Answer:
100 psia
Step-by-step explanation:
Applying,
Pressure law,
P/T = P'/T'................. Equation 1
Where P = initial pressure, P' = Final pressure, T = initial temperature, P' = Final temperature.
make P' the subject of the equation
P' = PT'/T............ Equation 2
From the question,
Given: P = 50 psia, T = 300°R = (300×5/9)K = 166.66 K, T' = 600°R = (600×5/9)K = 333.33 K
Substitute these values into equation 2
P' = (50×333.33)/166.66
P' ≈ 100 psia
P ≈ 100 psia
the time it takes a runner to complete a race is inversely related to the speed of the runner if a runner can complete a race in 40 minutes while running at 8 mph how long will it take the runner to complete the race running at 9 mph t
In May 2010, the Pew Research Center for the People & the Press carried out a national survey to gauge opinion on the Arizona Immigration Law. Responses (Favor, Oppose, Don’t Know) were examined according to groups defined by political party affiliation (Democrat, Republican, Independent). Which of the following would be appropriate for displaying these data?
a. Pie charts
b. Segmented bar chart.
c. Side by side bar chart.
d. Contigency table
Explanation:
It's most effective to use a contingency table because we have two variables here: 1) the responses, and 2) the party affiliation.
We can have the responses along the rows and the party affiliation along the columns, or vice versa.
See the example below. The values are completely random simply for the purpose of the example (and not drawn from any real life data source).
As per the given options, the appropriate for the displaying these data will be contingency table. Hence, option D is correct.
What is a Pie chart?A pie chart is a visual depiction of information in the shape of a pie, where the pieces of the pie represent the magnitude of the data. To depict data as a pie chart, you need a list of quantitative variables as well as categorical variables.
As per the given information in the question,
A contingency table in statistics is a particular kind of matrix-style table that shows the frequency of the variables. They are extensively utilized in scientific, engineering, business intelligence, and survey research.
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2x+2y=38 y=x+3 solve by the solution
Answer:
x = 8 , y = 11
Step-by-step explanation:
[tex]2x + 2y = 38 => x + y = 19 - -- ( 1 ) \\\\y = x + 3 ---- ( 2 ) \\\\Substitute \ ( 2 ) \ in \ ( 1) :\\\\ x + y = 19\\\\x + ( x+ 3) = 19\\\\2x + 3 = 19\\\\2x = 19 - 3 \\\\2x = 16 \\\\x = \frac{16}{2} = 8\\\\Substitute \ x = 8 \ in \ ( 1 ) : \\\\x + y = 19\\\\8 + y = 19\\\\y = 19 - 8 = 11[/tex]
Cathy is planning to take the Certified Public Accountant Examination (CPA exam). Records kept by the college of business from which she graduated indicate that 73% of students who graduated pass the CPA exam. Assume that the exam is changed each time it is given. Let n = 1, 2, 3, ... represent the number of times a person takes the CPA test until the first pass. (Assume the trials are independent).
(a) What is the probability that Cathy passes the CPA test on the first try?
(b) What is the probability that Cathy passes the CPA test on the second or third try?
Answer:
The responses to these question can be defined as follows:
Step-by-step explanation:
For point a:
[tex]\to P(1) = 0.73[/tex]
For point b:
[tex]\to P(2\ or\ 3) = P(2) + P(3)[/tex]
[tex]= 0.27 \times 0.73 + 0.27\times 0.27\times0.73\\\\=0.1971+0.1971\times 0.27\\\\=0.1971+0.053217\\\\=0.250317[/tex]
Solve the inequality. |X-15|>9
Answer:
X<6 or X>24
Step-by-step explanation:
find the equation of the line shown
Answer:
y=1/2x+1/2
Step-by-step explanation:
In order to find the slope, you can use rise/run, in this case, the slope is 1/2 and the y-intercept is at (0, 0.5)
Please help!! How do I solve for x?
The line in the middle is half the length of the line on the outside. Multiply the middle line by 2 and set it equal to the outside line.
