Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c. (Let x, y, and z be the dimensions of the rectangular box.)
Answers
The dimensions say (x,y,z) of a rectangular box of maximum volume is ( c/12, c/12, c/12).
What is Dimensions by Derivatives ?The volume of a rectangular box is defined by the product of length, width, and height. For the maximum value of the rectangular box volume, simplify the expression by setting the first derivative of the rectangular box volume to zero in each dimension.
Let x, y, and z be the dimensions of the rectangular box. Since, the sum of the lengths of its 12 edges is a constant c. This implies
4x + 4y + 4z = c
=> x + y + z = c/4 --(1)
The volume of rectangular box = xyz
Consider xyz + lamda(x + y + z - c/4) where λ is Lagrange multiplier.
let f( x,y,z) = xyz + λ(x + y + z - c/4) --(*)
compute the partial derivatives of f( x,y,z) as
fx = yz + λ(1) = yz + λ
fy = xz + λ(1) = xz + λ
fz = xy + λ(1) = xy + λ
For critical points, set the partial derivatives equals to zero.
fx = yz + λ = 0
fy = xz + λ = 0
fz = xy + λ =0
=> xz = yz = xy = - λ
So, x = y = z
then, from equation (1) we get,
x + x + x = c/4
=> 3x = c/4 => x = c/12
thus, x = c/12, y = c/12, z = c/12
Thus, the dimensions of the rectangular box are x = y = z = c/12 or ( c/12, c/12, c/12) and
Maximum volume of rectangular box = xyz
= c³/1728
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Related Questions
For f(x) = -4x + 1, what is f(-2)?
Answers
Answer:
Step-by-step explanation:
f(-2) = -4(-2) + 1 = 8 + 1 = 9
Answer:
− 2 f
Step-by-step explanation:
Move
− 2 to the left of f .
A square wrestling mat has a perimeter of (12x−32)
feet. Write an expression in simplest form that represents the side length (in feet) of the mat.
Answers
Answer:
The perimeter of a square is given by the formula P = 4s, where s is the side length. In this case, we are given that the perimeter is (12x - 32) feet, so we can set up an equation:
P = 4s
(12x - 32) = 4s
To solve for s, we can divide both sides by 4:
(12x - 32)/4 = s
Simplifying the expression on the left:
3x - 8 = s
Therefore, the expression in simplest form that represents the side length of the square wrestling mat is 3x - 8 feet.
Answer:
Step-by-step explanation:
the answer would be 3 x − 8 3x-8 3x−8 ft
please answer this question
Answers
itss 11
Step-by-step explanation:
no worries about it is the same time for me
Factorise
7lm - 21lmn
Answers
Answer:
7lm - 21lmn
taking common 7lm
7lm(1-3n)
awnser the qusetion 9iu523hb5hn33
Answers
The order of the matrices of each product is given as follows:
AB is nonexistent, BA is nonexistent.
How to apply multiplication of matrices?Multiplication of matrices is applied multiplying the rows of the first matrix by the columns of the second matrix, and hence the number of columns of the first matrix must be equal to the number of rows of the second matrix.
For the product AB, we have that:
A has four columns.B has two rows.Hence the product is nonexistent.
For the product BA, we have that:
B has four columns.A has two rows.Hence the product is nonexistent.
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Find the angle of taper on the steel bar shown if it is equal to twice.
Answers
The angle m can be found by noticing that:
cos m = 22.5/25
That's so because if we divide the triangle formed by the taper into two others, we get two right triangles.
Each right triangle has the angle m, the adjacent leg measuring 22.5 mm, and the hypotenuse measuring 25.0 mm.
So, using the formula for the cosine, we get the above relation. Then, solving this equation, we have:
cos m = 22.5/25
m = arccos 22.5/25
m ≅ 25.84º
Therefore, the angle of the taper is:
2 * m = 2 * 25.84º = 51.68º
Now, in order to convert this result using arc minutes, we need to remember that each 1º corresponds to 60' (60 arc minutes). Thus, we have:
1º --- 60'
0.68º --- x
So, cross multiplying those values, we find:
1º * x = 0.68º * 60'
x = (0.68º/1º) * 60'
x = 0.68 * 60'
x = 40.8'
x ≅ 41'
Therefore, the answer is 51º41'.
A swinging door opens to form a 90°angle. If the door is swung 36° closed, how many more degrees can the door close?
