The area enclosed by the ellipse [tex]\frac{x^2}{4}+\frac{y^2}{81}=1[/tex] is 60 square units.
The general form of the equation of ellipse is [tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]
Given the equation of ellipse is, [tex]\frac{x^2}{4}+\frac{y^2}{81}=1[/tex]
⇒ [tex]\frac{x^2}{2^2}+\frac{y^2}{9^2}=1[/tex]
That is, a = 2 and b = 9
Now, [tex]\frac{y^2}{9^2} = 1 -\frac{x^2}{2^2}[/tex]
⇒ [tex]y=\frac{9}{2} \sqrt{4-x^2}[/tex]
Area enclosed by the ellipse = 4 x Area of ellipse in the first quadrant
Therefore, Area of ellipse in the first quadrant, A = [tex]\int\limits^a_0{\frac{9}{2} \sqrt{4-x^2}} \, dx[/tex]
Using Substitution ,x = 2cosθ, then dx = -2sinθdθ and x = 0⇒ θ = π/2 and x = 2⇒ θ = 0
A = [tex]\int\limits^a_0{\frac{9}{2} \sqrt{4-4cos^2\theta}} \, dx[/tex]
= [tex]\int\limits^2_0{\frac{9}{2}\times2 \sqrt{1-cos^2\theta}} \, dx[/tex]
= [tex]\int\limits^2_0{9 sin^2\theta} \, dx[/tex]
= [tex]\int\limits^0_\frac{\pi}{2} {9 sin^2\theta} (-2sinx)\, d\theta[/tex]
= [tex]\int\limits^\frac{\pi}{2} _0{18 sin^3\theta} \, d\theta[/tex]
= [tex]\int\limits^\frac{\pi}{2} _0{18 (\frac{3sin\theta-sin3\theta}{4} )} \, d\theta[/tex]
= [tex]\frac{9}{2} \int\limits^\frac{\pi}{2} _b {3sin\theta} \, d\theta -\frac{9}{2} \int\limits^\frac{\pi}{2} _0 {sin3\theta} \, d\theta[/tex]
= 27/2 x (-cosθ) - 9/6 x (-cos3θ) | θ = 0 to θ = π/2
= 15 square units.
Therefore, the area enclosed by ellipse [tex]\frac{x^2}{4}+\frac{y^2}{81}=1[/tex] = 4 x A = 4 x 15 = 60 square units.
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Suppose that dollars in principal is invested for years at the given interest rates with continuous compounding. determine the amount that the investment is worth at the end of the given time period. p=1$2000, t=10yr
Amount after 10 years at 3%, 4% and 5.5% interest with principal $20000 is $26878.32, $29604.88 & $34162.88 respectively.
Although a part of your question is missing, you might be referring to this full question: Suppose that P dollars in principal are invested for years at the given interest rates with continuous compounding. Determine the amount that the investment is worth at the end of the given time period. p=$20000, t=10yr
a. 3% interest
b. 4% interest
c. 5.5% interest
P = $2000
T = 10 years
Formula to be used:
A = P(1+r/100)^n
A = total amount after n years
P = principal amount invested
r = rate of interest
n = number of years the money is invested
Solving for part a:
A = 20000(1+3/100)^10
= 20000*1.34
A = 26878.32
Solving for part b:
A = 20000(1+4/100)^10
= 20000*1.48
A = 29604.88
Solving for part c:
A = 20000(1+5.5/100)^10
= 20000*1.70
A = 34162.88
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The difference of a number p and 1 is greater than 6
Answer: p - 1 > 6, or p > 7
Step-by-step explanation:
We will write an expression from the sentence given.
difference of a number (p) and 1 ➜ p - 1
is greater than ➜ >
The difference of a number p and 1 is greater than 6 ➜ p - 1 > 6
To solve for the possible values of p, we will add 1 to both sides of the equation;
p > 7
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Write Each Number As A Power Of 5
• 0.2
• 0.04
• 125^-1
• 15^0
Urgent Plss Thank You !!