2(x-3) = x + 6
Simplify:
2x -6 p x + 6
Add 6 to both sides
2x = x + 12
Subtract x from both sides:
X = 12
The answer is B) 12
Suppose the mean income of firms in the industry for a year is 95 million dollars with a standard deviation of 11 million dollars. If incomes for the industry are distributed normally, what is the probability that a randomly selected firm will earn less than 114 million dollars
Answer:
95.73%
Step-by-step explanation:
Given data:
mean μ= 95
standard deviation, σ = 11
to calculate, the probability that a randomly selected firm will earn less than 114 million dollars;
Use normal distribution formula
[tex]P(X<114)=P(Z<\frac{X-\mu}{\sigma} )[/tex]
Substitute the required values in the above equation;
[tex]P(X<114)=P(Z<\frac{114-95}{11} )\\P(X<114)=P(Z<1.7272)\\P(X<114)=0.9573[/tex]
Therefore, the probability that a randomly selected firm will earn less than 114 million dollars = 95.73%
determine the general solution of cos2X -7cosX -3=0
Answer:
x=2pi/3 +2pi n, 4pi/3 +2pi n for all integar of n.
Step-by-step explanation:
A papaya is accidentally dropped from a bridge, which is 30 m above the water. Ignoring air resistance, the papaya's speed just before it hits the water will be __________ m/s.
Answer:
24 .30ms is the answer I think so if the answer is correct plz mark me as brainliest.
The papaya's speed just before it hits the water will be 24.24 m/s.
What is speed?The rate at which objects moves is called speed. It is given by
[tex]s = \frac{d}{t}[/tex]
An object held above the ground has a potential energy related to the height at which it is held,
PE = mgh
If you drop the object, its potential energy will become the kinetic energy of motion:
KE = ½ mv²
½ mv² = mgh
v = √(2gh)
v = √(2*9.8*30)
v = 24.24 m/s
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In the following diagram HI || JK.
HELP MATES PLEASE WILL GIVE 15 POINTS
What is the measure of Zx?
Angles are not necessarily drawn to scale.
67°
H
K
46°
2°
I
A
Answer:
m∠ x = 67
Step-by-step explanation:
∠AJK = ∠AHI = 67 Corresponding Angles
180 - 67 - 46 = x
x = 67
Triangle Sum Theory - the sum of all angles in a triangle = 180
Also, when you see parallel lines look for Corresponding,
Alternate Interior or Same side Interiors.
6/10 > _ > 1/3 which fraction goes in the blank?
Step-by-step explanation:
6/10 > _ > 1/3
3/5 > _ > 1/3
Taking the average of both the fraction½(⅗+⅓)
½(9+5/15)
½(14/5)
=7/15
6/10 > 7/15 > 1/3Answer:
7/15
Step-by-step explanation: 10 and 3 LCM is 30
6/10 x 3 =18/30 and 1/3x 10= 10/30
10/30 and 18/30 average is 14/30 which simplified is 7/15
The answer is 7/15
Hope it helps
Use cylindrical shells to find the volume of the solid generated when the region
R under y = x2 over the interval (0,2) revolved about the line y = -1
Answer:
[tex]\displaystyle V = \frac{176 \pi}{15}[/tex]
General Formulas and Concepts:
Pre-Algebra
Equality PropertiesAlgebra I
Terms/CoefficientsExpandingFunctionsFunction NotationGraphingExponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Calculus
Integrals
Definite IntegralsArea under the curveIntegration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Shell Method:
[tex]\displaystyle V = 2\pi \int\limits^b_a {xf(x)} \, dx[/tex]
[Shell Method] x is the radius[Shell Method] 2πx is the circumference[Shell Method] 2πxf(x) is the surface area[Shell Method] 2πxf(x)dx is the volumeStep-by-step explanation:
Step 1: Define
Identify
Graph of region
y = x²
x = 2
y = 4
Axis of Revolution: y = -1
Step 2: Sort
We are revolving around a horizontal line.