Answers
Answer:
54 degrees
Step-by-step explanation:
subtract 36 from 90
The mean operating cost per hour for the current production method at S-E Electronics equals $220. A new production method will be implemented if ahypothesis test based on sample data suggests the new method reduces (v. the current method) the mean operating cost per hour.
Required:
a. State the null and alternative hypotheses associated with the test.
b. What is the Type I Error for the test and what are the business consequences of a Type I Error?
Answers
Answer:
H0 : μ = 220
H0 : μ < 220
Rejecting the null when it is actually true
We incorrectly drop a better method.
Step-by-step explanation:
The null hypothesis, H0
H0 : μ = 220
H0 : μ < 220
A type 1 error is committed when a true null hypothesis is incorrectly rejected. This means that we conclude that new method reduces the mean operating cost per hour when in fact it hasn't.
The impact of this on business is that the new method is adopted in place of the old method thinking it works better Than the old method.
How many grams are there in 4 kilograms
Answers
Mr. Gupta gave his students a quiz with three questions on it. Let
�
XX represent the number of questions that a randomly chosen student answered correctly. Here is the probability distribution of
�
XX along with summary statistics:
�
=
# correct
X=# correctX, equals, start text, \#, space, c, o, r, r, e, c, t, end text
0
00
1
11
2
22
3
33
�
(
�
)
P(X)P, left parenthesis, X, right parenthesis
0.05
0.050, point, 05
0.20
0.200, point, 20
0.50
0.500, point, 50
0.25
0.250, point, 25
Mean:
�
�
=
1.95
μ
X
=1.95mu, start subscript, X, end subscript, equals, 1, point, 95
Standard deviation:
�
�
≈
0.8
σ
X
≈0.8sigma, start subscript, X, end subscript, approximately equals, 0, point, 8
Mr. Gupta decides to score the tests by giving
10
1010 points for each correct question. He also plans to give every student
5
55 additional bonus points. Let
�
YY represent a random student's score.
What are the mean and standard deviation of
�
YY?
Answers
The mean score of a random student (YY) is 574.5. the standard deviation of the random student's score (YY) is 8.
How to answer the aforementioned questionGiven:
- Each correct question is worth 10 points.
- Every student receives an additional 555 bonus points.
Let's calculate the mean and standard deviation of YY:
Mean of YY:
The mean score, denoted as μY, can be calculated using the mean of XX (μX) and the scoring scheme:
μY = μX * 10 + 555
Substituting the value of μX from the given information:
μY = 1.95 * 10 + 555
μY = 19.5 + 555
μY = 574.5
Therefore, the mean score of a random student (YY) is 574.5.
Standard Deviation of YY:
The standard deviation of YY, denoted as σY, can be calculated using the standard deviation of XX (σX) and the scoring scheme:
σY = σX * 10
Substituting the value of σX from the given information:
σY = 0.8 * 10
σY = 8
Therefore, the standard deviation of the random student's score (YY) is 8.
Complete question: Mr. Gupta gave his students a quiz with three questions on it. Let X represent the number of questions
that a randomly chosen student answered correctly. Here is the probability distribution of X along with
summary statistics:
0
1
2
2.
3
X = # correct
P(X)
0.05
0.20
0.50
0.25
Mean: Hex = 1.95
Standard deviation: Ox 0.8
Mr. Gupta decides to score the tests by giving 10 points for each correct question. He also plans to give
every student 5 additional bonus points. Let Y represent a random student's score.
What are the mean and standard deviation of Y?
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will give brainliest
Answers
The possible rational roots of the given equation are 1 and -3
Solving polynomial equationsFrom the question, we are to determine all the possible rational roots of the given equation
The given equation is
x⁴ -2x³ -6x² +22x -15 = 0
To determine the rational roots, we will test for values that make the equation equal to zero
Test for -1(-1)⁴ -2(-1)³ -6(-1)² +22(-1) -15
1 + 2 - 6 - 22 -15
= -40
∴ -1 is not a root of the equation
Test for 1(1)⁴ -2(1)³ -6(1)² +22(1) -15
1 - 2 - 6 + 22 -15
= 0
∴ 1 is one of the roots of the equation
Test for -2(-2)⁴ -2(-2)³ -6(-2)² +22(-2) -15
16 + 16 - 24 - 44 -15
= -51
∴ -2 is not a root of the equation
Test for 2(2)⁴ -2(2)³ -6(2)² +22(2) -15
16 - 16 - 24 + 44 -15
= 5
∴ 2 is not a root of the equation
Test for -3(-3)⁴ -2(-3)³ -6(-3)² +22(-3) -15
81 + 54 - 54 -66 -15
= 0
∴ -3 is one of the roots of the equation
Test for 3(3)⁴ -2(3)³ -6(3)² +22(3) -15
81 - 54 - 54 + 66 -15
= 24
∴ 3 is not a root of the equation
The other roots of the equation are irrational roots.
