The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. (Let x, y, and z be the dimensions of the aquarium. Enter your answer in terms of V.)
Answer:
The dimensions of the aquarium that minimize the cost of the materials:
[tex]x=y=\sqrt[3]{\frac{2V}{5}}\\z=\sqrt[3]{\frac{25V}{4}}[/tex]
Step-by-step explanation:
Let x, y and z be the dimensions of aquarium .
Surface area of an aquarium = xy+2yz+2xz
Volume of aquarium V=[tex]Length \times Breadth \times Height=xyz[/tex] ----A
We are given that slate costs five times as much (per unit area) as glass
So, Cost function : C=5xy+2yz+2xz
Now we will use langrage multiplier to find the dimensions of the aquarium that minimize the cost of the materials.
[tex]\nabla C =\lambda \nabla V[/tex]
[tex](\frac{\partial C}{\partial x},\frac{\partial C}{\partial y},\frac{\partial C}{\partial z})= \lambda (\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z})[/tex]
[tex](5y+2z,5x+2z,2y+2x)=\lambda(yz,xz,xy)[/tex]
So,
[tex]5y+2z=\lambda yz[/tex] ----1
[tex]5x+2z=\lambda xz[/tex] -----2
[tex]2y+2x=\lambda xy[/tex] ----3
Multiply 1 ,2 and 3 by x,y and z respectively.
[tex]5xy+2xz=\lambda xyz[/tex] ----4
[tex]5xy+2yz=\lambda xyz[/tex] -----5
[tex]2yz+2xz=\lambda xyz[/tex] ----6
Now equate 4 and 5
5xy+2xz=5xy+2yz
x=y
Substitute y=x in 5 and 6 and equate them
[tex]5x(x)+2(x)z=2(x)z+2xz\\5x^2=2xz\\5x=2z\\\frac{5}{2}x=z[/tex]
Substitute the values in A
[tex]V = xyz = x \times x \times \frac{5}{2}xV=\frac{5}{2}x^3\\\sqrt[3]{\frac{2}{5}V}=x\\x=y=\sqrt[3]{\frac{2}{5}V}\\z=\frac{5}{2}x=\frac{5}{2}(\sqrt[3]{\frac{2}{5}})=\sqrt[3]{\frac{25V}{4}}[/tex]
Hence,
The dimensions of the aquarium that minimize the cost of the materials:
[tex]x=y=\sqrt[3]{\frac{2V}{5}}\\z=\sqrt[3]{\frac{25V}{4}}[/tex]
What is the solution to j + (-3) = 9?
Answer:
j = 12
Step-by-step explanation:
j + (-3) = 9
j = 12
Which shows one way the equation can be represented in words? Z-6=14
A.The dfference of a number and z is the same as one and four-tenths.
B.Anumber subtracted from one and four-tenths is equal to six.
C.Six less than a number is the same as one and four-tenths.
D.Sx decreased by a number is equal to one and four-tenths.
Answer: c
Step-by-step explanation:
c is the answer
A candy factory that produces chocolate bars claims each bar weighs 50 grams, at least that is what is printed on the label. Of course, there is bound to be a little variation. An inspector randomly chooses 16 bars from one day's output of 4000 bars. The average weight of the 16 bars is only 48.5 grams. The inspector wishes to test the null hypothesis that the factory is doing what it is supposed to on this day against the alternative that the company is cheating the consumer. Assume that the weights of all the candy bars that are produced follow the normal curve with a SD of 2 gram(s).
Which significance test should be used?
A z test because the SD of the population is known and the weights follow the normal curve. You are correct. Computer's answer now shown above Previous Tries Your receipt no. is 159-2320
Do you need to use SD+ o estimate the SD of the population?
No, we're given the SD of the population, we don't need to estimate it. Incorrect.
Computer's answer now shown above ries 1/1 Previous Tries
Compute SE. (Give your answer as a decimal to two places) Submit Answer Tries 0/5
Compute the test statistic. (Round your answer to two decimals, using previously rounded answer.) Submit Answer Tries 0/5
What do you conclude?
