)) / ((6 + h) - 6) = (f(6 + h) - f(6)) / h
We can find f(6) and f(6 + h) by plugging in x = 6 and x = 6 + h into the expression for f(x):
f(6) = 6² - 6(6) = -12
f(6 + h) = (6 + h)² - 6(6 + h) = h² - 6h + 36 - 36 - 6h = h² - 12h
Substituting these values into the slope formula, we get:
slope = (h² - 12h - (-12)) / h = h - 12
Therefore, the slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)) is h - 12.
c. The slope of the tangent line at (6, f(6)) is the derivative of f(x) at x = 6:
f'(x) = 2x - 6
f'(6) = 2(6) - 6 = 6
Therefore, the slope of the tangent line at (6, f(6)) is 6.
d. To find the equation of the tangent line at (6, f(6)), we use the point-slope form of a line:
y - f(6) = f'(6)(x - 6)
Substituting f(6) and f'(6) into this equation, we get:
y - (-12) = 6(x - 6)
Simplifying, we get:
y = 6x - 48
Therefore, the equation of the tangent line at (6, f(6)) is y = 6x - 48.
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a strawberry field, you will find 4 plants per square foot. How many strawberry plants will you find in a square field that has a length of 208 ft (approx 1 acre )?
Answer: 832 plants
Step-by-step explanation:
If there are 4 plants for every 1 square ft the ratio is 4:1.
This tells us to multiply 208x4 giving us 832.
5+sin(3x)=4
solve for x on the unit circle where x is between 0 and 2pi
The solutions for x between 0 and 2π are: x = π/2, (5π)/6, and (11π)/6.
To solve the equation 5 + sin(3x) = 4 for x on the unit circle, where x is between 0 and 2π, follow these steps:
1. Subtract 5 from both sides: sin(3x) = -1
2. Determine the angle for which sin is -1: sin(3x) = sin(3π/2)
3. Since the sine function has a period of 2π, the general solution is: 3x = 3π/2 + 2πk, where k is an integer.
4. Divide both sides by 3: x = π/2 + (2πk)/3
Now, find the values of x between 0 and 2π by trying different integer values of k:
- If k = 0, x = π/2
- If k = 1, x = π/2 + 2π/3 = (5π)/6
- If k = 2, x = π/2 + 4π/3 = (11π)/6
Thus, the solutions for x between 0 and 2π are: x = π/2, (5π)/6, and (11π)/6.
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a florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. Express the grower's total yield as a function of the number of additional trees planted, draw the graph and estimate the total number of trees the grower should plant to maximize yield.
Answer: 80 trees
Step-by-step explanation:
YIELD = (NUMBER OF TREES)*(NUMBER OF ORANGES PER TREE)
Let's assume NUMBER OF TREES = 60 + x, where x is the number of additional trees above 60
The NUMBER OF ORANGES PER TREE will = (400-4x). Hence:
YIELD = (60+x)*(400-4x) = 24000-240x+400x-4x2 = -4x2 + 160x + 24,000
To find the maximum YIELD, take the derivative of YIELD wrt x, set it to zero, and solve for x:
d(YIELD)/dx = -8x + 160
0 = -8x +160
8x = 160
x = 20
The grower should grow 60 + 20 = 80 trees to maximize yield.
Answer: 80 trees
Step-by-step explanation: just bc it is
Jesus works at a computer outlet. He receives a bi-weekly salary of
$300 plus 5. 5% commission on his sales. In the last two weeks, he sold
$16,200 of computer equipment. He pays 8% for State Income Tax,
12. 3% for Federal Income Tax, 6. 3% for Social Security, and 1. 45%
for Medicare. What steps did I take to find Jesus' net bi-weekly
pay? (Show your work)
Jesus' net bi-weekly pay is $856.69 after paying his taxes and deductions.
To find Jesus' net bi-weekly pay, I followed these steps:
Calculate Jesus' commission: Jesus sold $16,200 of computer equipment, so his commission is 5.5% of $16,200, which is $891.
Calculate Jesus' gross bi-weekly pay: Jesus receives a bi-weekly salary of $300 plus his commission of $891, so his gross bi-weekly pay is $1,191.
Calculate Jesus' deductions: Jesus pays 8% for State Income Tax, 12.3% for Federal Income Tax, 6.3% for Social Security, and 1.45% for Medicare. To calculate the deductions, I multiplied his gross bi-weekly pay by each percentage rate:
State Income Tax: 8% of $1,191 = $95.28
Federal Income Tax: 12.3% of $1,191 = $146.67
Social Security: 6.3% of $1,191 = $75.09
Medicare: 1.45% of $1,191 = $17.27
Subtract the deductions from the gross bi-weekly pay: To find Jesus' net bi-weekly pay, I subtracted the total deductions of $334.31 from his gross bi-weekly pay of $1,191:
Net bi-weekly pay = $1,191 - $334.31 = $856.69
Therefore, Jesus' net bi-weekly pay is $856.69 after paying his taxes and deductions.