[Function] Rewrite in terms of y: x = √y[Graph] Identify bounds of integration: [0, 4]Step 3: Find Volume Pt. 1
[Shell Method] Find distance of radius x: [tex]x = y + 1[/tex][Shell Method] Find circumference variable f(x) [Area]: [tex]\displaystyle f(x) = 2 - \sqrt{y}[/tex][Shell Method] Substitute in variables: [tex]\displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - \sqrt{y})} \, dy[/tex][Integral] Rewrite integrand [Exponential Rule - Root Rewrite]: [tex]\displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - y^\bigg{\frac{1}{2}})} \, dy[/tex][Integral] Expand integrand: [tex]\displaystyle V = 2\pi \int\limits^4_0 {(-y^\bigg{\frac{3}{2}} + 2y - y^\bigg{\frac{1}{2}} + 2)} \, dy[/tex][Integral] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle V = 2\pi \bigg( \frac{-2y^\bigg{\frac{5}{2}}}{5} + y^2 - \frac{2y^\bigg{\frac{3}{2}}}{3} + 2y \bigg) \bigg| \limits^4_0[/tex]Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle V = 2\pi (\frac{88}{15})[/tex]Multiply: [tex]\displaystyle V = \frac{176 \pi}{15}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Applications of Integration
Book: College Calculus 10e
Ben starts walking along a path at 3 mi/h. One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at 7 mi/h. How long (in hours) will it be before Amanda catches up to Ben?
Enter the exact answer.
Hint: The distance formula is that distance = rate * time, so for example in one and a half hours, Ben has walked 3 * 1.5 miles.
Amanda catches up to Ben in ____________ hours.
Answer:
1.125 hours
Step-by-step explanation:
Given :
Ben's speed = 3 mi/hr
Time before Amanda starts = 1.5 hours
Amanda's speed = 7 mi/hr
Time before Amanda catches up with Ben
Recall :
Distance = speed * time
Distance already covered by Ben before Amanda starts :
(3 * 1.5) = 4.5
Hence, we can setup the equation :
Ben's distance = Amanda's distance
Let time taken = x
4.5 + 3x = 7x
4.5 = 7x - 3x
4.5 = 4x
x = 4.5 / 4
x = 1.125 hours
1.125 * 60 = 67. 5 minutes
A cylinder has a radius of 2.5 inches (in.) and a height of 11 in., as shown.
2.5 in.
11 in.
What is the surface area, in square inches, of the cylinder?
Answer:
212.06
Step-by-step explanation:
can't really explain since the formula is fricking long but trust me that's uts 212.06 in²
what are the following proof triangle LMN equals triangle OPQ
Answer:
D. SSS
Step-by-step explanation:
Was given to us that the corresponding sides are congruent so is SSS.
Side Side Side Theorem tells us that if am the sides of a triangle are having the same measurement with the corresponding sides of another triangle then the two triangles are congruent.
A plumber charges $50 for the first visit plus $8 per hour of work. If the total bill is $290, how many hours did the plumber work?
30 hours
40 hours
80 hours
None of these choices are correct.
Answer:
Step-by-step explanation:
50 + 8x = 290
8x = 240
x = 30 hours
A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation.
Y
p(x,y) 0 1 2
x 0 0.10 0.04 0.02
1 0.08 0.20 0.06
2 0.06 0.14 0.30
a. What is P(X = 1 and Y = 1)?
b. Compute P(X ≤ 1 and Y ≤ 1).
c. Give a word description of the event {X ≠ 0 and Y ≠ 0}, and compute the probability of this event.
d. Compute the marginal pmf of X and of Y. Using pX(x), what is P(X ≤ 1)?
e. Are X and Y independent rv’s? Explain.
Answer:
a. 0.2
b. 0.42
c. 0.7
d. the solution is in the explanation
e. x and y are not independent
Step-by-step explanation:
a. from the joint probability mass function table,
p(x=1) and p(Y= 1)
= p(1,1) = 0.2
b. prob(0,0)+prob(0,1)+prob(1,0)+prob(1,1)
= 0.10 + 0.04 + 0.08 + 0.20
= 0.42
P(X ≤ 1 and Y ≤ 1) = 0.42
c. prob {X ≠ 0 and Y ≠ 0}
= prob(1,1) + prob(1,2) + prob(2,1) + prob(2,2)
= 0.20 + 0.06 + 0.14 + 0.30
= 0.7
d. we have to calculate the marginal pmf of x and y here.
we have the x values as 0,1,2
prob(x=0) = 0.1 + 0.04 + 0.02
= 0.16
prob(x=1) = 0.08 + 0.2 + 0.06
= 0.34
prob(x=2) = 0.06+0.14+0.3
= 0.50
we have y values as 0,1,2
prob(y=0) = .1+.08+.06
= 0.24
prob(y=1) = .04+.2+.14
= 0.38
prob(y = 2) = 0.02+0.06+0.3
= 0.38
P(X ≤ 1) = prob(x=0)+prob(x=1)
= 0.34+0.16
= 0.50
e. from the joint table we have this,
prob(1,1) = 0.2
prob(x=1) = 0.34
prob(y=1) = 0.38
then prob(x=1)*prob(y=1)
= 0.34*0.38
= 0.1292
therefore prob(1,1) is not equal to prob(x=1)*prob(y=1)
0.2≠0.1292
x and y are not independent
A(n) _____ is an expression that uses variables to state a rule.
plz help asap
Answer:
A FORMULA is an expression that uses variables to state a rule.
Assume that 300 births are randomly selected and 5 of the births are girls. Use subjective judgment to describe the number of girls as significantly high, significantly low, or neither significantly low nor significantly high.g
Answer:
Following are the response to the given choice.
Step-by-step explanation:
Please find the complete question in the attached file.
Subjective opinion = Question of opinion.
Therefore this requires only just opinion and we don't have to do any actual calculations.
Does this seem like a great number of girls, a little number of girls, or a decent number of girls to you but if 1,300 babies were born, 5 of whom were females?
This is a small number of beautiful gals, in my honest opinion. We anticipate boys and girls to be produced about the very same frequency, thus I expect some half of them to be females if there are 1 300 newborns. You should have roughly 650 girls if 50% of the infants are girls, but now we only have five. That appears to me to be considerably low. which is your own opinion.
hello can anyone help with this?
Answer:
<2 and <13 are alternate exterior angles.
In simple form, alternate exterior angles are the opposite angle on the opposing parallel line. So, to make you understand better, <4 and <15 are alternate exterior angles.
Hope this helps :D
add 7/8 + 2 3/24 + 6 1/6
Answer:
9 4/24 or 9 1/6
Find the LCM(lowest common multiple) of 8, 24 and 6.The LCM of 8, 24 and 6 is 24.We now want to turn all the denominators into 24 so we are going multiply 7/8 by 3 and 1/6 by 4. We won't need to turn the denominator of 3/24 into 24 because it's already 24Whatever you do to the denominator you have to do to the numerator, so you also have to multiply the numerator of 7/8 by 3 and the numerator of 1/6 by 4That now results in 21/8 + 2 3/24 + 6 4/24Now you have to add all the fractions together which is going to equal to 28/24Because 28/24 is more than the whole, subtract 28 from 24 which gives us 4. That 4 is now our new numeratorWe are now going to all the whole numbers 6+2+1=9. Incase you're wondering, the '1' came from the 28/24The answer you should get should be 9 4/24 or if it should be simplified it would be 9 1/6
If the domain of a function that is reflected over the x-axis is (1, 5), (2, 1), (-1, -7), what is the range?
A. (1, -5), (2, -1), (-1, 7)
B. (5, 1), (1, 2), (-7, -1)
C. (-5, -1), (-1, -2), (7, 1)
D. (-1, 5), (-2, 1), (1, -7)
Answer:
A. (1, -5), (2, -1), (-1, 7)
Step-by-step explanation:
Reflecting a function over the x-axis:
When a function is reflected over the x-axis, the x-value stays the same, while y changes the signal, so the transformation rule is:
[tex](x,y) \rightarrow (x,-y)[/tex]
To find the range:
We apply the transformation to the points in the domain. Thus:
[tex](1,5) \rightarrow (1,-5)[/tex]
[tex](2,1) \rightarrow (2,-1)[/tex]
[tex](-1,-7) \rightarrow (-1,-(-7)) = (-1, 7)[/tex]
Thus the correct answer is given by option a.
Answer:
It is letter A and please give me brainliest
Step-by-step explanation:
In a box of chocolates, 12 of the chocolates are wrapped in red foil. That is 30% of the chocolates in the box. How many chocolates are there?
Answer:
The answer is 40 chocolates in the box in total
If m and n are positive integers and m^2 - n^2 = 9, which of the following could be the value of
n?