Hence, the possible rational roots of the given equation are 1 and -3
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determine which of the equations below could be the class equation given in the proof of theorem 24.2. for each part, provide your reasoning. a. 9 5 3 1 3 1 3 b. 21 5 1 1 1 1 3 1 3 1 3 1 3 1 7 c. 10 5 1 1 2 1 2 1 5 d. 18 5 1 1 3 1 6 1 8
Answers
The class equation is if G is a nontrivial finite group whose order is a power of a prime p, is the centralizer of G, and is the centralizer of the element a. (1) Here, the sum runs over representatives of all conjugacy classes that have more than one element.
How can the class equation be found?The commutativity degree, which indicates the likelihood that two members of a group will commute between themselves, is related to the class equation in group theory [3]. The definition is as follows: d ( G ) = | { ( a , b ) ∈ G 2 ∣ a · b = b · a } | | G | 2 .
How can the equation for the class be found?In group theory, there is a connection between the class equation and the commutativity degree, which indicates the likelihood that two members of a group will commute between themselves [3]. The following is the definition: d ( G ) = | { ( a , b ) ∈ G 2 ∣ a · b = b · a } | | G | 2 .
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help me pls i need help
Answers
Answer:
answer is D
Step-by-step explanation:
hope this helps :-D
ayo someone pls respond , I’ll give that brain god prize thing to you
Answers
Answer: Choice A
Explanation:
\(\frac{h}{6}\) = quotient of h and 6
\(\frac{h}{6}+7.8\) = 7.8 more than the quotient of h and 6
\(\frac{h}{6}+7.8 = w\)
The "is" often refers to "equals" in math translation problems like this.
Another example "h is 10" means "h = 10".
Bob 's daily commute time is randomly distributed with a minimum of 20 minutes, a maximum of 61 minutes, and the most common length is 30 minutes. What is the probability that his commute today took more than 35 minutes?
Answers
Answer:
0.6341 = 63.41% probability that his commute today took more than 35 minutes
Step-by-step explanation:
Randomly distributed = Uniform distribution.
A distribution is called uniform if each outcome has the same probability of happening.
The uniform distributon has two bounds, a and b, and the probability of finding a value higher than x is given by:
\(P(X > x) = \frac{b - x}{b - a}\)
Randomly distributed with a minimum of 20 minutes, a maximum of 61 minutes.
This means that \(a = 20, b = 61\).
What is the probability that his commute today took more than 35 minutes?
\(P(X > 35) = \frac{61 - 35}{61 - 20} = 0.6341\)
0.6341 = 63.41% probability that his commute today took more than 35 minutes
You pay sales tax on your purchases at the rate of 0.07. What fraction of your money is this? Remember to reduce your fraction to its lowest terms.
0.07 is equal to
nothing
Answers
Answer:
This is \(\frac{7}{100}\) of your money.
Step-by-step explanation:
We have a decimal number, and we want the answer to be a fraction. To transform a decimal number into a fraction, we multiply both the numerator and the denominator by 10 until none of them have decimal parts. So
\(0.07\times \frac{10}{10} = \frac{0.7}{10}\times \frac{10}{10} = \frac{7}{100}\)
This is \(\frac{7}{100}\) of your money.
Evaluate the expression for m = 2.
-26
Answers
Answer:
-13?
Step-by-step explanation:
If m = 2, then.. -26/2 = -13
Answer:
-13
Step-by-step explanation:
I think its -13
try
y =
COS
X +
y
5
3
2
3:20
22
5-7
371
-1 -2
-3
5
Answers
Second box would be -2
Third box would be 7
i don’t understand the formula i need to solve this equation. is it density?
Answers
Answer:
Step-by-step explanation:
They are wanting to find out what the size of the state is. I look at the units.