A. Reject the null, it's very unlikely that the factory is making the candy bars 50 grams as they claim.
B. Cannot reject the null, it's plausible that the factory is making the bars 50 grams as they claim to be doing. Submit Answer Tries 0/1
Answer:
Question 1
The correct option is A
Question two
The correct option is B
Question three
[tex]SE= 0.5[/tex]
Question four
[tex]z = - 3 [/tex]
Question five
The correct option is A
Step-by-step explanation:
The population mean is [tex]\mu = 50 \ grams[/tex]
The sample size is n = 16
The population size is N = 4000
The sample mean is [tex]\= x = 48.5 \grams[/tex]
The standard deviation is [tex]s = 2 \ gram(s)[/tex]
Considering the first question
The correct option is A
z test because the SD of the population is known and the weights follow the normal curve
Considering the second question
The correct option is B
Yes, because with only 20 observations the sample SD is not a great estimate of the true population SD.
Considering the third question
Generally the standard error is mathematically represented as
[tex]SE= \frac{s }{\sqrt{n} }[/tex]
=> [tex]SE= \frac{2 }{\sqrt{16} }[/tex]
=> [tex]SE= 0.5[/tex]
The null hypothesis is [tex]H_o : \mu = 50[/tex]
The alternative hypothesis is [tex]H_a : \mu < 50[/tex]
Generally the test statistics is mathematically represented as
[tex]z = \frac{ \= x - \mu }{ SE }[/tex]
=> [tex]z = \frac{ 48.5 - 50 }{ 0.5}[/tex]
=> [tex]z = - 3 [/tex]
Generally the p-value is mathematically represented as
[tex]p-value = P(Z < z )[/tex]
=> [tex]p-value = P(Z < - 3 )[/tex]
From the z-table the
[tex]P(Z < - 3 ) =0.0013499[/tex]
=> [tex]p-value = 0.0013499[/tex]
From the obtained values we see that
[tex]p-value < \alpha[/tex]
So we reject the null hypotheses
The conclusion
it's very unlikely that the factory is making the candy bars 50 grams as they claim
Testing the hypothesis, we have that:
The standard error is of 0.5.The test statistic is z = -3.A. Reject the null, it's very unlikely that the factory is making the candy bars 50 grams as they claim.At the null hypothesis, we test if the factory is doing what it is supposed to, that is, the mean is of 50 grams, hence:
[tex]H_0: \mu = 50[/tex]
At the alternative hypothesis, we test if the factory is cheating the customer, that is, the mean is of less than 50 grams, hence:
[tex]H_1: \mu < 50[/tex]
We have the standard deviation for the population, thus, the z-distribution is used. The test statistic is given by:
[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
The parameters are:
[tex]\overline{x}[/tex] is the sample mean. [tex]\mu[/tex] is the value tested at the null hypothesis. [tex]\sigma[/tex] is the standard deviation of the sample. n is the sample size.For this problem, the values of the parameters are: [tex]\overline{x} = 48.5, \mu = 50, \sigma = 2, n = 16[/tex]
Hence, the standard error is:
[tex]S_e = \frac{2}{\sqrt{16}} = 0.5[/tex]
The value of the test statistic is:
[tex]z = \frac{\overline{x} - \mu}{S_e}[/tex]
[tex]z = \frac{48.5 - 50}{0.5}[/tex]
[tex]z = -3[/tex]
The critical value for a left-tailed test, as we are testing if the mean is less than a value, with a significance level of 0.05, is of [tex]z^{\ast} = -1.645[/tex]
Since the test statistic is less than the critical value for the left-tailed test, we can reject the null hypothesis.
A similar problem is given at https://brainly.com/question/14983950
slove the equation
7h-5(3h-8)=-72
Answer:
Step-by-step explanation:
You can break it down into steps if you'd like.
