Show that Φ(u,v)=(7u+4,u−v,13u+v) parametrizes the plane 2x−y−z=8. Then: (a) Calculate Tu , Tv, and n(u,v). (b) Find the area of S=Φ(D), where D=(u,v):0≤u≤5,0≤v≤6. (c) Express f(x,y,z)=yz in terms of u and v and evaluate ∬sf(x,y,z)ds(a) Tu= , Tv= , n(u,v)=_______________(b) Area(S)=_______(c) ∬sf(x,y,z)ds=___________
Answers
Answer:
Step-by-step explanation:
1.Φ(u,v) gives the parametrization with its x, y, and z components. So, x=7u+4, y=u-v, and z=13u+v. We can use this to get the tangent vectors by taking the respective partial derivatives. Tu=<7, 1, 13> by taking the u derivatives of the components and Tv=<0, -1, 1> by taking the v derivatives of the components.
2. Taking the cross product of Tu and Tv will give the normal vector n(u,v), which is <14, -7, -7>
3. To find Area(S), you have to multiply the area of D by the magnitude of the normal vector from the previous step. D is the region defined by 0<=u<=5 and 0<=v<=6, so the area of D is 5*6=30. Multiply this by the magnitude of the normal vector to find that Area(S)=30sqrt(294)
4. To integrate, we first must put the initial function in terms of the parameters we found in the first step. Replace y with u-v and z with 13u+v. Next, multiply this integrand by the magnitude of the normal vector (sqrt294) and apply the given bounds for u (0<=u<=5) and v (0<=v<=6). From here the problem can be integrated like any other double integral, the final answer being 190sqrt(294)
Related Questions
suppose we play a game where we roll a six sided dice. if a 4,5 or 6 is rolled, i get 1 point if a 1,2,3 is rolled,you get 1 point. we play first to 5 but the winner must also win by 2. if we are tied at 5, what's the probability that you win the game?
Answers
The probability that you win the game is 50%.
The game is symmetrical for you and your opponent. Thus, your probability of winning is as good as your opponent's, so, you have a 50% of winning.
What does symmetric mean in probability?Symmetry goes into play in probability theory in this way: if different outcomes are equivalent, they would have the same probability.
A symmetrical distribution happens when the values of variables come at regular frequencies and usually the mean, median, and mode all happen at the same point.
In general, if there are N symmetrical outcomes, the probability of any given one of them happening is taken to be 1/N.
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1
5
Lian's dog eats of a bag of dog food each week. Lian currently has of a bag
20
8
remaining.
How many weeks will it take Lian's dog to finish this remaining food? .
weeks
Answers
It will take Lian's dog 12.5 weeks to finish the remaining food
How to determine how many weeks will it take Lian's dog to finish the remaining food?Fractions are numbers written as a quotient, in which a numerator is divided by a denominator
Given that:
Lian's dog eats 1/20 of a bag of dog food each week
Lian currently has of a bag 5/8 remaining
Since Lian's dog eats 1/20 in one week and 5/8 is remaining.
In order to find the number of weeks it will take the dog to finish the remaining food. We can say:
Number of weeks = remaining food / food for one week
Number of weeks = (5/8) / (1/20) = 12.5 weeks
Therefore, it will take Lian's dog 12.5 weeks to finish the remaining food
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Complete Question
Lian's dog eats 1/20 of a bag of dog food each week. Lian currently has 5/8 of a bag remaining.
How many weeks will it take Lian's dog to finish this remaining food?
If units of a certain commodity are produced, then the marginal cost is dc/da = 9x^2+4x+15x ^1/4 + 10. Suppose the total cost of producing the 1st unit is 100. Calculate the total cost of producing the first 16 units. (calculus 1)
Answers
If a certain unit of a commodity is produced with a marginal cost of [tex]\frac{dc}{da} =9x^{2}+4x+15x^{1/4} +10[/tex], the cost of producing the first 126 units is 289.4
How to differentiate a function?
Differentiation is a way of showing rate of change of a function at a given time.
