Divide 2/8 by 9/18 . Input your answer as a reduced fraction.\
Answers
Answer:
1/2
Step-by-step explanation:
There is a couple ways of thinking about this. Probably the fastest way is the rule, but the rule is not intuitive. The rule is when you divide fraction, you keep the first fraction the same and then multiple and flip the second fraction.
2/8 x 18/9 = 36/72 To reduce the fraction, could keep dividing the top and the bottom by the same number until it is no longer possible, I could use: 2, 4 3, 9, 18, or 36. If I used 36, I would only have to divide once. 36/16 is 1 and 72/36 is 2 so my answer would be 1/2.
I could have divided the top and bottom by 2 and got 18/36 and then divide by 2 again and get 9/18 and then by 9 and get 1/2. As you can, see there are a lot of option.
The second method is in the picture that I sent you, it is a little more work, but it makes sense.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{2}{8}\div\frac{9}{18} }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf Simplify \ \frac{2}{8} \ to \ \frac{1}{4}. \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1}{4}\div\frac{18}{9} }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf Use \ this \ rule: a\div \frac{b}{c}=a\times \frac{c}{b}a \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1}{4}\times\frac{18}{9} }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf Use \ this \ rule: \frac{a}{b} \times \frac{c}d{}=\frac{ac}{bd}. \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1\times18}{4\times9} \ \to \ \ Multiply }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{ \frac{18}{36}= }}\boxed{\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1}{2} }} \end{gathered}$}} \end{gathered}$}[/tex]
Related Questions
On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (2, negative 1). Everything to the left of the line is shaded.
Which linear inequality is represented by the graph?
y > 2x + 3
y < 2x + 3
y > −2x + 3
y < −2x + 3
Answers
The linear inequality of the given graph is: D. y < -2x + 3.
How to Determine the Linear Inequality of a Graph?First, using two points on the line, (0, 3) and (1.5, 0), find the slope (m).
Slope (m) = change in y / change in x = (3 - 0) / (0 - 1.5)
Slope (m) = 3/-1.5 = -2.
Next, substitute (x, y) = (0, 3) and m = -2 into y = mx + b to find the value of b.
3 = -2(0) + b
3 = 0 + b
b = 3
Substitute m = -2 and b = 3 into y < mx + b (the shaded part is at the left, so we use the "<" sign)
y < -2x + 3
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David invested $230 in a savings account that offers a 3% return on the investment. the value of david's investment will be at least $415 after a
period of years.
hint: use the formula a = a1 + ', where a is the amount after tyears, pis the amount invested, r is the rate of interest, and is the time period.
use a calculator to compute the answer, and round it off to the nearest year.
Answers
20 years is the answer.
A= R(1+r)^ t
A-415
R=230
r=3%
415=230 (1+3%)^ t
t=log1.03 315/230
t=19.89 (use calculator)
David will invest at least 20 years.
An investment is a dedication of an asset to achieve an increase in value over a period of time. Investing requires the sacrifice of current assets such as time, money and effort. In finance, the purpose of an investment is to generate a profit on the invested asset
The most common example of an investment type. Investment is generally what you want to use in the future with the aim of generating regular cash flow or increasing the value of something over time so that you can sell it at a higher price than you purchased.
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PLEASE HELP I HAVE AN HOUR LEFT!!
Which statement correctly identifies an asymptote of g (x) = StartFraction 42 x cubed minus 15 Over 7 x cubed minus 4 x squared minus 3 EndFraction using limits?
Limit of g (x) as x approaches plus-or-minus infinity= 5, so g(x) has an asymptote at x = 5.
Limit of g (x) as x approaches plus-or-minus infinity= 6, so g(x) has an asymptote at x = 6.
Limit of g (x) as x approaches plus-or-minus infinity= 5, so g(x) has an asymptote at y = 5.
Limit of g (x) as x approaches plus-or-minus infinity = 6, so g(x) has an asymptote at y = 6.
Answers
The statement that correctly describes the horizontal asymptote of g(x) is:
Limit of g (x) as x approaches plus-or-minus infinity = 6, so g(x) has an asymptote at y = 6.
