The area of the surface obtained by rotating the circle [tex]x^2 + y^2 = r^2[/tex] about the line y = r is π²r² square units.
To find the area of the surface obtained by rotating the circle
[tex]x^2 + y^2 = r^2[/tex] about the line y = r, we can use the method of cylindrical shells.
The circle [tex]x^2 + y^2 = r^2[/tex] is centered at the origin (0, 0) with a radius r. The line y = r is the line y-axis but shifted up by r units.
When we rotate the circle about the line y = r, it forms a 3D shape called a torus or a donut shape.
Consider a small strip on the circle at a distance y from the line y = r.
This small strip is at a distance r - y from the y-axis.
The length of this strip is the circumference of the circle at y, which is 2πy (since the circumference of a circle is 2π times its radius).
The width of this strip is the change in x, which we can denote as dx.
The area of this small strip is then given by the product of its length and width, which is 2πy dx.
Now, to find the total surface area, we integrate this area over the range of y values from -r to r (since the circle is symmetric about the y-axis):
Total Surface Area = ∫[from -r to r] 2πy dx
Now, we need to express y in terms of x using the equation of the circle [tex]x^2 + y^2 = r^2:\\y^2 = r^2 - x^2[/tex]
y = ±√(r² - x²)
Since we are considering the upper half of the circle, we take the positive square root:
y = √(r² - x²)
Now, we can rewrite the integral with respect to x:
Total Surface Area = ∫[from -r to r] 2π√(r² - x²) dx
To solve this integral, we can make a trigonometric substitution:
Let x = r sin(θ), then dx = r cos(θ) dθ.
When x = -r, θ = -π/2, and when x = r, θ = π/2.
Now the integral becomes:
Total Surface Area = ∫[from -π/2 to π/2] 2π√(r² - (r sin(θ))²) (r cos(θ)) dθ
Total Surface Area = 2πr² ∫[from -π/2 to π/2] √(1 - sin²(θ)) cos(θ) dθ
Now, we can use the trigonometric identity:
sin²(θ) + cos²(θ) = 1
√(1 - sin²(θ)) = cos(θ)
Total Surface Area = 2πr² ∫[from -π/2 to π/2] cos²(θ) dθ
Now, use the trigonometric identity: cos²(θ) = (1 + cos(2θ))/2
Total Surface Area = 2πr² ∫[from -π/2 to π/2] (1 + cos(2θ))/2 dθ
Total Surface Area = 2πr² [θ/2 + (sin(2θ))/4] [from -π/2 to π/2]
Total Surface Area = 2πr² [(π/2 + sin(π) - (-π/2 + sin(-π)))/4]
Since sin(π) = 0 and sin(-π) = 0:
Total Surface Area = 2πr² [(π/2 - (-π/2))/4]
Total Surface Area = 2πr² (π/2)
Total Surface Area = π²r²
So, the area of the surface obtained by rotating the circle [tex]x^2 + y^2 = r^2[/tex] about the line y = r is π²r² square units.
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Anybody got an answer for this
Answer:
11
Step-by-step explanation:
brainliest me pls
A basking shark measures 25 feet long. A giant squid measures
33 feet long. How many inches longer is the basking shark than
the giant squid?
ounds and
Answer: 8 feet
Step-by-step explanation:
you take 33-25 to get the sum of 8. so the squid is 8 feet longer than the shark.
Dalis' employer pays $2,150 in health insurance and $86
in life insurance per year. She also gets $3,520 in paid time
off per year. Her monthly gross pay is $4,370. What are
Dalis' total job benefits?
Answer: chiki
Step-by-step explanation:
A jar of coins contains only quarters and dimes. The total amount of money in the jar is $4.40, and there are 23 coins.
A. Write a system of equations that can be used to find d, the number of dimes, and q, the number of quarters, in the jar.