The given function [tex]\frac{dc}{da} =9x^{2}+4x+15x^{1/4} +10[/tex],
Differentiate with respect to x
[tex]\frac{dc}{da} =18x+4+\frac{15}{4} x-^{3/4}[/tex]
input the value 100 for the value of x,
This implies that [tex]\frac{dc}{da} =18x+4+\frac{15}{4} x-^{3/4}[/tex]
[tex]\frac{dc}{da}[/tex]=1800+4+[tex]\frac{15}{4}[/tex] x[tex]\frac{1}{\sqrt[4]{100^{3} } }[/tex]
[tex]\frac{dc}{da}=1800+4+\frac{15}{4}x\frac{1}{3.16}[/tex]
Simplifying further
[tex]\frac{da}{da} =1804+4.75[/tex]
[tex]\frac{dc}{da} =1808.75\\[/tex]
To produce the first 16 units,
[tex]\frac{1808.75}{100} }[/tex] x [tex]\frac{16}{1}[/tex]
The cost of producing the first 16 units is 289.4
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which equation represents a proportional relationship A. y=4x B. 5x+2 C. y= 7 D. y= 3x²
Answers
Answer:
The answer is A) [tex]y=4x[/tex]
Step-by-step explanation:
This is because y directly corresponds to x, if x is 1, then y = 4, if x is 2, then y = 8 and so on.
if the mean result from the four tests is compared to the criterion 140 mg/dl, what is the probability that shelia is diagnosed as having gestational diabetes?
Answers
The probability that Shelia is diagnosed as having gestational diabetes is 0.003.
In the given question,
A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested.
Mean 129 mg/dl and standard deviation 9 mg/dl.
I have to find the probability that Shelia is diagnosed as having gestational diabetes
Z value corresponding to the mean result of 140 mg/dl is
(140-129)/4 = 2.75
Thus, the probability that Shelia is diagnosed as having gestational diabetes = P(Z > 2.75) = 0.003
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The right question is:
Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 129 mg/dl and standard deviation 9 mg/dl. Let LL denote a patient's glucose level.
If the mean result from the four tests is compared to the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes?
If staff salaries were $44,000/month last year and $53,000/month this year, what is the approximate labor cost increase in percentage?.
Answers
Given that the pay was $44,000 last year and $53,000 this year, the compensation will have increased by 20.45%.
What is percent?In essence, percentages are fractions with a 100 as the denominator. We place the percent sign (%) next to the number to indicate that the number is a percentage. For instance, you would have received a 75% grade if you answered 75 out of 100 questions correctly on a test (75/100). Percentage is defined as "out of 100." Similar to fractions and decimals, percentages are used in mathematics to represent subsets of a whole. When expressing a sum as a percentage, 100 equal parts are considered to make up the whole.
Here,
Salary of staff last year=$44,000
Salary of staff this year=$53,000
The percent increase,
=(53000-44000)/44000*100
=20.45%
The percent increase in the salary will be 20.45% as the salary last year was $44,000 and the salary this year is $53,000.
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mary spent a total of $322.58 for a party. she spent $200.14 on food, plus an additional $30.61 for each hour of the party. how long was the party?
Answers
Answer: 4 hours
Step-by-step explanation:
322.58 - 200.14 = 122.44
122.44 divided by 30.61 = 4
20 students selected macaroni and cheese and 16 students chose the cajun potato barrels. each student was asked to rate the item they chose on a scale from 1 (extremely dislike) to 9 (extremely like). which of the following prevents this study from being a well-designed experiment?
Answers
The students weren't assigned the treatments (new menu item) at random.
In an experiment, randomization refers to assigning volunteers randomly to the treatment. For example, we may define randomization as setting people to therapy at random.
The results are fair because randomization eliminates bias. To make the results of the experiment precise, it ensures that the groups created for experimenting are as similar to one another as possible.
In the given example, the menu items are not randomly assigned. Because the students were given a choice to choose a menu item instead of selecting the new menu item randomly. If randomly assigned all the new menu items will have an equal probability of being chosen.
Therefore, option b is the correct option.
The complete question is -
Thirty-six students at a local middle school were randomly selected to participate in a taste test to select a new menu item for a cafeteria. Each student was given a choice between the macaroni and cheese and the Cajun potato barrels. From the sample, 20 students selected macaroni and cheese and 16 students chose the Cajun potato barrels. Each student was asked to rate the item they chose on a scale from 1 (extremely dislike) to 9 (extremely like). Which of the following prevents this study from being a well-designed experiment?
a) There was no replication because students did not eat both menu items.
b) The treatments (new menu item) were not randomly assigned to the students.
c) An item that is currently on the menu should have been used as a control.
d) Students should not have been randomly selected.
e) There were no problems, and the study was a well-designed experiment.
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3. The function h(x) is a transformed function of f(x) = |x|. The transformation is as follows: 3 units vertical shift down, 2 units horizontal shift right.
a. Write the transformed equation, h(x).
b. Graph f(x) and h(x) on the same coordinate plane. Be sure to label the functions f(x) and h(x). This must be graphed by hand or by using the tools in Word. You may not use a web-based graphing program.
Answers
a. The transformed equation is given as follows: h(x) = |x - 2| - 3.
b. The graph of the two functions is shown by the image presented at the end of the answer.
What are translations?The translations are movements to the triangle in each of these directions:
Up.Down.Left.Right.The parent function in this problem is given as follows:
f(x) = |x|.
The first translation was a vertical translation three units down, hence we have that:
g(x) = f(x) - 3 = |x| - 3.
The second translation was a horizontal translation two units right, hence the have that the definition of the transformed equation h(x) is given as follows:
h(x) = g(x - 2) = |x - 2| - 3.
The graph of the functions is given by the image shown at the end of the answer, and the translations are verified on every point of the function, but especially the vertices.
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The sum of the measures of the angles of any triangle is 180 degrees . In a certain triangle, the largest angle measures 55degrees less than twice the medium angle, and the smallest angle measures 25 degrees less than the medium angle. Find the measures of each of the three angles.
Answers
Each of the three angles of the triangles whose sum of angles is 180 degree measures 75°, 65°, and 40°
What is meant by measurement of angles?An easy geometric instrument like a protractor is used to measure angles. A protractor is employed more frequently than a compass. Angle measurement is important for determining the precise angle. In geometry, there are several sorts of angles that we face in our daily lives, and we may measure angles using these tools.
Basic geometric instruments like a protractor and a compass are used to measure angles. These instruments aid in determining an angle's precise measurement. A protractor aids in producing an accurate angle measurement, while a compass aids in creating the angle. Angles can be measured in three different ways: in degrees, radians, and revolutions.
Here, we have
Let the largest angle be x degrees, the median y, and the smallest z.
x = 2y - 55
z = y - 25
x + y + z = 180
Substituting for x in the last equation:
2y - 55 + y + z = 180
3y + z = 235
From the second equation:
- y + z = -25 Subtract this from the last equation:
4y = 260
y = 65 degrees
z = 65 - 25 = 40 degrees.
and x = 180 - 65 - 40 = 75 degrees.
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QUESTION 3
The triangle ABC is taken to triangle A'B'C' by a dilation. Select all of the scale factors for the
dilation that would result in an image that was smaller than the original figure.
A) 1/2 B) 8/9
C) 1
D) 3/2
E) 2
Answers
A
A
A
A
A
A
A
Aa
A
Aa
A
an eyewitness to a crime said that the license plate of the getaway car began with the letters y and v, but he could not remember the order. the rest of the plate had the numbers 4, 1, 6, 0, and 3, but again he did not recall the order. how many license plates must the police investigate to find the car?
Answers
Police must investigate 50 cars to find the car.
What is permutation and combination?
A combination could be a choice of things from a collection that has distinct members, such the order of choice doesn't matter.