What are the asymptotes of a function f(x)?The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.The horizontal asymptote is the limit of f(x) as x goes to infinity, as long as this value is different of infinity.In this problem, the function is:
[tex]g(x) = \frac{42x^3 - 15}{7x^3 - 4x^2 - 3}[/tex]
The horizontal asymptote is given as follows:
[tex]y = \lim_{x \rightarrow \infty} g(x) = \lim_{x \rightarrow \infty} \frac{42x^3 - 15}{7x^3 - 4x^2 - 3} = \lim_{x \rightarrow \infty} \frac{42x^3}{7x^3} = \lim_{x \rightarrow \infty} 6 = 6[/tex]
Hence the correct statement is:
Limit of g (x) as x approaches plus-or-minus infinity = 6, so g(x) has an asymptote at y = 6.
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The ________ could be used to describe a data set by itself, while ___________ would almost never be used to describe a data set by itself.
Answers
The mean could be used to describe a data set by itself, while interquartile range would almost never be used to describe a data set by itself.
Brief Description of Mean and Interquartile Range
While interquartile range only assesses the middle half of the data, mean and range deal with the entire set of data. The mean is significant in statistics because it helps us determine where a dataset's "center" is. The mean contains information from each observation in a dataset as a result of how it is calculated.
The interquartile range in descriptive statistics reveals the spread of your distribution's middle half. Any distribution that is sorted from low to high is divided into four equal portions using quartiles. The second and third quartiles, or the center half of your data set, are contained in the interquartile range (IQR).
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Two poles, AB and ED, are fixed to the ground with the help of ropes AC and EC, as shown:
Two right triangles ABC and EDC have a common vertex C. Angle ABC and EDC are right angles. AB is labeled 12 feet, AC is labeled 14 feet, EC is labeled 10 feet, and ED is labeled 7 feet.
What is the approximate distance, in feet, between the two poles?
7.14 feet
7.21 feet
14.35 feet
15.59 feet
Answers
Answer:
14.35 ft
Step-by-step explanation:
We can use the pythagorean theorem to find the distance between the poles since both triangles are right triangles.
a^2 + b^2 = c^2
a^2 + 12^2 = 14^2
a^2 + 144 = 196
a^2 = 52
a = 7.21 ft
a^2 + b^2 = c^2
a^2 + 7^2 = 10^2
a^2 + 49 = 100
a^2 = 51
a = 7.14 ft
7.21 + 7.14 = 14.35 ft
Brainliest, please :)
Answer: c; 14.35
Step-by-step explanation: I need help on this and i saw this guy asked for brainlist and i think two people need to anser so please give him brainlist.
Given: mAngleTRV = 60°
mAngleTRS = (4x)°
Prove: x = 30
3 lines are shown. A line with points T, R, W intersects with a line with points V, R, S at point R. A line extends from point R to point Z between angle V R W. Angle V R T is 60 degrees and angle T, R, S is (4 x) degrees.
What is the missing reason in step 3?
A 2-column table with 6 rows is shown. Column 1 is labeled Statements with entries measure of angle T R V = 60 degrees and measure of angle T R X = (4 x) degrees, angle T R S and angle T R V are a linear pair, measure of angle T R S + measure of angle T R V = 180, 60 + 4 x + 180, 4 x =120, x = 30. Column 2 is labeled Reasons with entries given, definition of a linear pair, question mark, substitution property of equality, subtraction property of equality, division property of equality.
substitution property of equality
angle addition postulate
subtraction property of equality
addition property of equality
Mark this and return
Answers
https://brainly.com/question/68367When two lines intersect at a point, angles are formed. Some of these angles formed are vertically opposite and thus are equal.
Therefore, the required proof and answer to the question are stated below:
a) m< TRV = 60° (given)
m<TRS = 4x° (given)
Thus, it can be concluded from the diagram that:
<TRV ≅ m<BRW (vertically opposite angle property)
Also,
m<TRS ≅ m<VRW (vertically opposite angle property)
But,
m<VRW = m<VRZ + m<ZRW
Thus,
m<TRV ≅ m<BRW = 60°
m<TRV + m<BRW + m<TRS + m<VRW = [tex]360^{o}[/tex]
60° + 60° + m<TRS + m<VRW = [tex]360^{o}[/tex]
m<TRS + m<VRW = [tex]360^{o}[/tex] - [tex]120^{o}[/tex]
= [tex]240^{o}[/tex]
2m<TRS = [tex]240^{o}[/tex] (since m<TRB = m<VRW )
m<TRS = 120
4x = 120
x = [tex]\frac{120}{4}[/tex]
= [tex]30^{o}[/tex]
Thus, x = [tex]30^{o}[/tex]
b) The missing reason in step 3 is the angle addition postulate.
jhgvbiluhliuhnhl
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In the figure below, O is the center of the circle. Name a diameter of the circle.
Answers
The diameter of circle O in the image given is: AB.
What is the Diameter of a Circle?The diameter of a circle can be referred to as the largest chord in a circle which is the line segment that passes through the center of a circle with both ends on the circle.
In the image given, AB is the largest chord and also passes through the center of the circle, O.
Thus, the diameter is: AB.
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By use of technology, we investigated Mary’s investment and created the model, M(x) = 3.03(1.28)2x, in thousands of dollars. What was Mary’s initial investment? $4.96 $3,030 $4,960 $3.03
Answers
Using an exponential function, it is found that Mary's initial investment was of $3,030.
What is an exponential function?An exponential function is modeled by:
[tex]y = ab^x[/tex].
In which:
a is the initial value.b is the rate of change.Her investment model, in thousands of dollars, is:
[tex]M(x) = 3.03(1.28)^{2x}[/tex]
Then a = 3.03, since we measure the amount in thousands of dollars, Mary's initial investment was of $3,030.
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kathy uses a 1/2 cup of milk with every bowl of her favorite cereal if there are only 3 3/5 cups of milk left, then how many bowls of cereal would kathy have?
Answers
Taking a quotient, we will see that she can make 7 bowls of cereal (and some leftover milk).
How many bowls of cereal would kathy have?We know that for each bowl, she needs 1/2 cups of milk.
And we also know that she has a total of (3 + 3/5) cups of milk.
To know how many bowls she can make, we need to take the quotient between the total that she has and the amount that she needs for each bowl:
(3 + 3/5)/(1/2)
We can rewrite the total as:
3 + 3/5 = 15/5 + 3/5 = 18/5
Then the quotient becomes:
(18/5)/(1/2) = (18/5)*2 = 36/5 = 35/5 + 1/5 = 7 + 1/5
So she can make 7 bowls of cereal (and some leftover milk).
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Evaluate 5 - 3(a3 – b2)2 when a = 3 and b = 5.
Answers
Answer:
Step-by-step explanation:
Evaluate 5 - 3(a3 – b2)2 when a = 3 and b = 5.
5 - 3 × (a³-b²)² a=3 and b=5
5 - 3 × (3³ - 5²)² =
5 - 3 × (27 - 25)² =
5 - 3 × (2)² =
5 - 3 × 4 =
5 - 12 =
-7
On a coordinate plane, a curved line with a minimum value of (negative 2.5, negative 12) and a maximum value of (0, negative 3) crosses the x-axis at (negative 4, 0) and crosses the y-axis at (0, negative 3).
Which statement is true about the graphed function?
F(x) < 0 over the interval (–∞, –4)
F(x) < 0 over the interval (–∞, –3)
F(x) > 0 over the interval (–∞, –3)
F(x) > 0 over the interval (–∞, –4)
Answers
The points on the graph of (-4, 0), (-2.5, -12), and (0, -3), gives;
F(x) > 0 over the interval (-∞, -4)Which method can be used to find the true statement?From the description of the graph, we have;
Furthest point left of the graph = (-4, 0)
The furthest point right on the graph = (0, -3) = The maximum point
The minimum point = (-2.5, -12)
F(x) < 0 at the minimum point
The minimum point is to the right of x = -4
The point the graph crosses the y-axis = (0, -3)
Therefore;
The interval of the graph where F(x) is larger than 0 is to the left of (-4, 0), is the interval (-∞, -4)
The true statement is therefore;
F(x) > 0 over the interval (-∞, -4)Learn more about graphs of functions here:
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please help me im stuck on this
Answers
Answer:
Step-by-step explanation:
dddfykb
The sum of two odd numbers is 80 and their difference is 6. work out these numbers.
Answers
The two odd numbers are 43 and 37.