-5 ( 3h - 8) = -15h + 40
So,
7h - 5 ( 3h -8 ) = -72
7h - 15h + 40 = -72
Now you can combine the like terms
-8h + 40 = -72
Add 8h to both sides
40 = -72 + 8h
Add 72 to both sides
40 + 72 = 8h
112 = 8h
Divide both sides by 8 to find h.
h = 112/8 = 14
timed question need help
Step-by-step explanation:
m-7/3 = m+2
= m-7 = 3(m+2)
= m-7 = 3m+6
= -7-6 = 3m-m
= 2m = -13
= m = -13/2
can someone help me find angle 3?
Answer:
45
Step-by-step explanation:
corresponding angles are congruent
3 and 7 are corresponding
There are 3 different roads from city A to city B and 2 different roads from city B to city C. In how many ways can someone go from city A to city C passing by city B.
Answer: 6
Step-by-step explanation: Since there are three different ways to get from A to B, that creates three variants of each way to get from B to C.
Say the roads are numbered. From A to B, you can use roads 1, 2, or 3. From B to C, you can use roads 4 or 5. The different combinations are 1-4, 1-5, 2-4, 2-5, 3-4, and 3-5.
A biker rides for h hours at a speed of s miles per hour and travels d miles. Which equation models the relationship?
Answer:
hs=d
Step-by-step explanation:
Answer:
hs= d
Step-by-step explanation:
what is the least common denominator of 4/5 and 3/10?plssreal quick
Answer:
5
Step-by-step explanation:
hi, what is 1+1 and 2+3
Answer:
1+1 is 2 and 2+3 is 5
Step-by-step explanation:
Compute 49/24 of 15/24. Give your answer as a fraction in simplest form (not a mixed number).
Answer: 35/16
Step-by-step explanation:
Answer:
35/16
Step-by-step explanation:
49x15=735
Denominator will stay the same because both are same (24)
Then you simplify 735/24
Which will give you 35/16
The ratio of children to teachers at a preschool is 14:5. How many children are at the preschool if there are 30 teachers?
Answer:
14 30
----- = ------
5
cross multiple 30(5)=150
divide 150/14= 10.7142857
idk if they want you to round but
11 or
10.7
A recipe calls for 3 3/4 cups of flour. How many cups will you need if you are preparing 5 batches of the recipe?
Answer:
18.75 cups of flour
Step-by-step explanation:
The answer is 18.75 cups of flour as you have to multiply 5 by 3 3/4. The result will be 18.75. Hope it helps!
You need 18 3/4 cups of flour, to make 5 batches of the recipe.
What is a fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.
If one batch of the recipe calls for 3 3/4 cups of flour, then 5 batches of the recipe will require:
5 × 3 3/4 cups
To multiply a whole number by a mixed number, we first convert the mixed number to an improper fraction:
3 3/4 = (4 × 3 + 3) / 4 = 15/4
Now we can multiply:
5 × 15/4 = (5 × 15) / 4 = 75/4
To convert this fraction to a mixed number, we divide the numerator by the denominator and write the remainder as a fraction:
75 ÷ 4 = 18 remainder 3
So, 75/4 is equal to 18 3/4.
Therefore, 5 batches of the recipe will require:
18 3/4 cups of flour.
To learn more about fractions;
https://brainly.com/question/10354322
#SPJ2
Evaluate `int cosec^(2)(7-5x)dx
Answer:
∫ cosec²(7-5x) dx =[-cot(7-5x)]/-5 +c
=(1/5)cot(7-5x)+c
note: d/dx(cotx)=- cosecx
estimate the product choose the method 27 * 42
Answer:
About 1200
Step-by-step explanation:
I would round them to the nearest 10 so 27 would become 30 and 42 would be 40.
Then you would multiply to get the answer of about 1200.
