use spherical coordinates to find the volume of the solid. the solid between the spheres x2 y2 z2 = a2 and x2 y2 z2 = b2, b > a, and inside the cone z2 = x2 y2
Answers
The volume of the solid is π/6 (b³ - a³).
To solve this problem, we need to use the spherical coordinates, which are defined as:
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
where ρ is the distance from the origin, θ is the angle in the xy-plane (measured from the positive x-axis), and φ is the angle between the positive z-axis and the vector from the origin to the point (x, y, z).
The cone z² = x² y² can be expressed in spherical coordinates as:
z² = ρ² cos²(φ) = ρ² sin²(φ) sin²(θ) cos²(φ)
Simplifying this equation, we get:
tan²(φ) = sin²(θ) cos²(φ)
Therefore, the solid is defined by the inequalities:
a² ≤ ρ² ≤ b²
tan²(φ) ≤ sin²(θ) cos²(φ)
The volume of the solid can be computed as:
V = ∭ dV
where dV = ρ² sin(φ) dρ dφ dθ is the volume element in spherical coordinates.
Using the above inequalities, we can write the triple integral as:
V = ∫∫∫ dV
= ∫₀²π ∫0π/2 ∫a²b² ρ² sin(φ) dρ dφ dθ, subject to tan²(φ) ≤ sin²(θ) cos²(φ)
Integrating with respect to ρ, we get:
V = ∫₀²π ∫₀π/2 [1/3 ρ³sin(φ)]a²b² dφ dθ, subject to tan²(φ) ≤ sin²(θ) cos²(φ)
Integrating with respect to φ, we get:
V = ∫₀²π [1/3 (b³ - a³) cos(θ)² sin(θ)] dθ, subject to 0 ≤ θ ≤ π/2
Simplifying this equation, we get:
V = 4π/3 [(b³ - a³)/3] (1/2)
= π/6 (b³ - a³)
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Related Questions
Kevin has a checking account. This month he deposits three paychecks. Each paycheck is for $225.75. He also makes two payments for utilities. Each payment is for $75. He makes a third payment for $125. What is Kevin's new account balance?
Answers
Kelvin's new account balance is, $402.25
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
We have to given that;
Kevin has a checking account. This month he deposits three paychecks.
Here, Each paycheck is for $225.75. He also makes two payments for utilities. Each payment is for $75. He makes a third payment for $125.
Hence, Kevin's new account balance is,
⇒ 3 × $225.75 - 2 × $75 - $125
⇒ $677.25 - $150 - $125
⇒ $402.25
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Which of the following expressions is equivalent to −5/6−13?Select all that apply. A. (−5/6)⋅(−3) B. (−6/5)⋅(−3) C. (−6/5)⋅(−1/3) D. (5/6)÷(1/3) E. 5/6⋅(−3)
Answers
Expressions which are equivalent to −5/6−13 is none of the given expressions (A, B, C, D, E) are equivalent to -5/6 - 13.
The given expression is −5/6−13. To find the equivalent expressions, we need to simplify the given expression. −5/6−13 can be written as −5/6−78/6 (using a common denominator of 6).
This simplifies to −83/6. Now, let's check each option to see which are equivalent to −83/6:
A. (−5/6)⋅(−3) = 15/6 = 5/2, not equivalent to −83/6
B. (−6/5)⋅(−3) = 18/5, not equivalent to −83/6
C. (−6/5)⋅(−1/3) = 2/5, not equivalent to −83/6
D. (5/6)÷(1/3) = 5/2, not equivalent to −83/6
E. 5/6⋅(−3) = −5/2, not equivalent to −83/6
Therefore, none of the options are equivalent to −5/6−13 or −83/6.
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3.once we moved from the 5-kingdom scheme to the 3-domain scheme, which two kingdom names were completely eliminated?
Answers
The three domains are the Archaea, the Bacteria, and the Eukary.
What is 5 kingdom scheme?R.H. Whittaker proposed the five kingdom division in 1969. Based on traits including cell structure, manner of nutrition, source of nutrition, and body organization, the five kingdoms were created. The Kingdoms Monera, Protista, Fungi, Plantae, and Animalia are all included in it. Depending on factors like cell structure, mode of nutrition, mode of reproduction, and body organization, living things are categorized into five different kingdoms: Protista, Fungi, Plantae, Animalia, and Monera.The conclusions have since become accepted, leading to replacement of the kingdom Monera with the two domains Bacteria and Archaea.learn more about kingdom scheme click here:
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the foreclosure crisis has been particularly devastating in housing markets in much of the south and west united states, but even when analysis is restricted to relatively strong housing markets the numbers are staggering. for example, in 2017 an average of three residential properties were auctioned off each weekday in the city of boston, up from an average of one per week in 2011. what is the probability that no more than two foreclosure auctions occurred on a randomly selected weekday of 2017 in boston?
Answers
The probability that no more than two foreclosure auctions occurred on a randomly selected weekday of 2017 in boston is 0.4232
What is probability?Probability turned into added into arithmetic to are expecting the possibility of an occasion occurring. Probability essentially approach how in all likelihood it's miles that some thing will happen. This is the fundamental principle of possibility and is likewise utilized in possibility distributions to examine feasible consequences of random experiments.
Calculationcalculation has been attached in the picture below
the probability that no more than two foreclosure auctions occurred on a randomly selected weekday of 2017 in boston is 0.4232
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Write the equation of the line that is parallel to the line 14x - 2y = 51 and goes through the point(3, -11)
Answers
EXPLANATION: Parallel lines have the same slope but different y intercepts so first turn the eq of the first line into slope intercept form by subtracting -14x and dividing by -2, we get a slope of 7 and we know a parallel line has the same slope so y=7x+b, plug in the values of x and y in the coordinates and add like terms and we get b is equal to -33 so the equation is y=7x-33
Solve for the angles below. Assume that segments that appear to be tangent are tangent. Round your final answers to the nearest tenth, if necessary.
m∠NSR=______degrees
m∠RSP=______degrees
Answers
Answer:
m∠NSR = 113°
m∠RSP = 67°
Step-by-step explanation:
Angle formed between the intersecting chords is one half the sum of the measures of the arcs intercepted by the angles.
m∠NSR = \(\frac{1}{2}[\text{arc}(NR)+\text{arc}(PQ)]\)
= \(\frac{1}{2}(176+50)\)
= 113°
Since, m∠NSR + m∠RSP = 180° [Linear pair of angles are supplementary]
113° + m∠RSP = 180°
m∠RSP = 180° - 113°
= 67°
Help plz Help plz Help plz Help plz
Answers
Answer:
I wanted to let you know that you are NOT what your username suggests you are. Please do not think like that.
Step-by-step explanation:
Oakwood Inc. Is a public enterprise whose shares are traded in the over-the-counter market. At December 31, 2018, Oakwood had 6,000,000 authorized shares of $10 par value common stock, of which 2,000,000 shares were issued and outstanding. The shareholders' equity accounts at December 31, 2018, had the following balances: Common stock $20,000,000 Additional paid-in capital on common stock 7,500,000 Retained earnings 6,470,000 Transactions during 2019 and other information relating to the shareholders' equity accounts were as follows: On January 5, 2019, Oakwood issued at $54 per share, 100,000 shares of $50 par value, 9%, cumulative convertible preferred stock. Each share of preferred stock is convertible, at the option of the holder, into 2 shares of common stock. Oakwood had 600,000 authorized shares of preferred stock. On February 2, 2019, Oakwood reacquired 20,000 shares of its common stock for $16 per share. Oakwood uses the cost method to account for treasury stock. On April 27, 2019, Oakwood sold 500,000 shares (previously unissued) of $10 par value common stock to the public at $17 per share. On June 18, 2019, Oakwood declared a cash dividend of $1 per share of common stock, payable on July 13, 2019, to shareholders of record on July 2, 2019. On November 9, 2019, Oakwood sold 10,000 shares of treasury stock for $21 per share. On December 14, 2019, Oakwood declared the yearly cash dividend on preferred stock, payable on January 14, 2020, to shareholders of record on December 31, 2019. On January 18, 2020, before the books were closed for 2019, Oakwood became aware that the ending inventories at December 31, 2018, were understated by $300,000 (the after-tax effect on 2018 net income was $210,000). The appropriate correcting entry was recorded the same day. After correcting the beginning inventory, net income for 2019 was $4,500,000. Required: Question Content Area 1. Prepare a statement of retained earnings for Oakwood for the year ended December 31, 2019. Assume that only single-period financial stat
Answers
Retained earnings, December 31, 2018 $6,470,000
Add: Net income December 31, 2019 $4,500,000
Less: Preferred stock dividends declared and paid $4,500,000
Less: Common stock dividends declared and paid $1,000,000
Total adjustments $3,500,000
Retained earnings, December 31, 2019 $9,970,000
How do we compute the statement of retained earnings?The statement shows changes in the balance of retained earnings account during the year. The retained earnings balance at the beginning of the year is adjusted for the net income earned during the year and for the dividends declared and paid to the shareholders.
In this case:
retained earnings balance at December 31, 2018 was $6,470,000,net income for the year ended December 31, 2019 was $4,500,000, declared and paid preferred stock dividends of $4,500,000,common stock dividends of $1,000,000 during the year.So, total adjustments to the retained earnings balance:
= $4,500,000 + $1,000,000 - $4,500,000.
= $3,500,000
After adjustment on net income and dividends, the retained earnings balance at December 31 2019 is:
= $6,470,000 + $4,500,000 - $3,500,000
= $9,970,000
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Ben measures the length of the street he lives on to be 80 000 cm. What is the street length in kilometres?
(1km 100 000 cm)
Answers
Answer:
.8Km
Step-by-step explanation:
There is exactly 100,000 CM in every KM, if someone such as Ben was to measure out 80,000, to convert to KM you'd take the given value and divide by the appropriate values.
80,000cm / 100,000cm = .8 KM
Street length in kilometers = 80,000 cm / 100,000 cm = 0.8 km
Therefore, the street length is 0.8 kilometers.
2. What does the 'V' represent in the equation? 110 TENTION valence velocity voltage
Answers
Answer:
V represents final velocity
MORE TO KNOW -
Voltage is represented by W V is final velocity U is initial velocity Velocity is the distance travelled by a body in a unit time in a given direction. It is vector quantityIf the scale factor of two similar cylinders is 4:7,
what is the volume ratio?
A. 64:343
B. 12:21
C. 16:49
D. 256:2,401
Answers
Answer:
the answer is A. 64:343.
Step-by-step explanation:
The ratio of the volumes of two similar figures is equal to the cube of the ratio of their corresponding side lengths. Since the scale factor of the two cylinders is 4:7, the ratio of their corresponding radii is 4:7 and the ratio of their heights is also 4:7.
So, the ratio of their volumes is (4/7)^3 = (64/ 343) = 64:343
So, the answer is A. 64:343.
if 3 girls can send 30 text messages in 30 minutes how many texts can 6 girls send in 60 minutes
Answers
Answer:
120
Step-by-step explanation:
6 girls can send 60 text messages in 30 minutes, so 60 minutes woudl equal 120
Jennifer has $210 in her checking account. She writes
check for $250. What is the new balance in her checking
account?
Answers
Dos anyone know how much a is 2(a+7)-7=9
Answers
Answer:
a =1
Step-by-step explanation:
2(a+7)-7=9
Add 7 to each side
2(a+7)-7+7=9+7
2(a+7)=16
Divide by 2
a+7 = 16/2
a +7 = 8
Subtract 7
a+7-7 = 8-7
a = 1
Answer:
\(a=1\)
Step-by-step explanation:
So we have the equation:
\(2(a+7)-7=9\)
And we want to find the value of a.
So, add 7 to both sides. The left side cancels:
\(2(a+7)=16\)
Divide both sides by 2:
\(a+7=8\)
Subtract 7 from both sides:
\(a=1\)
Therefore, the alue of a is 1.
And we're done!
PLEASE AM TIMED STEP BY STEP
Answers
Answer:
Here you go :)
Step-by-step explanation:
If profits decrease by 13.8% when the degree of operating
leverage (DOL) is 3.8, then the decrease in sales is:
A) 0.28%
B) 0.52%
C) 3.63%
D) 10%
E) 52.44%
Answers
Given that profits decrease by 13.8% when the degree of operating leverage (DOL) is 3.8.
The decrease in sales is: We have to determine the percentage decrease in sales Let the percentage decrease in sales be x.
Degree of Operating Leverage (DOL) = % change in Profit / % change in Sales3.8
= -13.8% / x Thus, we have: x
= -13.8% / 3.8
= -3.63%Therefore, the decrease in sales is 3.63%.Hence, the correct option is C) 3.63%. Percentage decrease in sales = % change in profit / degree of operating leverage
= 13.8 / 3.8
= 3.63% The percentage decrease in sales is 3.63%.
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The temperature today in Chicago is 50°E
Instead of using the standard conversion
formula °C = x(°F - 32), Tommaso uses his
9
grandmother's rule, which is easier but gives
an approximate value: "Subtract 32 from the
value in °F and multiply the result by 0.5."
Answers
Error = (10-9)°C & Error percentage = 10%.
Given that temperature today in Chicago is 50°F
Standard conversion formula
°C = \(\frac{5}{9} *(T^{0}F-32)\) - 1
Given T = 50°F
Put the value in equation 1
\(^{0}C = \frac{5}{9} *(50-32) \\\\= \frac{5}{9}*18\\ \\= \frac{90}{9}\)
= 10°C
Tommasa's grandmothers rule is to subtract 32 from the value in °F and multiply the result by 0.5
i.e. \(^{0}C = 0.5(T^{0} F-32)\) - 2
Given temperature is T = 50°F
Put the value of T in equation 2
\(^{0}C = 0.5*(50-32)\)
= 0.5 * 18
= 1/2 * 18
= 9°C
Temperature obtained by standard conversion Formula is 10°C and the temperature obtained by Tammaso's grandmother rule is 9°C.
So error = (10-9)°C
= 1°C
Error percentage = \(\frac{1}{10}*100\)
= 10%
Hence the answer is Error = (10-9)°C & Error percentage = 10%.
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Please help ME for brainliest.
Answers
Answer:
-1.2y + -4.2y = -3.78
During one day in a hardware tore, 37 people came in and made a purchae, and 13 people looked around but didn’t purchae anything
Answers
The required ratios: a. Purchasers to non purchasers is 37:13 b. Non Purchasers to purchasers is 13:37 c. Purchasers to total customers is 37:50 d. Non Purchasers to total customers is 13:50
So, based on the information-
Non purchasers = 13
Purchasers = 37
Total customers = 50
Finding ratio by keeping the values as per the requirement -
a. Purchasers to non purchasers = 37:13
b. Non Purchasers to purchasers = 13:37
c. Purchasers to total customers = 37:50
d. Non Purchasers to total customers = 13:50
Hence, the ratios are a. 37:13, b. 13:37, c. 37:50 and d. 13:50.
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In a recent year, 28. 7% of all registered doctors were female. If there were 52,600 female registered doctors that year, what was the total number of registered doctors?
Round your answer to the nearest whole number.
Answers
The total number of registered doctors is 183,275.
What was the total number of registered doctors?
Percentage is a measure of frequency that determines the fraction of an amount as a number out of hundred. The sign that is used to represent percentages is %.
Total number of registered doctors = total number of female registered doctors / percentage of female registered doctors
52,600 / 28.7%
52,600 / 0.287 = 183,275
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Using the given percentage, we can see that the total number of registered doctors is 183,044
What was the total number of registered doctors?Let's say that the total number of registered doctors is D, we know that 28.7% are female.
Then the number of female registered doctors is given by:
F = (0.287)*D
We know that there are 52,600 female registered doctors, then we can write:
52,600 = (0.287)*D
Solving this for D, we get:
52,600/0.287 = D
183,044 = D
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If Alex spent $25 on his dinner and he wants to leave the waiter a 19% tip, how much would Alex give as TIP?
Answers
Answer:
$4.75
Step-by-step explanation:
Given:
$25
19%
let x = total amount of tip
x = ($25)(19%)
x = $4.75
Help if link i will report
Answers
Answer:
x=16
Step-by-step explanation:
Solve 10x-29=8x+3
the cost of first-class postage in 2013 was raised to 46 cents. according to the exponential regression model, what was the predicted cost for 2013? start by noting that 2013 is 63 years since 1950.
Answers
The predicted cost of postage for 2013 is 45 cents.
How to determine the predicted cost in 2013We need to use the exponential regression model to predict the cost of postage for the year 2013, given that it was 63 years since 1950.
Let's assume that the cost of postage can be modeled using the exponential function:
C(t) = a * e^(kt)
where:
C(t) is the cost of postage at time ta is the initial cost of postage (in 1950)k is the growth rate of the cost of postage (in decimal form)We can use the two data points we have to solve for a and k:
In 1950, the cost of postage was 3 cents (a = 0.03)In 2013, the cost of postage was 46 cents (C(63) = 0.46)Using the above data points, we can solve for k as follows:
0.03 = a * e^(0 * k)
0.46 = a * e^(63k)
Dividing the second equation by the first equation, we get:
15.333 = e^(63k)
Taking the natural logarithm of both sides, we get:
k = ln(15.333)/63
Evaluate
k = 0.043
Now that we have k, we can use it to predict the cost of postage in 2013:
C(63) = a * e^(63k)
C(63) = 0.03 * e^(63 * 0.043)
C(63) = 0.45
Hence, according to the exponential regression model, the cost is 45 cents.
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Complete question
The cost of first-class postage in 2013 was raised to 46 cents.
According to the exponential regression model, what was the predicted cost for 2013 if the cost of postage was 3 cents in 1950
Start by noting that 2013 is 63 years since 1950.