What are whole numbers?Whole numbers are positive numbers belonging to the set W ∈ {1, 2,3, 4, ...}
The two odd numbers can be represented as (2m-1) and (2n-1) respectively sum=80 and difference = 6
Let m and n be two whole numbers
Therefore
,[tex]2m-1) + (2n-1)= 80\\(2m-1) - (2n-1)= 6\\\\2(m+n-1)=80\\2(m-n)=6\\\\m+n = 41\\m-n=3\\[/tex]
Adding the two equations
[tex](m+n)+(m-n)=2m=44\\m=22\\n=m-3=19\\[/tex]
so m=22 ad n=19. our two odd numbers are 2m-1 = 43 and 2n-1 = 37
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The square of y varies directly as the cube of x. when x = 4, y = 2. which equation can be used to find other combinations of x and y?
Answers
The equation can be used to find other combinations of x and y is y^2 = (1/16)x^3
How to determine the equation?The direct variation from the square of y to the cube of x is represented as:
y^2 = kx^3
Where k represents the variation constant.
When x = 4, y = 2.
So, we have:
2^2 = k * 4^3
This gives
4 = 64k
Divide both sides by 64
k = 1/16
Substitute k = 1/16 in y^2 = kx^3
y^2 = (1/16)x^3
Hence, the equation can be used to find other combinations of x and y is y^2 = (1/16)x^3
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In a group of 55 people, 3x+5 people like banana and x+5 people like apple. If all the people who like apple also like banana and if 2x+15 people like at least one of fruit then find how many like
(i) bananas only,
(ii) none of the fruits
(iii) at most one fruit
also show subset
Answers
i. The number of people who like banana only is 20.
ii. The number of people who like none of the fruits is zero.
iii. The number of people who like at most one fruit is 30.
The question has to do with sets.
What is a set?A set a collection of well ordered items
i. How to find how many people like banana only.Since we have 55 people, 3x + 5 people like banana and x + 5 people like apple. If all the people who like apple also like banana and if 2x + 15 people like at least one of fruit then, we have that
3x + 5 + x + 5 + 2x + 15 = 55
3x + 2x + x + 5 + 5 + 15 = 55
6x + 30 = 55
6x = 55 - 25
6x = 30
x = 30/6
x = 5
Since the number of people who like banana only is 3x + 5.
So, number of people who like banana only is n = 3x + 5
= 3(5) + 5
= 15 + 5
= 20
So, the number of people who like banana only is 20.
ii. The number of people who none of the fruits.Since from the question, a person likes at least one fruit, either banana, apple or both. No one likes none of the fruits.
So, the number of people who like none of the fruits is zero.
iii. The number of people who like at most one fruit?To like at most one fruit, the person either likes apple or banana.
So, their sum is n = 3x + 5 + x + 5
= 4x + 10.
Since x = 5, we have n = 4x + 10.
= 4(5) + 10
= 20 + 10
= 30
So, the number of people who like at most one fruit is 30.
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The roots of the equation (p+2)x^2-2px=5-p are complex; find the range of values of p
Answers
Range of values of the variable p is [tex]0\leq p\leq \frac{10}{3}[/tex]
Step-by-step explanation:
From general quadratic equation [tex]ax^{2} +bx+c[/tex], for imaginary case-
[tex]D\leq 0\\b^2-4ac\leq 0[/tex]
In given question, [tex]a= p+2[/tex], [tex]b=2p[/tex], [tex]c= 5-p[/tex]
using above condition-
[tex]b^2-4ac = 4p^2-4(2p)(5-p)\leq 0\\4p^2 - 40p+8p^2\leq 0\\12p^2-40p\leq 0\\4p(3p-10)\leq 0[/tex]
Here, [tex]4p\leq 0, p\leq 0[/tex]
and
[tex]3p-10\leq 0\\p\leq \frac{10}{3}[/tex]
Therefore, range of values of the variable p is [tex]0\leq p\leq \frac{10}{3}[/tex].
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A rectangular gate measures 1.2 m by 2.3 m with a 2.4 m diagonal. Is the gate
square? If not, should the diagonal be longer or shorter?
Answers
Using Pythagoras Theorem
a^2+b^2=c^2
(1.2)^2+(2.3)^2= c^2
c^2= 6.73
c= 2.594224354214569 (Negative value rejected, length cannot be negative)
Therefore the diagonal should be longer.