A) 1
B) 16
C) 9
D) 4
Answer:
4
Problem:
If m and n are positive integers and m^2 - n^2 = 9, which of the following could be the value of
n?
A) 1
B) 16
C) 9
D) 4
Step-by-step explanation:
One approach would be to plug in the choices and see.
If n=1, then we have m^2-1=9.
This would give m^2=10 after adding 1 on both sides. There is no integer m when squared would give us 10. ( Square root of 9 is a decimal )
If n=16, then we would have m^2-256=9.
This would give m^2=265 after adding 256 on both sides. There is no integer m when squared would give us 265. ( Square root of 265 is a decimal )
If n=9, then we would have m^2-81=9.
This would give m^2=90 after adding 81 on both sides. There is no integer m when squared would give us 90. ( Square root of 90 is a decimal )
If n=4, then we would have m^2-16=9.
This would give m^2=25 after adding 16 on both sides. There is an integer m when squared would give us 25. ( Square root of 25 is a 5)
If 5x = 3x+12 then x = …..
↦ [tex]\huge\underline{ \underline{Answer:-}}[/tex]
[tex]5x = 3x + 12 \\ 5x - 3x = 12 \\ 2x = 12 \\ x = \frac{12}{2} \\ x = 6[/tex]
ʰᵒᵖᵉ ⁱᵗ ʰᵉˡᵖˢ
꧁❣ ʀᴀɪɴʙᴏᴡˢᵃˡᵗ2²2² ࿐
Step-by-step explanation:
Explanation is in the attachmenthope it is h helpful to you
SOMEONE PLS HELP ME I WILL MAKE U BRAINLIST ! In a survey sample of 83 respondents, about 30.1 percent of the samplework less than 40 hours per week. What is the estimated standard error for the group of respondents who work 40 hours or more per week?
(*round to two decimal places)
Answer:
Answer = √(0.301 × 0.699 / 83) ≈ 0.050
A 68 percent confidence interval for the proportion of persons who work less than 40 hours per week is (0.251, 0.351), or equivalently (25.1%, 35.1%)
Step-by-step explanation:
√(0.301 × 0.699 / 83) ≈ 0.050
We have a large sample size of n = 83 respondents. Let p be the true proportion of persons who work less than 40 hours per week. A point estimate of p is because about 30.1 percent of the sample work less than 40 hours per week. We can estimate the standard deviation of as . A confidence interval is given by , then, a 68% confidence interval is , i.e., , i.e., (0.251, 0.351). is the value that satisfies that there is an area of 0.16 above this and under the standard normal curve.The standard error for a proportion is √(pq/n), where q=1−p.
Hope this answer helps you :)
Have a great day
Mark brainliest
Will give brainliest answer
Answer:
the x-intercepts are at
x = -3
x = 0
x = 1
Step-by-step explanation:
ask the points, where the functional value is 0.
2x³ + 4x² - 6x = 0
we see that every term contains an expression of x. so, we can simplify this
x × (2x² + 4x - 6) = 0
so, one solution is plainly visible : x=0
for the other solutions we need to solve the square equation
2x² + 4x - 6 = 0
or even simpler
x² + 2x - 3 = 0
the solution of a square equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a=1
b=2
c=-3
x = (-2 ± sqrt(2² - 4×1×-3))/(2×1) = (-2 ± sqrt(4 + 12))/2 =
= (-2 ± sqrt(16))/2 = (-2 ± 4)/2 = -1 ± 2
x1 = -1 + 2 = 1
x2 = -1 - 2 = -3
A student takes an exam containing 16 true or false questions. If the student guesses, what is the probability that he will get exactly 14 questions right
Answer:
0.001831055
Step-by-step explanation:
Here, n = 16, p = 0.5, (1 - p) = 0.5 and x = 14
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)
We need to calculate P(X = 14)
P(x =14) = 16C14 * 0.5^14 *(1-0.5)^2
= 120 * 0.5^14 *(1-0.5)^2
= 0.001831055
Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint.