Main body:
there are 2 possible position for the first letter.
there is 1 possible position for the second letter.
there are 5 possible position for the first number.
there are 4 possible position for the second number.
there is 3 possible position for the third number.
there are 2 possible position for the second number.
there is 1 possible position for the third number.
the total possibilities are 2 * 1 * 5 * 4* 3 * 2 * 1 = 50
Or we can solve by saying:
There are 2 letters T and V that can be in the first two positions in any order that is 2!, and there are 5 numbers (1,4 , 6 , 0 and 3) that can be in the last 5 positions in any order that is 5!.
Therefore possible arrangement
2! * 5! = 2 * 1 * 5 * 4 * 3 * 2 * 1 = 50
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Denise works for a company where the IT department charges her department for actual usage of a SharePoint server, determining how often users log in and how much storage space her department consumes. What type of IT funding model is the company deploying?
chargeback
Answers
The company is deploying chargeback type of IT funding model where the IT department charges her department for actual usage of a SharePoint server, determining how often users log in and how much storage space her department consumes.
What is IT model funding?Traditional IT and project finance methods offer a key business risk and, if not modified, jeopardize the future of technology-based enterprises.
This excerpt from the whitepaper Changing the Finance Method by Keanen Wold, Levi Geinert, Shaun Norris, Charles Betz, Beth Dumke, Sam Guckenheimer, and Troy Magennis discusses that contradiction over how traditional financing methods handle ethical concerns and also what Agile and DevOps teams require to execute successfully.
Balancing the high-risk investment in innovation (research and development department) with regular business-as-usual responsibilities has always been difficult for organizations. Too much risk might, in the case of an unexpected failure, result in a business's dissolution; too little innovation could ultimately cause the company to be overtaken and lose market share to rivals and new upstarts.
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evaluate the expression x(2 - 4y^2) when x = -2 and y = -2
Answers
Answer:
28
Step-by-step explanation:
substitute the given values into the expression
x(2 - 4y² )
= - 2(2 - 4(- 2)² )
= - 2(2 - 4(4))
= - 2(2 - 16)
= - 2(- 14)
= 28
Carlton is a project manager for a landscape design company. He is analyzing the duration of four current projects. Which graph best shows how many days each project has lasted? A. B. C. D.
Answers
Answer:i am pretty sure it is c
Step-by-step explanation:
this is because it says Which graph best shows how many days each project has lasted? and c is the only one that is talking about days
Which number set includes only RATIONAL numbers? O a O b O c Od {-4/5, 0, 0.121314..., 97/3) (-0.123123...., 0, n/2, V100, 77/4} (-66/7, 0, 0.13, V45, 98/5} (-2/3, 0, 0.321, 149, 101/3}
PLEASE HELP
Answers
The number form that includes only rational numbers are
(-2/3, 0, 0.321, √49, 101/3}
What is rational number?A rational number is one that has the form p/q, where p and q are both integers and q is not zero
Rational function means that the equation will be a fraction
examining the numbers
(-2/3, 0, 0.321, √49, 101/3}
-2/3 is already a fraction
0 = 0/1 hence can be written as a fraction
√49 = 7 = 7/1
101/3 already a fraction
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what times 5 gives me 1,000
Answers
Answer:
200
Step-by-step explanation:
To find that out, you divide 1,000 by 5.
1,000 ÷ 5 = 200
Could you please mark me as Brainliest? Thank you!
Answer: 200
Step-by-step explanation: 1000 divided by 5 = 200
Kai is using 16-inch square tiles to tile a square area that is 9 feet long on each side. What is the least number of tiles he can use for the square area?
Answers
Answer:
144 if u multiply u will get ur answer because it is asking for the least number if 144 is to high that its got to be 1 which it can't that's if u divide it answer - 144
Which equations are true equations? Select all correct answers. 48 ÷ 8 + 2³ = 5 + 3²/ 48 ÷ 8 + 2³ = 5 + 3² / 15052=30−42−12 150 /12 8⋅72−12=22−2⋅3
Answers
The equations out of the following that are true would be:
A) and B): [tex]48[/tex] ÷ [tex]8+2^3=5+3^2[/tex] ←(Options A and B were the same.)
The remaining two I am not certain if they are true or not. The way they were laid out made little to no sense. If you could maybe evaluate further on the other options, perhaps I can double check them for you and see if there are any other remainder options that are true. However, I know for certain that A and B are true.
Which equation has the solution x = 5? Select each correct answer. Responses 32−4x=12 32 minus 4 x equals 12 25x+4=9 25 over x end fraction plus 4 equals 9 x5+5=6 x over 5 end fraction plus 5 equals 6 18−2x=9 18 minus 2 x equals 9 11 + 6x = 22
Answers
Answer:
32−4x=12,
25 over x end fraction plus 4 equals 9 ((25)/(x)+ 4 equals 9), and
x over 5 end fraction plus 5 equals 6 ((x)/(5) plus 5 equals 6)
Step-by-step explanation:
hope this help
does the series converge? if the series does converge, find the sum. (if the quantity diverges, enter diverges.)
Answers
The given series [tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ][/tex] is convergence and the sum of the series is 33/28.
In the given question we have to check whether the series converge or not. If the series does converge, then we have to find the sum.
The given series is [tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ][/tex].
Firstly we check the convergence of the given series.
When a series approaches a limit, it is considered to be convergent if both convergent and divergent behaviour occur. The deletion of a finite number of terms from the beginning of a series has no impact on convergence or divergence.
We can write the series as:
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ]=\sum_{n=1}^{\infty} (0.3)^{n-1}-\sum_{n=1}^{\infty}(0.2)^{n} \right[/tex]
As |0.3|<1 and |0.2|<1
Then [tex]\sum_{n=1}^{\infty}(0.3)^{n-1}\text{ and }\sum_{n=1}^{\infty}(0.2)^n[/tex] is convergent.
So [tex]\sum_{n=1}^{\infty} (0.3)^{n-1}-\sum_{n=1}^{\infty}(0.2)^{n} \right[/tex] is also convergent.
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ]=\sum_{n=1}^{\infty} (0.3)^{n-1}-(0.2)\sum_{n=1}^{\infty}(0.2)^{n-1} \right[/tex]
Series [tex]\sum_{n=1}^{\infty}(x)^{n-1}[/tex] converges to 1/x-1 for |x|<1.
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ]=\frac{1}{1-0.3}-(0.2)\frac{1}{1-0.2}[/tex]
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ]=\frac{1}{0.7}-\frac{0.2}{0.8}[/tex]
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ][/tex] = 10/7 - 1/4
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ][/tex] = 33/28
Hence, the sum of the given series is 33/28.
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The right question is:
Determine whether the series is convergent or divergent.
[tex]\sum_{n=1}^{\infty}\left [ (0.3)^{n-1}-(0.2)^{n} \right ][/tex]
If is is convergent, find the sum. (If the quantity diverges, enter diverges.)
research question: is the average number of people who visit the er every day increasing? it is known the average number of people who visit the er is 145. a sample is collected of 20 hospitals. the average number of people who came to the ers per day was 162, with a sample standard deviation of 45. conduct a hypothesis test based on the research question presented. using an alpha of .05
Answers
The result of hypothesis test on the given data null hypothesis is accepted and alternate hypothesis is rejected.
What is hypothesis test?Hypothesis testing is a type of measurable deduction that utilizes information from an example to make determinations about a populace boundary or a populace probability distribution. Initial, a conditional supposition that is made about the boundary or dispersion. This supposition that is known as the invalid speculation and is meant by H0.
According to question:We have,
mean(X) = 162, n = 20 , standard deviation(σ) = 45,
Let us consider Null hypothesis H₀: μ=145
And alternate hypothesis Hₐ : μ[tex]\neq[/tex] 145
So, Z = X - μ/σ/√n
Z = 162 - 145/45/√20 = 1.68
Now Level of significance(α)= 0.05
From the Z table,
Z = 1.96
So, 1.68 < 1.96
Thus, Null hypothesis is accepted and alternate hypothesis is rejected.