B. How many quarters are in the jar?
Answer:
(1) d+q=23 (the total number of coins is 23)
(2) 10d+25q = 440 (the total value of the coins, at 10 cents for each dime and 25 cents for each quarter, is 440 cents)
there are 14 quarters and 9 dimes
Step-by-step explanation:
Please help me with this
Answer:
9) C: x=4, y=2rt3
10) A: XY/YZ
11) A: XY/XZ
Step-by-step explanation:
9 - that is a 30, 60, 90 triangle so the ratios will be a, root3a, 2a
10 - tan = opposite / adjacent
11 - cos = adjacent/hypotenuse
Answer:
9. (c)
10. (a)
11. (a)
................
Pls answer! this problem
Answer:
9k + 36 represent the perimeter of the triangle. (I think)
Step-by-step explanation:
3k + 5 + 15 + 4k + 2k + 16 = the perimeter of the triangle
reorganize:
(3k + 4k + 2k) + (5 + 15 + 16)
9k + 36
Have a nice day!
A salesperson at a jewelry store earns 8% commission each week. Last week, Jarrod sold $720 worth of jewelry. How much
did he make in commission? How much did the jewelry store make from his sales?
Jarrod earned $ in commission.
Mr. Thompson earns $ 11.52 per hour. Last week he worked 37 hours how much F.I.C.A tax did he have to pay at 7.65%?
Answer:
$32.61
Step-by-step explanation:
First, we multiply the amount he earns per hour by the amount of hours he worked. $11.52 * 37 = $426.24. Then, we figure out what 7.65% of $426.24 is, which we can do by multiplying 426.24 by 0.0765. This gives us $32.60736, but since there is no form of US money that is worth less than 1 cent, we round to the nearest cent and get $32.61.
Hope this helps!
Jon flips a coin and writes down H if heads comes up or T if tails comes up. He does this 3 times. If the order of the results matters, how many possible outcomes are there?
Answer:
There are always two possible outcomes H (heads) and T (tails) but the possible combination outcomes are HH, TT, HT, and TH
Step-by-step explanation:
It is just my common sense~
There is 8 possible outcomes.
y = 3x + 13
y = -2x - 22
substitution method show work
pls someone help quick
Answer:
x = -7
Step-by-step explanation:
3x + 13 = -2x - 22
subtract 13 from both sides
3x = -2x - 35
Add 2x to both sides
5x = -35
Divide each side by 5
x = -7
Algebra 2
1) What is the multiplicative inverse of 300?
a) 300
b) 0.03
c) 1/300
d) None of the above
e) a and b
f) b and c
g) a and c
Answer:
Step-by-step explanation:
Option C is the correct answer
Answer:
the multiplicative inverse of 300 is 300Step-by-step explanation:
answer 300Which matrix represents the system of equations shown below?
2x - 9y = 12
-5x - 7y = 15
O A.
[2 -9 12
5 -7 15
O B.
1-2 -9 127
5 -7 15
O C.
2 -9 127
-5 -7 15
O D. 13:53
1-2 -9 -121
-5 -7 -15
Answer:
c because c is c kkjwjwjjjwjwjsjssjsjjssmsswnsnsnsnnsns
The matrix that is represented by the system of equations is:
[tex]\left[\begin{array}{ccc}2&-9&12\\-5&-7&15\end{array}\right][/tex]
Option C is the correct answer.
What is a matrix?A matrix is a set of numbers arranged in rows and columns such that it forms a rectangular array.
We have,
Here, the coefficient matrix is the matrix on the left-hand side, the variable matrix is the column matrix containing the variables x and y, and the constant matrix is the column matrix on the right-hand side containing the constants 12 and 15.
To solve the system of equations, we can use matrix operations to transform the coefficient matrix into the identity matrix, while performing the same row operations on the variable matrix and the constant matrix.
Once we have transformed the coefficient matrix into the identity matrix, the variable matrix will be transformed into the solution vector containing the values of the variables x and y that satisfy the system of equations.
Thus,
The matrix that is represented by the system of equations is:
[tex]\left[\begin{array}{ccc}2&-9&12\\-5&-7&15\end{array}\right][/tex]
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Find the equation of the line
Step-by-step explanation:
Use photo math in the AppStore
Answer: y=1/4x-6
Step-by-step explanation:
The slope formula is y=mx+b
M- represents the rise over run.
B- represents where the line crosses over the y-axis.
So once you input the numbers you get 1/4 for m and -6 for b.
Which ordered pair would form a proportional relationship with the point graphed below? GIVING 30 POINTS FOR CORRECT ANSWER!!!
Answer:
c
Step-by-step explanation:
Which ordered pair would form a proportional relationship with the point graphed below?
[tex]( - 30,10 \: )[/tex]HOPE IT WILL HELP YOU
PLEASE ANSWE ASAP IF YOU’RE AN EXPERT AT MATH !!!
Answer:
13.67
Step-by-step explanation:
At a local college, 118 of the male students are smokers and 472 are non-smokers. Of the female students, 50 are smokers and 450 are non-smokers. A male student and a female student from the college are randomly selected for a survey. What is the probability that both are smokers? Do not round your answer.
Answer:
The probability that both are smokers is 0.02
Step-by-step explanation:
Prob. of male smoker [tex]P(ms)= \frac{118}{590} = 0.2[/tex]
Prob. of female smoker [tex]P(fs) = \frac{50}{500} = 0.1[/tex]
Prob. that both ar smokers [tex]P(bs)=P(ms) * P(fm) =0.2 * 0.1 = 0.02[/tex]
What is the value of x?
Answer:
x=9
Step-by-step explanation:
2x=x+9
Subtract x from both sides;
2x-x=x+9-x
Simplify;
x=9
PLEASE HELP ASAP- What is the value of AB?
Answer:
the answer would be 19 because these are congruent shapes
Hope this helps:)
Step-by-step explanation:
find the indicated sum
Step-by-step explanation:
option no 2 is correct
36 I hope is helpful
Question 3 A student solved a quadratic equation as shown here
Answer:
Didn't move the 12 on the other side of the equal sign to make
2x^2 + 5x - 12 = 0
the other steps error city.
Step-by-step explanation:
2x^2 + 5x - 12 = 0
(2x - 3)(x + 4)= 0
2x - 3 = 0
2x = 3
x = 3/2
x + 4 = 0
x = -4
1.If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4?
Answer:
6
Step-by-step explanation:
5 × 24= 120 and 24 - 4 = 20 and 120 ÷ 20 = 6
I hope this helps!!
Find the measure of the given angle. Round to the nearest degree.
(O
6
?
13
15 degrees by eye i think
9/10 - 3/10
?????…..
Answer:
Step-by-step explanation:
9 - 3 = 6
6/10 = 3/5
Answer:
6/10 and/or 3/5(this is the answer simplified
Step-by-step explanation:
Subtract: 9/10 - 3/10
= 9 - 3/10
= 6/10
= 2 · 3/ 2 · 5
= 3/5
It is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common multiple of both denominators - LCM(10, 10) = 10. It is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 10 × 10 = 100. In the following intermediate step, cancel by a common factor of 2 gives 3/5. In other words - nine tenths minus three tenths is three fifths.
What multiplication expression can I use to find 3/8 divided 3/4
Answer: [tex]\frac{3}{8}[/tex]➗[tex]\frac{3}{4}[/tex]
Step-by-step explanation: Are you looking to work out and calculate how to multiply 3/8 by 3/4? In this really simple guide, we'll teach you exactly what 3/8 times 3/4 is and walk you through the step-by-process of how to multiply two fractions together.
WASTE Suppose the waste generated by nonrecycled paper and cardboard products in tons y after x days can be approximated by the function y=1000(2)0.3x.
a. Identify key features.
y-intercept: , 1 of 8.
domain: , 2 of 8.
range: , 3 of 8.
b. Identify the relevant domain and range.