Population density is in People/mi²
Population is people
and they are looking for _____ mi²
Population density = population / size of state
Let's rework that equation so we can solve for size of state
size of state = population/population density
size of Texas = (29145505 people) / (108.51 people/mi²)
size of Texas = 268597 mi²
f(x) = х2. What is g(x)?
Answers
Answer:
C
Step-by-step explanation:
Ordenered pairs of g(x):
(0,0); (2,1); (- 2,1)
g(x) = ax² + bx + c
a.0² + b.0 + c = 0
c = 0
a.2² + b.2 + c = 1
4a + 2b = 1
a.(- 2)² + b.(- 2) + c = 1
4a - 2b = 1
4a + 2b = 4a - 2b
2b + 2b = 0
4b = 0
b = 0
4a + 2b = 1
4a = 1
a = 1/4
g(x) = (1/4)x² + 0x + 0
g(x) = (1/4)x²
g(x) = (1/2x)²
I hope I've helped you.
Christian has a collection of 18 shark teeth. He identified them as 6 tiger shark teeth, 8 sand shark teeth, and the rest as bull shark teeth. What does the ratio 6:8 represent in this situation? What does the ratio 4:18 represent in this situation. Explain your reasoning. Include a model in your explanation. I
Answers
Answer:
Step-by-step explanation:
There are 18 teeth
6 tiger + 8 sand = 14
18-14 = 4
This means there are 4 bull shark teeth
4:18
4 bull shark teeth: 18 total shark teeth
Answer:
Step-by-step explanation: ratio 6:8 is tiger shark:sand shark
the ratio 4:18 is bull shark teeth:all the teath
Ihope this helps :D
Math how to learn more abt it
Answers
Answer:
To learn more about math, you need to practice it, and when you have mistakes, understand it better then you'll get it. And ask a teacher or adult for help when needed.
Step-by-step explanation:
If m∠J = 44°, what is m∠I?
Answers
Answer:
6666
Step-by-step explanation:
factorise x³-4x²+x+6
Answers
The binomial factors of x³- 4x²+x+6 are (x+2), (x+3), and (x-1).
Using the splitting and grouping the terms:
x³ + 4x² + x - 6
= x³ + 2x² + 2x² + x - 6 [Splitting 4x² = 2x² + 2x²]
= (x³ + 2x²) + (2x² + x - 6)
= x² (x + 2) + (2x² + 4x - 3x - 6)
= x² (x + 2) + [ 2x (x + 2) - 3 (x + 2)]
= x² (x + 2) + (x + 2) (2x - 3)
= (x + 2) ( x² + 2x - 3)
= (x + 2) ( x² + 3x - x - 3)
= (x + 2) [x (x + 3) - 1 (x + 3)]
= (x + 2) (x + 3) (x - 1)
Hence, the binomial factors are (x + 2), (x + 3) and (x - 1)
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Need the correct answer to this question please
Answers
Answer:
3x-10=7x+5
Step-by-step explanation:
David and John own 37-1/2 acres of land. David owns 3/5 and John owns 2/5 of the land. They decide to section it off into ¾ acre lots to rent out as garden space for people who do not have garden space at their homes. The price for the entire summer for the rent of each lot is $50. Since there is no water supply to water the gardens, David and John have asked their renters if they would like a 100-gallon water tank or a 70-gallon water tank. 60% of the renters chose 100 gallons and 40% chose 70 gallons. David calculated that they would need to fill each tank 2 times and the cost of a gallon of water is $.05.
Answers
The total spent amount on water is $220
The total area is 37.5=75/2
The number of lots is 75/2/3/4=50 nos
Total rent is 50×50=2500
So 3/5 will go to david 2500×3/5=1500
2/5 will go to john 2500×2/5=1000
So 60% Renters choose 100 gallons
Total renters is 50
60% renters is 50×60%=30 nos
40% renters choose 70 gallons
40% of 50 is 20 nos
The tank fills twice
100×30+70×20=4400 to be filled
Per gallon cost is $0.05
So the total cost is 4400×0.5=$220
So they spent $220 on water
A fraction shows part of a whole. This entire may refer to a place or a group of places.A fraction can be a portion or section of any quantity out of a whole, where the whole can be number of any value.To learn more about fraction visit:
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Perform the indicated operation:
(4 + 2i) (1 + 5i)
-6 +22i
6-221
-14 + 18i
6+22i
Answers
Answer:
-235+84i
Step-by-step explanation:
I hope this is what you were looking for.
find lim x approaches 3
f(x) = x^2 - 2
f(x) = -3x + 15. x> 3
Answers
The value of the limit of the function f(x) = x² - 2 is 7.