The numbers as powers of 5 are 5^-1, 5^-2, 5^-3 and 5^0
How to rewrite the numbers as a power of 5?The numbers are given as:
• 0.2
• 0.04
• 125^-1
• 15^0
Number 1: 0.2
Express as fraction
1/5
Apply the power rule of indices
5^-1
Number 2: 0.04
Express as fraction
1/25
Express 25 as 5^2
1/5^2
Apply the power rule of indices
5^-2
Number 3: 125^-1
Express as fraction
1/125
Express 125 as 5^3
1/5^3
Apply the power rule of indices
5^-3
Number 4: 15^0
Evaluate the exponent
15^0 = 1
Express 1 as 5^0
15^0 = 5^0
Hence, the numbers as a power of 5 are 5^-1, 5^-2, 5^-3 and 5^0
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Solve the system:
c+12d=-66
-4c+3d=-42
Answer: c=6, d=-6
Step-by-step explanation:
c+12d=-66
-4c+3d=-42
(-4c+3d=-42)*4
-16c+12d=-168 ==> subtract this
c+12d=-66 ==> by this
-17c=-168-(-66) ==> subtract the top two equations
-17c=-168+66
-17c=-102
c=6
c+12d=-66
6+12d=-66
12d=-72
d=-6
Simplify (2m6)3.
Please I need the answer fast!
Answer:
6m sqared to 6
Step-by-step explanation:
Answer:
Step-by-step explanation:
(2m6)3
- if you mean (2m^6)^3
it is 8m^18.
What are the coordinates of the point on the directed line segment from (-4, 1)(−4,1) to (1, 6)(1,6) that partitions the segment into a ratio of 4 to 1?
ARITHMATIC SEQUENCE
Grade 11 - one question
Answer:
a₁ = - 11 and d = 3
Step-by-step explanation:
the sum to n terms of an arithmetic sequence is
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]
where a₁ is the first term and d the common difference
given S₄₀ = 1900 , then
S₄₀ = [tex]\frac{40}{2}[/tex] [ 2a₁ + 39d ] = 1900 , that is
20(2a₁ + 39d) = 1900 ( divide both sides by 20 )
2a₁ + 39d = 95 → (1)
the nth term of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
given a₄₀ = 106 , then
a₁ + 39d = 106 → (2)
subtract (2) from (1) term by term to eliminate d
a₁ + 0 = - 11 , so
a₁ = - 11
substitute a₁ = - 11 into (2) and solve for d
- 11 + 39d = 106 ( add 11 to both sides )
39d = 117 ( divide both sides by 39 )
d = 3
i need help in algebra gang i pay 30 a month help mee
Answer:
2c+2
The number that will go in the green box is 2.
Step-by-step explanation:
3c-1-c+3
Combine the like terms.
3c-1-c+3=2c+2
Hope this helps!
Please mark as brainliest if correct!
Find the volume of the solid.
The volume of the solid is
cubic centimeters.
Answer:
27
Step-by-step explanation:
4.5*3*2 = 27
hi,
You just have to multiply the three measures.
So :
2*3*4.5 = 27 cm³
Easy, is it not ?
-7,5 + 4,2 =?
-0,9 * 2,7 =?
[tex]-7,5 + 4,2 = -3.3[/tex]
[tex]-0,9 \cdot 2,7 = -2.43[/tex]
Write an equation for each line.
y-intercept-5/2, x-intercept-1/3
a.y = -5/2x -1/3
c. y = 2.5x + 7.5
b. y = 7.5x + 2.5
d. y = "-7.5x" - 2.5
The equation of the line is y = -7.5x - 2.5
How to determine the equation of the line?The given parameters are
y-intercept = -5/2
x-intercept = -1/3
A linear equation is represented s
y = mx + c
Where
c = y-intercept = -5/2
So, we have:
y = mx - 5/2
The equation x-intercept = -1/3 means that
y = 0 and x = -1/3
So, we have
0 = -1/3m - 5/2
This gives
1/3m = -5/2
Evaluate
m = -15/2
Substitute m = -15/2 in y = mx - 5/2
y = -15/2x - 5/2
Rewrite as
y = -7.5x - 2.5
Hence, the equation of the line is y = -7.5x - 2.5
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Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. ages of children: 5, 6, 7, 8, and 9
There are four levels of measurements in statistics, namely nominal, ordinal, interval, ratio. These levels are important when it comes to analyzing data, since it helps us determine the technique that we can use to support or refute our study.
In the nominal level, we can categorize data but they cannot be ranked. An example would be hair color. In an ordinal data, the data can be both categorize and ranked, but doing mathematical calculation may not make sense. Also, the intervals between rankings doesn't necessarily dictate how close or far apart the data are.