Gina is having a party and wants to spend less than $580. She has already spent $400. The only item left on her list is pizzas, which cost $15 each. How many pizzas, x, can she purchase and stay under her budget? Select the inequality that includes the largest number of pizzas she can purchase and still stay under her budget.
Answers
Answer:
. .
Step-by-step explanation:;
Write the expression 5x(x + 9) − 7(x + 9) in complete factored form. How do you do this step by step
Answers
Answer:
(x+9) are the like terms so u just put 5x and -7 in brackets
the answer is
(5x-7) ( x+9)
Help! Hurry! NO SPAM!!!! I will mark brainliest if you get it correct.
Identify the first five terms of the sequence in which a1 = 1 and an = 3an −1 + 2 for n ≥ 2.
Answers
Answer:
1, 5, 17, 53, 161
Step-by-step explanation:
Using the recursive rule and a₁ = 1 , then
a₂ = 3a₁ + 2 = 3(1) + 2 = 3 + 2 = 5
a₃ = 3a₂ + 2 = 3(5) + 2 = 15 + 2 = 17
a₄ = 3a₃ + 2 = 3(17) + 2 = 51 + 2 = 53
a₅ = 3a₄ + 2 = 3(53) + 2 = 159 + 2 = 161
The first 5 terms are 1, 5, 17, 53, 161
1, 1, 1, 1, 1 are the first 5 terms of the given sequence.
What is inequality?In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.
We can find the first few terms of the sequence using the recursive formula:
a₁ = 1
aₙ = 3aₙ₋₁ + 2 for n ≥ 2
Using this formula, we can find the second term:
a₂ = 3a₁ - 2 = 3(1) - 2 = 1
We can then find the third term:
a₃ = 3a₂ - 2 = 3(1) - 2 = 1
Continuing this pattern, we find:
a₄ = 3a₃ - 2 = 3(1) - 2 = 1
a₅ = 3a₄ - 2 = 3(1) - 2 = 1
So the first five terms of the sequence are:
1, 1, 1, 1, 1
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√6.76 +5.4=
evaluate
Answers
Is v=8u+8 linear equation
Answers
[1] u = v
// Plug this in for variable u in equation [2]
[2] 8•(v ) - 4v = -32
[2] 4v = -32
// Solve equation [2] for the variable v
[2] 4v = - 32
[2] v = - 8
// By now we know this much :
u = v
v = -8
// Use the v value to solve for u
u = (-8) = -8
Solution : {u , v } = { -8,-8}
Answer:
no its not
Step-by-step explanation:
if the area of circular pond is 169pi square metres,find the radius of its base.
Answers
Answer:
The radius of its base is 13.
Step-by-step explanation:
\(A = \pi r^{2} \\169\pi = \pi r^{2} \\\\\frac{169\pi}{\pi} = \frac{\pi r^{2}}{\pi} \\169 = r^{2}\\\sqrt{169} = r\\ 13 = r\)
Find the values of a,b, and c for which the quadratic equation ax^(2)+bx+c=0 has the given numbers as solutions. Then use those values to write a quadratic equation. -4,6
Answers
To find the values of a, b, and c for which the quadratic equation ax^2 + bx + c = 0 has the solutions -4 and 6, we can use the fact that the sum and product of the solutions of a quadratic equation are related to the coefficients of the equation.
Given that the solutions are -4 and 6, we know that the sum of the solutions is:
-4 + 6 = 2
The quadratic equation with solutions -4 and 6 is:
a(x^2 - 2x - 24) = 0
The sum of the solutions is also related to the coefficient b. Therefore, we have:
2 = -b/a (Equation 1)
Similarly, the product of the solutions is:
-4 * 6 = -24
The product of the solutions is related to the coefficient c. Therefore, we have:
-24 = c/a (Equation 2)
To write the quadratic equation using the values of a, b, and c, we substitute the values obtained from Equations 1 and 2.
From Equation 1, we have:
2 = -b/a
Multiplying both sides by -a, we get:
-2a = b
Substituting this into the quadratic equation, we have:
ax^2 + (-2a)x + c = 0
Now, let's substitute the value of c from Equation 2:
ax^2 + (-2a)x + (-24a) = 0
Simplifying further, we have:
ax^2 - 2ax - 24a = 0
Factoring out 'a' from the equation, we get:
a(x^2 - 2x - 24) = 0
Finally, the quadratic equation with solutions -4 and 6 is:
a(x^2 - 2x - 24) = 0
Note that the value of 'a' can be any non-zero real number, as long as it satisfies the condition.
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