The value of the limit of the function f(x) = -3x + 15 is -6.
We have,
To evaluate the limit as x approaches 3, we need to find the limit of function f(x) as x approaches 3 from both sides of 3.
First, let's evaluate the limit from the left side of 3:
lim x → 3⁻ f(x) = lim x → 3⁻ (x^2 - 2)
Substituting x = 3 - h (where h approaches zero) gives:
lim h → 0⁺ [(3 - h)^2 - 2]
= lim h → 0⁺ [9 - 6h + h^2 - 2]
= lim h → 0⁺ [h^2 - 6h + 7]
= 7
(since the limit of a quadratic function as h approaches zero is the constant term)
Next, let's evaluate the limit from the right side of 3:
lim x → 3⁺ f(x)
= lim x → 3⁺ (-3x + 15)
= -6
Since the limit from the left side is not equal to the limit from the right side, the limit of f(x) as x approaches 3 does not exist.
Thus,
The value of the limit of the function f(x) = x² - 2 is 7.
The value of the limit of the function f(x) = -3x + 15 is -6.
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Rent or buy analysis
• If they rent, the builder will require monthly rental payments of $1,100 and a security deposit equal to two months of rent.
• Since they want to be protected against the possible loss of their possessions, they will purchase a renters’ policy of $200 every six months, while a more comprehensive homeowners’ policy will cost 0.5% of the home’s value per year.
• Money used to fund the house’s security deposit could otherwise be invested to earn 5% per year after taxes. Funds expended for a home’s down payment and closing costs also incur an opportunity cost.
• If the house is purchased, it will cost $110,000 and will require a 20% down payment. The loan will carry an interest rate of 6%, a term of 30 years, and monthly payments of $528. The closing costs associated with the house’s mortgage will be $3,500.
• The property taxes and the maintenance and repair expenses on the house are estimated to be 3% and 1% of the house’s total price, respectively.
• Your ordinary income is taxed at the rate of 28%, and you’ll be willing to itemize your tax deductions in the event that you purchase your new home.
• Financial publications report that home values are expected to increase by 3% this year due to inflation.
Answers
The key factors to consider include rental payments, security deposit, insurance costs, investment opportunity cost, down payment, mortgage interest rate, loan term etc.
Rental Costs: Monthly rental payments of $1,100 and a security deposit equal to two months of rent.
Insurance Costs: Renters' policy of $200 every six months and a homeowners' policy costing 0.5% of the home's value per year.
Investment Opportunity Cost: Money used for security deposit could be invested to earn 5% per year after taxes.
Purchase Costs: House price of $110,000 with a 20% down payment, a 6% interest rate, a 30-year term, and monthly payments of $528. Closing costs of $3,500.
Additional Expenses: Property taxes estimated to be 3% of the house's total price and maintenance and repair expenses estimated at 1%.
Tax Considerations: Ordinary income taxed at a rate of 28%, with the possibility of itemizing tax deductions for homeowners.
Home Value Appreciation: Expected home value increase of 3% due to inflation.
The analysis involves comparing the total costs and benefits associated with renting and buying a home, taking into account factors such as cash flow, investment returns, tax implications, and future home value. The decision will depend on individual financial circumstances and preferences.
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if it takes three people 4 hours to clean a warehouse how long will it take 4 people to clean the warehouse
Answers
The solution is ,if it takes three people 4 hours to clean a warehouse , then 3 hours will it take 4 people to clean the warehouse.
Here, we have,
given that.
if it takes three people 4 hours to clean a warehouse
now, we have to find that how long will it take 4 people to clean the warehouse.
i.e. we have,
3 people take to clean = 4 hours
let, 4 people will take to do it x hours.
now, we have,
3 people take to clean = 4 hours
so, 1 people take to clean = 4*3 hours
so, 4 people take to clean = 4*3 / 4 hours
i.e. we get,
x = 4*3 / 4 hours
or, x = 12/4 hours
= 3 hours
Hence, if it takes three people 4 hours to clean a warehouse , then 3 hours will it take 4 people to clean the warehouse.
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