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The fifth and tenth terms of an arithmetic sequence,
respectively, are -2 and 53. What is the seventh
term of the sequence?
If the fifth and tenth terms of an arithmetic sequence, respectively, are -2 and 53, the seventh term of the arithmetic sequence is 20.
To find the seventh term of the arithmetic sequence, we need to first find the common difference (d) of the sequence. We know that the fifth term is -2 and the tenth term is 53.
The formula for the nth term of an arithmetic sequence is: an = a1 + (n-1)d
Using this formula, we can set up two equations:
-2 = a1 + 4d (since the fifth term is a1 + 4d)
53 = a1 + 9d (since the tenth term is a1 + 9d)
We now have two equations with two variables (a1 and d). We can solve for either variable using substitution or elimination. I'll use elimination:
-2 = a1 + 4d
53 = a1 + 9d
Subtracting the first equation from the second equation, we get: 55 = 5d
Therefore, d = 11
Now that we know the common difference is 11, we can use the formula for the nth term again to find the seventh term:
a7 = a1 + (7-1)d
a7 = a1 + 6d
We still don't know a1, but we can solve for it using one of the previous equations:
-2 = a1 + 4d
-2 = a1 + 4(11)
-2 = a1 + 44
a1 = -46
Now we can substitute a1 and d into the formula for the seventh term:
a7 = -46 + 6(11)
a7 = -46 + 66
a7 = 20
Therefore, the seventh term of the arithmetic sequence is 20.
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How do I do number 3?
3.) The volume of the triangular prism = 6,480 mi³
The surface area of the triangular prism = 1,872mi²
How to calculate the surface area of the triangular prism?To determine the surface area of the given triangular prism, the formula that should be used is given as follows:
Surface area = bH + (b1+b2+b3)×l
where ;
b= 8 mi
b1 = 18 mi
b2 = 24 mi
b3 = 30 mi
Height = 18 mi
length = 24 mi
Surface area = 8×18 + ( 18+24+30)× 24
= 144 + 1728
= 1,872mi²
To calculate the volume of a triangular prism, the formula the should be used is given as follows;
Volume = 1/2 × b× h × L
where;
b = 24 ni
h = 18 mi
l = 30 mi
volume = 1/2 ×24×18×30
= 6,480 mi³
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Taylor would like to have a karaoke deejay at her graduation party. her three sisters volunteered to split the cost of hiring the deejay. they need to rent a tent for $45 and a microphone system for $60 and then pay the deejay $30 an hour for four hours. how much do each of the sisters owe?
write out all the work used to determine the answer to the question.
Each of the three sisters owes $75 to cover the cost of hiring the karaoke deejay for Taylor's graduation party.
To determine how much each sister owes, we need to first calculate the total cost of the party and then divide that cost by three, since there are three sisters splitting the cost.
1. Tent rental: $45
2. Microphone system: $60
3. Deejay cost: $30/hour × 4 hours = $120
Now, we'll add these costs together to find the total cost:
Total cost = $45 (tent) + $60 (microphone) + $120 (deejay) = $225
Finally, we'll divide the total cost by the number of sisters (3) to find out how much each sister owes:
Amount owed per sister = $225 (total cost) ÷ 3 (sisters) = $75
So, each sister owes $75 for the karaoke deejay at Taylor's graduation party.
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PLEASE HELP ASAP 3 PART QUESTION
Answer:
that is really hard but im pretty sure one of the answers to the first one is -16? for the second x
Step-by-step explanation:
Ide whole numbers
1
franny spent 35 minutes walking around a track. she made 7 laps around the track.
it took franny the same amount of time to walk each lap. how many minutes did it take her to walk each lap?
оа.
28
ов.
5
oc. 245
od. 1
reset
submit
It took Franny 5 minutes to walk each lap. The correct option is B.
Franny spent a total of 35 minutes walking around the track and made 7 laps around the track, so the total time for all 7 laps is 35 minutes. Let's assume it took Franny t minutes to walk each lap. Then, we can set up the following equation:
7t = 35
We can solve for t by dividing both sides of the equation by 7:
t = 35/7 = 5
Therefore, the correct option is B.
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Determine if the sequence below is arithmetic or geometric and determine the common difference / ratio in simplest form.
12
,
8
,
4
,
.
.
.
12,8,4,...
This is sequence and the is equal to
Answer: arithmetic. Common difference is -4
Step-by-step explanation:
constantly subtract four to get to the next
solve quadratic equation 6x²-11x-35= 0 pls needed urgently
Answer:
Step-by-step explanation:To solve the quadratic equation 6x²-11x-35= 0, we can use the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
In this case, we have:
a = 6
b = -11
c = -35
Substituting these values into the quadratic formula, we get:
x = (-(-11) ± sqrt((-11)² - 4(6)(-35))) / 2(6)
Simplifying this expression:
x = (11 ± sqrt(121 + 840)) / 12
x = (11 ± sqrt(961)) / 12
x = (11 ± 31) / 12
So, we have two solutions:
x = (11 + 31) / 12 = 3
and
x = (11 - 31) / 12 = -5/2
Therefore, the solutions to the equation 6x²-11x-35= 0 are x = 3 and x = -5/2.