A good example is level of education. In an interval level, the data can be categorized, ranked, and measured but they do not have a true zero. An example could be a range of values that does not include zero. Lastly, in the Ratio level the data can be categorized, rank, and measured, and it has a true zero.
So ratio level is most appropriate for ages of children. Note that ages can be categorized, rank, and measured .Moreover, an age equal to zero means that there is no age or the absence of age.
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What feature in excel allows you to begin with a 1 in a1 and a 2 in a2 and complete the sequence through the number 30 in a30 without having to type in the entire sequence?
Answer:
type 1 in a1 type 2 in a2, (or whatever ur preferred no.s are),then heighlight both cells, then go to the bottom right corner of a2 and drag down or double click
Step-by-step explanation:
Drag down or double-click the bottom right corner of a2 to begin with a 1 in a1 and a 2 in a2 and complete the sequence.
What is excel?Excel is a software in which we can process large amount of data and data handling easily and effectively, Excel can be used in Data analysis.
in a1 type 1 Type 2 in cell a2 (or whatever your chosen numbers are), highlight both cells, and then drag down or double-click the bottom right corner of a2.
Thus, drag down or double-click the bottom right corner of a2 to begin with a 1 in a1 and a 2 in a2 and complete the sequence.
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Is the expression (5x5)^3 equivalent to 5x5^3
Answer:
No.
Step-by-step explanation:
The order of operations in both problems are different, giving two very different results. Therefore, they are unequal.
Why it is very important to follow the steps in solving an equation with 2 or more operations involve?
Is 168 divisible by 2,3,4,5,6,8,9 or 10 explain why please
Help ASAP
Answer:
2,3,4,6,8
Step-by-step explanation:
prime factorization: 2^3×3×7 which has: 2, 3, 4, 6, 8
Answer:
It is divisible by 1, 2, 3, 4, 6, 7, 8
Step-by-step explanation:
Everything is divisible by one, so this is obvious.
168 is an even number, therefore being able to be divided by 2
1+6+8=15 which is in the 3 times tables (3x5=15)
If the last two digits are divisible by 4 (which 68 is) then you can divide it by 4.
168 doesn't end in 5 or 0 so it can't be divided by 5
This number can be divisible by 2 and 3 so it can be divided by 6
160 is divisible by 8 so if you add 8 it is divisible.
1+6+8=15 again which is not divisible by 9
It doesn't end in 0 meaning it cannot be divisible by 10
Hope this helped! <3
1
tion
What fraction represents the shaded part of the circle?
Select one:
O a. 12
O b. 5
4555/5
O d.
12
O c. 7
12
7
Answer:
c
Step-by-step explanation:
because 7 of the 12 sections are shaded in blue.
What is the value of the "?" in the figure below
Answer:
I think it is none of the choices.
Step-by-step explanation:
Triple a number decreased by 25 is equivalent to five less than half the same number.
The number is??
The unknown number is 8.
Let the unknown number be x.
According to the given question.
We have a phrase "Triple a number decreased by 25 is equivalent to five less than half the same number".
Therefore, the equation for the phrase "Triple a number decreased by 25 is equivalent to five less than half the same number" is given by
3x - 25 = x/2 - 5
To find the unknown number we have to solve the above equation for x.
Therefore,
3x - 25 = x - 10/2
⇒ 6x - 50 = x - 10
⇒ 6x - x = -10 + 50
⇒ 5x = 40
⇒ x = 40/5
⇒ x = 8
Hence, the unknown number is 8.
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luna wants to sell a Tshirt
In her shop making a profit of
25%
if the cost of the Tshirt
is 100$, in the same time
the customer will have a discut
of 30%, what shall she put
on a Sale Tag
Answer:
Price: $178.57
30% off: -$53.57
Sale Price: $125.00
Step-by-step explanation:
She buys each shirt for $100.
She wants to make a profit of 25%.
25% of $100 = 0.25 × $100 = $125
She must sell the shirt at $125.
She wants to mark the shirt at a higher price than $125, so that after the 30% discount, she will still sell it for $125.
Call the higher price x.
x - 30% of x = 125
x - 0.3x = 125
0.7x = 125
x = 125/0.7
x = 178.57
The sale tag can read:
Price: $178.57
30% off: -$53.57
Sale Price: $125.00
2 mandy is playing a game. early in the game she had - 250 points. now
mandy has 350 points. what was the change in mandy's score?