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x+y=8 now let x=0 what is y
Answers
Answer: Y = 8
Step-by-step explanation: If x = to 0 then 0+y=8 therefore y=8
Hope this helps
Answer:
y = 8
Step-by-step explanation:
Given information,
→ x + y = 8
→ x = 0
Now the value of y will be,
→ x + y = 8
→ 0 + y = 8
→ y = 8 - 0
→ [ y = 8 ]
Hence, the value of y is 8.
what is the answer i need answers now
Answers
Answer:
The answer is third one, 6 and 7/20.
Step-by-step explanation:
First, rewrite the decimal number as a fraction with 1 as the denominator.
[tex]\frac{6.35}{1}[/tex]
Then, multiply to remove 2 decimal places.
[tex]\frac{6.35}{1} *\frac{100}{100} = \frac{635}{100}[/tex]
Next, find the GCF(Greatest common factor) of 635 and 100, which is 5.
Divide both the numerator (top) and denominator (bottom) by the GCF (5)
Simply the improper fraction.
There you go!
Find the difference between 54 and 9. Is the
answer even or odd?
Answers
Answer:
45, odd
Step-by-step explanation:
54-9=45
a person stands on a scale in an elevator. as the elevator starts, the scale has a constant reading of 590 n. as the elevator later stops, the scale reading is 386 n. assume the magnitude of the acceleration is the same during starting and stopping.
Answers
Assuming the person's weight and magnitude, we can estimate their weight to be 491N.
What do we mean by magnitude?A mathematical object's magnitude or size defines whether it is greater or smaller than other items of its kind.
In formal terms, an object's size can be thought of as the outward manifestation of an ordering—of the class of objects to which it belongs.
So, the person's weight will be:
The scale calculates the floor's usual upward force on the passenger.
The most force is produced when going up (when starting an upward trip or ending a downward trip).
The normal force is lowest when there is a downward acceleration.
Regarding any such situation
∑F y =ma y changes to +391Nmg = ma for halting and +591Nmg=+ma for starting. (Where a represents the size of the acceleration.)
Add these two simultaneous equations to eliminate a and determine mg:
+591N−mg+391N−mg=0
OR
982N–2mg=0
Fg = mg = 982N/2 = 491N
Therefore, assuming the person's weight and magnitude, we can estimate their weight to be 491N.
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Complete question:
A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591N. As the elevator later stops, the scale reading is 391N. Assuming the magnitude of the acceleration is the same during starting and stopping, determine the weight of the person.
The image below is my question
Answers
The value of the side c which is the hypotenuse of the given right angle triangle using Pythagoras theorem is; c = √148
How to use Pythagoras Theorem?
Pythagoras Theorem is simply the way that we can find the missing length of a right angled triangle. The right angle triangle has three sides, namely the hypotenuse (which is the longest side), Opposite (which does not touch the hypotenuse) and then the adjacent side (which is between the opposite side and the hypotenuse).
The formula is literally;
c² = a² + b²
where;
c is hypotenuse and a and b are opposite and adjacent sides. Thus;
c² = 2² + 12²
c² = 4 + 144
c² = 148
c = √148
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A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 19 feet. What should the rectangle's dimensions be in order to maximize the area of the window and, therefore, allow in as much light as possible? (Round your answers to two decimal places.)
Answers
The dimensions of the rectangle that will maximize the area of the window are length, L = 3.41 feet and width, w = 3.41 feet
What rectangle dimensions will maximize the area of the window?