Hope it helps :)
A manufacturer produces a commodity where the length of the commodity has approximately normal distribution with a mean of 6.5 inches and standard deviation of 0.5 inches. If a sample of 46 items are chosen at random, what is the probability the sample's mean length is greater than 6.3 inches? Round answer to four decimal places.
Answers
The probablity that the sample's mean length is greate than 6.3 inches is0.8446.
Given mean of 6.5 inches,standard deviation of 0.5 inches and sample size of 46.
We have to calculate the probability that the sample's mean length is greater than 6.3 inches is 0.8446.
Probability is the likeliness of happening an event. It lies between 0 and 1.
Probability is the number of items divided by the total number of items.
We have to use z statistic in this question because the sample size is greater than 30.
μ=6.5
σ=0.5
n=46
z=X-μ/σ
where μ is mean and
σ is standard deviation.
First we have to find the p value from 6.3 to 6.5 and then we have to add 0.5 to it to find the required probability.
z=6.3-6.5/0.5
=-0.2/0.5
=-0.4
p value from z table is 0.3446
Probability that the mean length is greater than 6.3inches is 0.3446+0.5=0.8446.
Hence the probability that the mean length is greater than 6.3 inches is 0.8446.
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2. A high-speed train travels at a speed of 200 km/h. If the train sets off from Station A at 12 24 and reaches Station B at 14 12, find the distance between the two stations, giving your answer in metres.
Answers
The distance between the two stations is 360km.
What is speed?Speed is the rate of change of distance.
Rate is a measure of one quantity against another in this case distance and time.
Analysis:
time at station A = 12:24
time at station B = 14:12
time spent = 14:12 - 12:24 = 1 hour 48 minutes = convert 48 minutes to hour
we divide 48 by 60 = 0.8
Total time = 1.8 hours
Distance = speed x time = 200 x 1.8 = 360 km
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Solve the quadratic equation numerically (using tables of x- and y- values). x(x + 6) = 0 a. x = -1 or x = 3 c. x = 0 or x = -3 b. x = 0 or x = 6 d. x =0 or x = -6
Answers
Answer:
d. x =0 or x = -6
Step-by-step explanation:
x(x + 6) = 0
This is telling us that either x is 0 or x is -6, because:
1) when x = 0, 0*(0+6)=0, and
2) when x = -6, -6*(-6+6) = 0; -6(0) = 0
The width of a rectangle is 4 less than twice its length. if the area of the rectangle is 75 cm 2 , what is the length of the diagonal?
Answers
Answer:
Step-by-step explanation:
BRAINLIEST ASAP!!!
For which interval(s) is the function increasing and decreasing? y=3x^3 -16x+2
Answers
Considering the critical points of the function, we have that:
The function is increasing for |x| > 1.63.The function is decreasing for |x| < 1.63.What are the critical points of a function?The critical points of a function are the values of x for which:
[tex]f^{\prime}(x) = 0[/tex]
In this problem, the function is:
[tex]f(x) = 3x^3 - 16x + 2[/tex]
The derivative is:
[tex]f^{\prime}{x} = 6x^2 - 16[/tex]
The critical points are given as follows:
[tex]6x^2 - 16 = 0[/tex]
[tex]x^2 = \frac{16}{6}[/tex]
[tex]x = \pm \sqrt{\frac{16}{6}}[/tex]
[tex]x = \pm 1.63[/tex]
For x < -1.63, one example of the derivative is:
[tex]f^{\prime}{-2} = 6(-2)^2 - 16 = 8[/tex]
Positive, hence increasing.
For -1.63 < x < 1.63, one example of the derivative is:
[tex]f^{\prime}{0} = 6(0)^2 - 16 = -16[/tex]
Negative, hence decreasing.
For x > 1.63, one example of the derivative is:
[tex]f^{\prime}{2} = 6(2)^2 - 16 = 8[/tex]
Positive, hence increasing.