The change of Mandy's score is 600 points.
Here,
Mandy is playing a game.
Early in the game she had - 250 points.
Now, Mandy has 350 points.
We have to find the change of Mandy's score.
What is difference of two number?
To find the difference between two numbers, subtract the number with the smallest value from the number with the largest value.
Now,
Early in the game she had - 250 points.
And, now Mandy has 350 points.
So, The change of Mandy's score = 350 - (-250) = 350 + 250 = 600
Hence, The change of Mandy's score is 600 points.
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Which number doesnt belong?
Answer:
-5
Step-by-step explanation:
-5 is the only integer. The other numbers have a decimal that repeats forever.
distribution consists of 30 scores of 10 and 30 scores of 20. Find the ratio of the range to the standard deviation.
I would love these answer please
The relation 1 is not a function, while the relation 2 is a function
How to determine if each relation is a function or not?For a relation to be a function, the following must be true:
Each y value in the function must have a unique x value
i.e. Each y value must have exactly one x value
Using the above as a guide, we have
Relation 1: This relation is not a function, because the x value 5 has two y values 2 and 4
Relation 2: This relation is a function, because each y value must have exactly one x value
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Any tips for doing complicated order of operation problems. Algebra 2
A business has a budget of $480 to spend on their marketing campaign. Each month, x, the business spends $30 on different advertisements. The amount of money remaining in the budget for marketing can be modeled by the function M(x) = −30x + 480. Based on the graph of the linear function M(x) and the context of the problem, what is the domain?
[0, ∞)
[0, 16)
[0, 480]
ℝ
This shows that a reasonable domain for this function is [0, 16)
Domain of a functionThe domain of a function is a value of the independent variable of the function for which it exists.
Given the equation below:
M(x) = −30x + 480
Since the function represents the The amount of money remaining in the budget for marketing then the function f(x) must be greater than zero;
−30x + 480 > 0
-30x> -480
x < 480/30
x < 16
This shows that a reasonable domain for this function is [0, 16)
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____Is used for examining data with more than two variables, and it includes a different vertical axis for each variable.
A Scatter plot is used for examining data with more than two variables, and it includes a different vertical axis for each variable.
What is the Scatter plot?The variables in the scatter plot made from the data are described by the independent variable, time, which is graphed along the horizontal axis. The independent variable will be time, and the dependent variable will be the number of calories provided.
When analyzing data with more than two variables, a scatter plot is used, which has a separate vertical axis for each variable.
Therefore, a scatter plot is used for examining data with more than two variables, and it includes a different vertical axis for each variable.
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1.
Write the numbers of dots in the next three steps of
this pattern:
3
6
10
15
2. What i
Answ
Answer:
your answer is 21
Step-by-step explanation:
The gap between 3 and 6 is 3 the gap between 6 and 10 is 4 the gap between 10 and 15 is 5 so your answer will be 6 more than 15 which is 21
your welcome
Graph the inequality x<3.
To graph an inequality, select the correct type of ray and then select the correct endpoint on the number line.
Answer:
You would start at 3 and then go backwords till the end.
So you'd choose the 3 option. <-o
Only use the filled in o if the equation is <_ with the line underneath, if its like < then its o
Hope this helps! <3
Quadrilateral ABCD has vertices A = (2, 5), B = (2, 2), C = (4, 3) and D = (4, 6). Quadrilateral A'B'C'D' is formed when Quadrilateral ABCD is dilated by a scale factor of 2. Which statement is true? Select all that apply
can someone answer this please
Considering that the transformation is a dilation, the correct statement is given by:
The angles of Quadrilateral ABCD and A'B'C'D' are the same.
What are the statements?There are three statements:
None of the answers apply.The side lengths of Quadrilateral ABCD and A'B'C'D' are the same.The angles of Quadrilateral ABCD and A'B'C'D' are the same.What is a dilation?A figure is said to go through a dilation is when it's side lengths are changed, either enlarged or diminished, but the angles remain constant.
For this problem, the scale factor is of 2, meaning that the side lengths are doubled, while the angles remain constant and the correct statement is:
The angles of Quadrilateral ABCD and A'B'C'D' are the same.
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