Given: Perimeter (P) = 19 feet
Perimeter of window = Perimeter of rectangle + Perimeter of semicircle
P = 2(2r + L) + πr
where r = radius of the semicircle and L = length of the rectangle
width(w) of the rectangle = 2r
Note: the diameter of the semicircle is the width of the rectangle
19 = 2(2r + L) + πr
19 = 4r + 2L + πr
2L = 19-4r-πr
L = 19/2 -2r- πr/2
Find the area (A) of the window:
A = Area of Rectangle + Area of Semicircle
A = 2rL + 1/2πr²
Put L into A = 2rL + 1/2πr²:
A = 2r(19/2 -2r- πr/2) + 1/2πr²
A = 19r - 4r² - πr² + 1/2πr²
A = 19r - 4r² - 1/2πr²
To maximize the area, differentiate both sides and set A' = 0
A = 19r - 4r² - 1/2πr²
A' = 19 - 8r - πr
0 = 19 - 8r - πr
8r + πr = 19
r(8 + π) = 19
r = 25/(8 + π)
width(w) = 2r
w = 2[19/(8 + π)]
w = 38/(8 + π)
w = 3.41 feet
put r = 19/(8 + π) into L = 25/2 -2r- πr/2:
L = 19/2 -2(19/(8 + π))- π(19/(8 + π))/2
L = 19/2 - 38/(8 + π) - 19π/2(8 + π)
L = [19(8 + π) - 19(2)- 19π] / [2(8 + π)]
L = 38/(8 + π)
L = 3.41 feet
Therefore, the dimensions of the rectangle are length of 3.41 feet and width of 3.41 feet
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a group of clerks must sort 1775 files. each clerk sorts at the constant rate of 30 files per hour. some of the clerks stop working at the end of the first hour; the same number of clerks stop working at the end of the second hour; and the same number of clerks stop working at the end of the third hour. the group finishes the sorting in 3 hours and 10 minutes. find the number of files sorted during the first one and a half hours of sorting.
Answers
The number of files sorted in the first one and a half hours of sorting is 945.
Here, we are given that a group of clerks must sort 1775 files.
Each clerk sorts at a constant rate of 30 files per hour.
Let the number of clerks who start working initially be x.
Thus, they will sort 30x files in the first hour.
After the first hour, let the number of clerks who stop working be a
Thus, the number of files sorted in the second hour will be 30(x - a)
Similarly, the number of files sorted in the third hour will be 30(x - 2a)x
In the last 10 minutes, the number of files sorted will be 5(x - 3a)
Thus, we can form the following equation-
30x + 30(x - a) + 30(x - 2a) + 5(x - 3a) = 1775
30x + 30x - 30a + 30x - 60a + 5x - 15a = 1775
95x - 105a = 1775
19x - 21a = 355
x = (355 + 21a)/ 19
since x and a both have to be integers, (355 + 21a) must be divisible by 19.
This is the case when a = 3
⇒ x = (355 + 21*3)/ 19
x = 418/ 9
x = 22
Now, the number of files sorted in first one and a half hours will be-
30x + 15(x - a)
= 30*22 + 15(22 - 3)
= 660 + 15*19
= 660 + 285
= 945
Thus, the number of files sorted in the first one and a half hours of sorting is 945.
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Emma is going to invest in an account paying an interest rate of 5. 5% compounded annually. How much would emma need to invest, to the nearest hundred dollars, for the value of the account to reach $480 in 8 years?.
Answers
Emma would need to invest about $320 to increase her value by $480 over the course of 8 years.
We may use the compound interest formula to determine how much Emma would need to invest:
[tex]\mathrm{A = P(1 + r/n)^{(nt)}}[/tex]
Where:
A is equal to the final sum, in this case $480.
P is the principal sum, which represents the initial investment that we're looking for.
r = the yearly interest rate (0.055 in decimal form, or 5.5%)
n = the number of times the interest is compounded annually (in this case, n = 1 because it is compounded annually).
t = the number of years, in this instance 8 years.
Let's substitute the known values into the formula and solve for P:
[tex]\mathrm{480 = P(1 + 0.055/1)^{(1\times8)}}[/tex]
Simplifying:
[tex]480 = \mathrm P(1.055)^8[/tex]
Dividing both sides by (1.055)⁸:
[tex]\mathrm {P = \frac{480 }{ (1.055)^8}}[/tex]
P ≈ $480 / 1.50
P ≈ $320
Hence the amount to be invested is $320.
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