Hence:
The function is increasing for |x| > 1.63.The function is decreasing for |x| < 1.63.More can be learned about critical points at https://brainly.com/question/2256078
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Which of the following are identities? I. y=x II. 4=3x2+2 III. x=x−−√ A. II and III B. I and III C. all D. none
Answers
None of the expressions is an identity
How to determine the identities?The expressions are given as:
I. y=x
II. 4 = 3x^2 + 2
III. x=x
In algebra, there are three identities; and they are
(x+y)^2 = x^2 + y^2 + 2xy(x-y)^2 = x^2 + y^2 – 2xyx^2 – y^2 = (x+y) (x-y)None of the given expression take the above forms
Hence, none of the expressions is an identity
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Which equation is equivalent to the given equation?
−4x = x² + 3
x² - 4x +3=0
x² - 4x - 3=0
x² + 4x - 3=0
x² + 4x +3=0
Answers
Answer: x² + 4x +3=0
Step-by-step explanation:
Add 4x to both sides.
which of the following statements would be most similar to an electiral current passing through a metal wire
Answers
The correct option is A. hot water moving through a pipe.
Electrical current passing through a metal wire is similar to hot water moving through a pipe.
What is Electrical current?Electrical charge carriers, often electrons or atoms deficient in electrons, travel as current. The capital letter I is a typical way to represent current. The ampere, denoted by the letter A, is the common unit.
The method of flowing of electrical current in metal wire is-
Faraday's Law established that when spinning magnets are close to a coil of wire, a voltage results. With the help of that voltage, you can force electrons through wires, and the moving electrons will travel to their intended locations and do useful tasks. In essence, that is how the electrical grid functions.Comparison of flow of electricity with flow of water in pipe-
Electrical charge (a current) flowing through a wire is comparable to water flowing through a conduit without any bubbles or leaks. The flow of charge is resisted by a resistor, just as the flow of water is resisted by a constriction in a pipe. A circuit's voltage can be compared to a pipe's pressure.To know more about Electric current, here
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The complete question is-
which of the following situations would be mist similar to an electrical current passing through a metal wire in a closed circuit?
A. hot water moving through a pipe
B. an automobile traveling through a tunnel
C. an airplane flying through clouds
D. a flying pan heating up on a hot plate
E. a bird building a nest
the half life of an element in the periodic table measured over a period of time, t, is modeled by the function f(t)=16(1/2)^t. what is the initial amount of the element
Answers
The initial amount of the element is 16
How to determine the initial amount?The function is given as:
f(t) = 16(1/2)^t
Set t = 0, to determine the initial amount
f(0) = 16(1/2)^0
This gives
f(0) = 16 * 1
Evaluate
f(0) = 16
Hence, the initial amount of the element is 16
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A house is octagon-shaped, and each side measures 22 feet long. How many lineal feet of exterior wall does this house have
Answers
Length of each side is 176 feet ² .
What is octagon ?A polygon of eight angles and eight sides.It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° ( radians). The central angle is 45° ( radians).each side of house = 22 feet long
Perimeter of octagon = 8 × sides
Length of each side of house = 8 × 22 ⇒ 176 feet²
Therefore, length of each side is 176 feet ² .
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Use cylindrical or spherical coordinates, whichever seems more appropriate. Evaluate z dV E , where E lies above the paraboloid z
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The resulted integral is [tex]I=\frac{8}{3} \times \frac{5 \pi}{16}=\frac{5 \pi}{6}[/tex].
What is integrals?In mathematics, an integral is either a number representing the region under a function's graph for a certain interval or just an added to the initial, the derivative of which is initial function (indefinite integral).
Computation of the integrals:
Step 1: We employ the equations in cylindrical coordinates.
[tex]x=r \cos \theta, y=r \sin \theta, z=z[/tex]
Thus, in cylindrical coordinate system,
E lies above the paraboloid [tex]z=r^{2}[/tex] and below the plane [tex]z=2 r \sin \theta[/tex] .
Therefore, the top part E is [tex]z=2 r \sin \theta[/tex] is the cross-section between paraboloid and the plane.
Now, at the cross-section use, [tex]r^{2}=2 r \sin \theta[/tex] and [tex]z=2 r \sin \theta[/tex] .