(a) f (x, y) = x^2 - y^2; x^2 + y^2 = 1
Max of 1 at (plusminus 1, 0), min of - 1 at (0, plusminus l)
(b) f (x, y) = 3x + y; x^2 + y^2 = 10
Max of 10 at (3, 1), min of - 10 at (- 3, - 1)
(c) f (x, y) = xy; 4x^2 + y^2 = 8
Max of 2 at plusminus (1, 2), min of - 2 at plusminus (l, - 2)
Answer:
a) f(x,y) = - 1 minimum at P ( 0 ; -1 )
b) f (x,y) = 10 maximum at P ( 3 , 1 ) and f (x,y) = - 10 minimum at Q ( - 3 , - 1 )
c) Max f ( x , y ) = 2 for points P ( 1, 2 ) and T ( -1 , -2 )
Min f ( x , y ) = -2 for points Q ( 1 , - 2 ) and R ( -1 , 2 )
Step-by-step explanation:
A) f(x,y) = x² - y² subject to x² + y² = 1 g(x,y) = x² + y²- 1
δf(x,y)/ δx = 2*x δg(x,y)/ δx = 2*x
δf(x,y)/ δy = - 2*y δg(x,y)/ δy = 2*y
δf(x,y)/ δx = λ* δg(x,y)/ δx
2*x = λ*2*x
δf(x,y)/ δy = λ* δg(x,y)/ δy
- 2*y = λ*2*y
Then, solving
2*x = λ*2*x x = λ*x λ = 1
- 2*y = λ*2*y y = - 1
x² + y²- 1 = 0 x² + ( -1)² - 1 = 0 x = 0
Point P ( 0 ; -1 ) ; then at that point
f(x,y) = x² - y² f(x,y) = 0 - ( -1)² f(x,y) = - 1 minimum
b) f( x, y ) = 3*x + y g ( x , y ) = x² + y² = 10
δf(x,y)/ δx = 3 δg(x,y)/ δx = 2*x
δf(x,y)/ δy = 1 δg(x,y)/ δy = 2*y
δf(x,y)/ δx = λ * δg(x,y)/ δx ⇒ 3 = 2* λ *x (1)
δf(x,y)/ δy = λ * δg(x,y)/ δy ⇒ 1 = 2*λ * y (2)
x² + y² - 10 = 0 (3)
Solving that system
From ec (1) λ = 3/2*x From ec (2) λ = 1/2*y
Then (3/2*x ) = 1/2*y 3*y = x
x² + y² = 10 ⇒ 9y² + y² = 10 10*y² = 10
y² = 1 y ± 1 and
y = 1 x = 3 P ( 3 , 1 ) y = - 1 x = -3 Q ( - 3 , - 1 )
Value of f( x , y ) at P f (x,y) = 3*x + y f (x,y) = 3*(3) +1
f (x,y) = 10 maximum at P ( 3 , 1 )
Value of f( x , y ) at Q f (x,y) = 3*x + y f (x,y) = 3*(- 3) + ( - 1 )
f (x,y) = - 10 minimum at Q ( - 3 , - 1 )
c) f( x, y ) = xy g ( x , y ) = 4*x² + y² - 8
δf(x,y)/ δx = y δg(x,y)/ δx = 8*x
δf(x,y)/ δy = x δg(x,y)/ δy = 2*y
δf(x,y)/ δx = λ * δg(x,y)/ δx ⇒ y = λ *8*x (1)
δf(x,y)/ δy = λ * δg(x,y)/ δy ⇒ x = λ *2*y (2)
4*x² + y² - 8 = 0 (3)
Solving the system
From ec (1) λ = y/8*x and From ec (2) λ = x/2*y Then y/8*x = x/2*y
2*y² = 8*x² y² = 4*x²
Plugging that value in ec (3)
4*x² + 4*x² - 8 = 0
8*x² = 8 x² = 1 x ± 1 And y² = 4*x²
Then:
for x = 1 y² = 4 y = ± 2
for x = -1 y² = 4 y = ± 2
Then we get P ( 1 ; 2 ) Q ( 1 ; - 2)
R ( - 1 ; 2 ) T ( -1 ; -2)
Plugging that values in f( x , y ) = xy
P ( 1 ; 2 ) f( x , y ) = 2 R ( - 1 ; 2 ) f( x , y ) = - 2
Q ( 1 ; - 2) f( x , y ) = -2 T ( -1 ; -2 ) f( x , y ) = 2
Max f ( x , y ) = 2 for points P and T
Min f ( x , y ) = -2 for points Q and R