Because time , 4 of 8. , the relevant domain is , 5 of 8. .
Because the amount of recycled paper and cardboard , 6 of 8. , and the amount when x=0
is , 7 of 8.tons, the relevant range is , 8 of 8. .
a. The key features of the exponential function y = 1000(2)^(0.3x) is given as follows:
y-intercept: y = 100.Domain: all real values.Range: y ≥ 1000.b. The relevant domain and range are given as follows:
Domain: x ≥ 0.Range: y ≥ 1000.What are the features of the exponential function?The exponential function for this problem is modeled as follows:
y = 1000(2)^(0.3x).
The features are given as follows:
Initial value, when x = 0, of y = 1000, representing the y-intercept.Rate of change of 2 for each period of 0.3 units.The domain of the function is the set that composes all input values of the function, and as the exponential function does not have any restrictions such as a fraction or an even root, it is composed by all real values.
However, the relevant domain is x ≥ 0, as the number of days does not assume negative values.
The range is the set containing all output values of the exponential function. As the initial value of the exponential function is of 1000 and the function is increasing, the range is y ≥ 1000.
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Work out the lengths of sides a and b.
Give your answers to 1 decimal place.
Step-by-step explanation:
[tex] {a}^{2} = {8}^{2} + {5}^{2} \\ {a }^{2} = 64 + 25 \\ {a}^{2} = 89 \\ a = \sqrt{89} = 9.4 \: cm[/tex]
[tex] {17}^{2} = {12}^{2} + {b}^{2} \\ {b}^{2} = {17}^{2} - {12}^{2} \\ {b}^{2} = 289 - 144 \\ {b}^{2} = 145 \\ b = 12 \: cm[/tex]
help w one question plz
Answer:
B is the right answer
Step-by-step explanation:
The diagram below represents a street intersection. The dashed lines are new streets that are being proposed. If the turn represented by
Answer:
Step-by-step explanation:
Evaluate the following:
c) The height of each cylinder in a set of food-storage containers is 30 cm.
The radius of the largest container is 10 cm.
The volume of the smallest container is one-third the volume of the largest container.
The volume of the middle-sized container is two-thirds the volume of the largest
container. What is the volume of each container?
Answer:
Volume of smallest container = 3141.6 cm³
Volume of middle sized container = 6283.2 cm³
Volume of largest container = 9424.8 cm³
Step-by-step explanation:
The formula to find the volume of a cylinder = πr²h
The height of each cylinder in a set of food-storage containers is 30 cm.
The radius of the largest container is 10 cm.
Step 1
We can find the volume of the largest cylinder
= r = 10cm
h = 30cm
Volume of largest cylinder = π × 10² × 30
= 9424.8 cm³
Step 2
We find the volume of the smallest container
We are told in the question:
The volume of the smallest container is one-third the volume of the largest container.
Hence
Volume of Smallest container = 1/3 (9424.8 cm³)
= 3141.6 cm³
Step 3
The volume of the middle-sized container is two-thirds the volume of the largest container.
Hence,
Volume of the middle sized container =
2/3 (9424.8 cm³)
= 6283.2 cm³
Therefore, the volume of each container is
Volume of smallest container = 3141.6 cm³
Volume of middle sized container = 6283.2 cm³
Volume of largest container = 9424.8 cm³
What are the values for the coefficients and constant term of 0 2 3x² 5x?
The coefficient of the expression 3x² is the number 3 and constant term is 20
What are the coefficients?
The coefficients are the numbers on the left side or before each variable.
What is an algebraic expression?
Is a set of numbers and letters that make up an expression that has a meaning, the letters are variables and the numbers are coefficients or independent terms, algebraic expressions can be part of an equation and model mathematical processes.
If the expression of a single term is:. 3x²
Then the only coefficient will be the number that is placed before it and it is:
Number 3
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