Thus, the limits are given as ;
[tex]r^{2} \leq z \leq 2 r \sin \theta \quad 0 \leq r \leq 2 \sin \theta[/tex]
Apply the limits as compute the integration;
[tex]\begin{aligned}I=\iiint_{E} z d V &=\int_{0}^{\pi} \int_{0}^{2 \sin \theta} \int_{\tau^{2}}^{2 r \sin \theta} z r d r d z d \theta \\&=\int_{0}^{\pi} \int_{0}^{2 \sin \theta}\left[\frac{z^{2}}{2}\right]_{r^{2}}^{2 r \sin \theta} r d r d \theta \\&=\frac{1}{2} \int_{0}^{\pi} \int_{0}^{2 \sin \theta}\left[4 r^{2} \sin ^{2} \theta-r^{4}\right] r d r d \theta\end{aligned}[/tex]
[tex]\begin{aligned}&=\frac{1}{2} \int_{0}^{\pi} \int_{0}^{2 \sin \theta}\left[4 r^{3} \sin ^{2} \theta-r^{5}\right] d r d \theta \\&=\frac{1}{2} \int_{0}^{\pi}\left[r^{4} \sin ^{2} \theta-\frac{r^{6}}{6}\right]_{0}^{2 \sin \theta} d \theta \\&=\frac{8}{3} \int_{0}^{\pi} \sin ^{6} \theta d \theta\end{aligned}[/tex]
Step 2: Now, calculate for the [tex]I_{1}=\int_{0}^{\pi} \sin ^{6} \theta d \theta[/tex].
[tex]\begin{aligned}\sin ^{6} \theta &=\left(\sin ^{2} \theta\right)^{2} \times \sin ^{2} \theta \\&=\left[\frac{1-\cos 2 \theta}{2}\right]^{2} \times\left[\frac{1-\cos 2 \theta}{2}\right] \\&=\frac{1}{8}\left(1-2 \cos 2 \theta+\cos ^{2} 2 \theta\right)(1-2 \cos 2 \theta) \\&=\frac{1}{8}\left(1-2 \cos 2 \theta+\frac{1+\cos 4 \theta}{2}\right)(1-2 \cos 2 \theta)\end{aligned}[/tex]
[tex]\begin{aligned}&=\frac{1}{16}(3-4 \cos 2 \theta+\cos 4 \theta)(1-2 \cos 2 \theta) \\&=\frac{1}{32}(10-15 \cos 2 \theta+6 \cos 4 \theta-\cos 6 \theta)\end{aligned}[/tex]
Further compute the value of
[tex]\begin{aligned}I_{1} &=\int_{0}^{\pi} \sin ^{6} \theta d \theta \\&=\frac{1}{32} \int_{0}^{\pi}(10-15 \cos 2 \theta+6 \cos 4 \theta-\cos 6 \theta) d \theta \\&=\frac{1}{32}\left[10 \theta-\frac{15 \sin 2 \theta}{2}+\frac{3 \sin 4 \theta}{2}-\frac{\sin 6 \theta}{6}\right]_{0}^{\pi} \\&=\frac{5 \pi}{16}\end{aligned}[/tex]
Therefore, the obtained integral is [tex]I=\frac{8}{3} \times \frac{5 \pi}{16}=\frac{5 \pi}{6}[/tex].
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The complete question is -
Use cylindrical or spherical coordinates, whichever seems more appropriate. Evaluate ∫∫∫E z dV, where E lies above the paraboloid
z = x² + y²
and below the plane z = 2y. Use either the Table of Integrals or a computer algebra system to evaluate the integral.
A sphere of radius 222 inches is cut by three planes passing through its center. This partitions the solid into 888 equal parts, one of which is shown above. The volume of each part is t\pitπt, pi cubic inches. What is the value of ttt?
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The value of t based on the information about the sphere is 1.3π.
How to calculate the value?It should be noted that the volume of a sphere is 4/3πr³. In this case, it's divided into 8 equal parts.
Volume of each part = 1/8 × 4/3πr³
= 1/8 × 4/3 × π × 8
= 4/3π
= 1.3πin³
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Measure the angle shown below. a protractor showing an angle going through the tenth tick mark after ten degrees 15° 17° 20° 22°
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The value of the angle will be C. 20°
How to calculate the angle?From the information given, it was stated that the protractor showed an angle going through the tenth tick mark after ten degrees.
This means the value of the angle will be:
= 10° + 10°
= 20°
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