Consider the following series: (-1)^ 71n n We will test this series for convergence or divergence. (i) What test(s) is(are) applicable to test this series? Click for List (i) Determine whether this series converges or diverges. O Converges O Diverges (iii) What is the sum of the series? Note: Write the exact answer not the decimal approximation (for example write 1 / not 0.8). Answer: The sum of the series is PO
The series given is (-1)^71n*n. To test for convergence or divergence, the alternating series test as the series alternates in sign. Also, the absolute value of the terms of the series is decreasing as n increases. In conclusion, the given series (-1)^71n * n diverges, and it does not have a finite sum.
According to the alternating series test, if a series alternates in sign, and the absolute value of its terms is decreasing as n increases, then the series converges.
Thus, the series converges.
To find the sum of the series, we use the formula for the sum of an alternating series:
Sum = (-1)^71*1 - (-1)^71*2 + (-1)^71*3 - (-1)^71*4 + ...
= (-1)^71*(1 - 2 + 3 - 4 + ...)
= (-1)^71*(n(n+1)/2)
= 0
Therefore, the sum of the series is 0.
I'd be happy to help you with your question. To determine the convergence or divergence of the series (-1)^71n * n, we can follow these steps:
(i) Since the series has terms that alternate in sign, we can use the Alternating Series Test to test for convergence.
(ii) To apply the Alternating Series Test, we must first verify that the sequence of positive terms is decreasing and has a limit of zero. In this case, the sequence of positive terms is given by "n". As "n" goes to infinity, the sequence increases, not decreases, and the limit is not zero. Therefore, the series (-1)^71n * n does not pass the Alternating Series Test, and it diverges.
(iii) Since the series diverges, it does not have a finite sum.
In conclusion, the given series (-1)^71n * n diverges, and it does not have a finite sum.
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Find the height of a cone with a diameter of 12m whose volume is 226m3. Use 3. 14, and round your answer to nearest meter
The height of a cone with a diameter of 12m whose volume is 226m³ is 6 meters.
The formula for the volume of a cone is
V = (1/3) * π * r^2 * h
where V is the volume, r is the radius, h is the height, and π is approximately equal to 3.14.
We know the diameter of the cone is 12m, which means the radius is 6m.
We also know that the volume of the cone is 226m^3.
Substituting these values into the formula, we get:
226 = (1/3) * π * 6^2 * h
Simplifying:
226 = (1/3) * 3.14 * 36 * h
226 = 37.68h
h = 226/37.68
h ≈ 6
Therefore, the height of the cone is approximately 6 meters.
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Add the polynomials. (5x3+x)+(3x3+8) enter the answer in the box, in standard form (highest exponent to lowest).
The sum of the polynomials (5x³ + x) + (3x³ + 8) is 8x³ + x + 8
(5x³ + x) + (3x³ + 8) can be simplified by adding the coefficients of the like terms. The like terms are 5x³ and 3x³, which can be combined to give 8x³. The single terms are x and 8, which can be combined to give 8 + x. Therefore, the polynomials add up to
8x³ + x + 8
In standard form (highest exponent to lowest), the answer is:
8x³ + x + 8
Therefore, the sum of the polynomials (5x³ + x) + (3x³ + 8) is 8x³ + x + 8 in standard form (highest exponent to lowest)
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Find the smallest number of terms of the series ∑ n = 1 (-1)^n+1/2^n you need to be certain that the partial sum Sn is within 1/100 of the sum.n=2 n=4 n=6 n=8 n=7
We need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.
We want to find the smallest value of n such that the absolute value of the difference between the sum of the first n terms and the sum of the entire series is less than 1/100.
The sum of the first n terms of the series is given by:
Sn = ∑_(k=1[tex])^n[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]
We can write the sum of the entire series as:
S = ∑_(k=[tex]1)^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]
The absolute value of the difference between the sum of the first n terms and the sum of the entire series is:
|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] [tex](-1)^(k+1)/2^k|[/tex]
We want to find the smallest value of n such that |S - Sn| < 1/100.
Let's start by evaluating the sum of the series:
S = ∑_(k=1) (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex] = 1/2 - 1/4 + 1/8 - 1/16 + ...
This is a geometric series with first term a = 1/2 and common ratio r = -1/2. The sum of the series is:
S = a/(1-r) = (1/2)/(1+1/2) = 1/3
Now we can write:
|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k|[/tex] <= 1/[tex]2^(n+1)[/tex]
The last inequality is true because the terms of the series are decreasing in absolute value, and we are summing an infinite number of terms.
Therefore, we need to find the smallest value of n such that 1/2^(n+1) < 1/100. This gives:
n+1 > log2(100)
n > log2(100) - 1
n > 6.64
The smallest integer value of n that satisfies this inequality is n = 7.
Therefore, we need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.
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