Hope that helps and have a great day!
ayuda nose ayudaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
Charlie bought hamburgers and drinks for his coworkers. Each hamburger cost $3.75 and each drink cost 50 cents. In total, Charlie paid $25.50 for his order. Write an equation that represents the relationship between the number of hamburgers he bought, h, and the number of drinks he bought, d.
Answer:
25.50=3.75h+0.50d
Step-by-step explanation:
(4 to the 4th power)to the 3rd power
What is the midpoint of the segment shown below?
10
A. (10,1)
B.(5,1)
c.(5,1/2)
d. (10,1/2)
Solve by taking the square too of both sides.
3(c+3)^2-81=0
Answer:
x = -3 ±3sqrt(3)
Step-by-step explanation:
3(x+3)^2-81=0
Add 81 to each side
3(x+3)^2=81
Divide each side by 3
3/3(x+3)^2=81/3
(x+3)^2=27
Take the square root of each side
sqrt ((x+3)^2)=±sqrt(27)
x+3 =±sqrt(9*3)
x+3 =±sqrt(9)*sqrt(3)
x+3 =±3sqrt(3)
Subtract 3 from each side
x = -3 ±3sqrt(3)
What is the product of
5
7
and
14
10
A
1
4
O B.
2
4
oC.
2
C D 3
Answer:
1
Step-by-step explanation:
5/7 *14/10 = 14/7 * 5/10 = 2 * 1/2 =1
Hope I understood the question correctly.
Susan works part-time in a local video store. Her salary varies directly as the number of hours
worked. Last week she earned $144 for working 16 hours. This week she earned $378. How
many hours did she work this week?
Answer:
Around 2.63
Step-by-step explanation:
144 x 2.63 = 378.72
I would put 2.60, but you do you
a spinner is split into eight wedges numbered 0-7
Answer:
The last one is 8
Step-by-step explanation:
I'm unsure of the question.
What is the distance between 4.5 and 3.75
Answer:
0.75
Step-by-step explanation:
The 7th grade team went out for lunch. Their server brought the bill that totaled $72. The team wanted to tip the server 20% and there is also a 9% sales tax. How much will the 7th grade team spend after the tip and tax are added? Round your answer to the nearest cent if necessary.
Answer:
I think it's 92.88
Step-by-step explanation:
add up 20 and 9 to get a total of 29%
then multiply .29 and 72 to get 20.88
then add 20.88 and 72 to get the total cost including tips and tax
Determine the slope given the two points:
(4, 9) and (1, 6)
m=
Answer:
1
Step-by-step explanation:
y2-y1
/
x2-x1
6 - 9
/
1-4
= -3/-3
which simplifies to 1
The Empire State building is 1250 feet tall maria wants to make a model that has a scale of 1/4 foot equals 50 feet how tall with her model B
Answer:
6 1/4
Step-by-step explanation:
Emily bought 2 kilograms of rice for $6.25.
Select True or False for each statement.
O True
O False
The product (25) will give the unit price of one kilogram of rice.
0 True
O False
Emily paid less than $1.50 per kilogram of rice.
O True
O False
If Emily has $1.00 to spend on rice, she can get io of a kilogram of rice.
#CarryOnLearning
Answer:
1.) False
The product [tex]\bold{\frac{5}{2}}[/tex]([tex]\bold{\frac{4}{25}}[/tex]) will give the unit price of one kilogram of rice.
2.) False
Emily paid less than $1.50 per kilogram of rice.
3.) True
If Emily has $1.00 \:to\: spend\: on\: rice, she\: can\: get [tex]\frac{4}{10}[/tex]$ of a kilogram of rice.
#CarryOnLearning
A minor league baseball office sells admission tickets for $5.75 for Wednesday night games. The ticket
office can sell a maximum of 500 tickets. The function M(t) = 5.75t represents the amount of money
(M) the ticket office brings in on Wednesday nights, where t is the number of tickets sold. Which best
describes the domain of M(t) in this context?
Answer:
Its B because 500 tickets is the maximum amount
Step-by-step explanation: