GEOMETRY!! Provide the missing reasons for the proof.
Given: AB ≅ CD
BA ⊥ AC
DC ⊥ AC
Prove: BC ≅ DA
Answers
b) Reflexive Property
c) Given
d) Perpendicular lines form right angles. All right angles have an angle measure of 90°.
f) All right angles are congruent.
g) SAS Postulate
h) CPCTC
The triangle ΔABC is congruent to the triangle ΔCDA. Then the side BC is equal to the side DA.
What is the triangle?The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.
The triangles are said to be identical if the two sides and that included aspect of one triangle match the two sides and that includes the degree of the other triangle.
The ratio of the matching sides will remain constant if two triangles are comparable to one another.
Statements Reason
AB = CD (Given)
AC = AC (Reflexive Property)
BA ⊥ AC and DA ⊥ AC (Given)
∠BAC = ∠DCA (Perpendicular makes right angle)
ΔABC ≅ ΔCDA (Congruent triangle)
BC = DA (Corresponding sides are equal to a congruent triangle)
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Related Questions
Which equation can be used to find the answer? A playground with four sides has a perimeter of 52 ft. Three of the sides have lengths of 9 ft, 16 ft, and 19 ft. What is the length of the fourth side? A. s – 9 + 16 + 19 = 52 s =26 26 ft B. s + 9 + 16 + 19 = 52 s = 8 8 ft C. s – 52 = 9 + 16 – 19 s = 58 58 ft D. s – 9 – 16 – 19 = 52 s = 96 96 ft
Answers
Perimeter=52 ft
Perimeter=s1+s2+s3+s4
s1=9ft
s2=16ft
s3=19ft
s4=s
What we need to find: equation for given information and solution to s4=s
Perimeter=s1+s2+s3+s4
52=9+16+19+s
52=9+16+19+s
52=44+s like terms added
52-44=44-44+s additive inverse
8=s
Solution: B. s + 9 + 16 + 19 = 52 s = 8 8 ft
Option B)s + 9 + 16 + 19 = 52. Solving this gives the fourth side as 8 ft.
To find the length of the fourth side, you need to know the equation that correctly represents the given information. The perimeter of the playground is the sum of all four sides. Therefore, the correct equation to find the fourth side (s) is:
B. s + 9 + 16 + 19 = 52
9 + 16 + 19 = 44.
s + 44 = 52.
s = 52 - 44.
s = 8.
Therefore, the length of the fourth side is 8 ft.
So, the correct option is B. s + 9 + 16 + 19 = 52.
Evaluate 3 to the power of 3 + 6 (2 + 3/6)
Answers
Thus, 3 + 6(2 + 3/6) = 3 + 6 (15/6)
= 3 + 15 = 18.
Then: 3^18 is the answer. You could expand this if you wanted, but what would be the point of that?
What is the probability of selection for any man in a proportionate random sample, where a sample of 100 will be drawn from a population of 1,000 that is 50% male and 50% female?
Answers
nx = (Nx/N)*n
where:
nx = sample size for stratum x
Nx = population size for stratum x
N = total population size
n = total sample size
Given:
Nx = 100
N = 1000
n = 0.5*(1000) = 500
Required: Probability of Man to be selected
Solution:
nx = (Nx/N)*n
nx = (100/1000)*500 = 50 men
ny = (Nx/N)*n
ny = (100/1000)*500 = 50 women
Probability of Man to be selected = nx/(nx + ny)*100 = 50/(50+50)*100 = 50%
ANSWER: 50%
The salaries of the president and the Vice President of a certain country total $562,500 a year. If the president makes $237,400 more than the Vice President , find each of their salaries.
Answers
I need to solve and show work for 3x over4 divided by 5-1 over 8
Answers
4
Is the answer
3x
---
4
over
5x-1
------
8
To accomplish this, invert the 2nd fraction and then multiply:
3x 5x-1
--- * ------ = 15x^2-3x divided by 32.
4 8
Which of the following represents the zeros of the function
g(x) = x3 - 9x2 + 2x + 48 ?
A. x= -8, x= 2 , and x= -3
B. x = 8, x = -2 , and x = 3
C. x = 6, x = -4 , and x = 9
D. x = -6, x = 4 , and x = -9
Answers
Probe some roots. When you use x = - 2
you will have: (-2)^3 - 9(-2)^2 + 2(-2) + 48 = -8 - 36 - 4 + 48 = 0
So, - 2 is a root
From that you can divide x^3 - 9x^2 + 2x + 48 by x + 2 and you will get
x^2 - 11x + 24
Then you can factor that: (x - 8)(x - 3)
So, the three roots are x = - 2, x = 3 and x = 8, which is the option B.
Answer: option B. x = 8, x = -2 , and x = 3
PLEASE HELP ASAP DUE IN A DAY What is the unit rate for 768.57 m 41.1 h? Enter your answer, as a decimal, in the box.
Answers
The unit rate for 768.57 meters in 41.1 hours is approximately 18.68 meters per hour. This calculation is derived by dividing the distance by the time to find the rate of movement.
To find the unit rate, divide the distance by the time:
[tex]\[ \text{Unit rate} = \frac{\text{Distance}}{\text{Time}} \][/tex]
[tex]\[ \text{Unit rate} = \frac{768.57 \, \text{m}}{41.1 \, \text{h}} \][/tex]
[tex]\[ \text{Unit rate} \approx \frac{768.57}{41.1} \, \text{m/h} \][/tex]
[tex]\[ \text{Unit rate} \approx 18.68 \, \text{m/h} \][/tex]
So, the unit rate is approximately 18.68 meters per hour.
Johns collection contains US, Indian and British stamos. If the stamps ratios of US to Indian stamps is 5 to 2, and the ratio of Indian stamps to British stamps is 5 to 1, what is the ratio of US to British stamps
Answers
The ratio of US to British stamps is 25 to 2.
Explanation:To find the ratio of US to British stamps, we need to consider the ratios provided in the question. The ratio of US to Indian stamps is 5 to 2, and the ratio of Indian stamps to British stamps is 5 to 1. We can combine these ratios by multiplying the two ratios together. 5/2 multiplied by 5/1 is equal to 25/2. Therefore, the ratio of US to British stamps is 25 to 2.
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Over the last three evenings, Raina received a total of 83 phone calls at the call center . The first evening, she received 7 more calls than the third evening. The second evening, she received 2 times as many calls as the third evening. How many phone calls did she receive each evening ?
Answers
x + y + z = 83
x = z + 7
y = 2z
z + 7 + 2z + z = 83
4z + 7 = 83
4z = 83 - 7
4z = 76
z = 76/4
z = 19 <=== 3rd evening calls
x = z + 7
x = 19 + 7
x = 26 <=== 1st evening calls
y = 2z
y = 2(19)
y = 38 <=== 2nd evening calls
Toby, a toy store buyer, ordered 2 gross rubber spiders, 60 teddy bears, and 5 dozen video games. The spiders cost 7 cents each, the bears cost 59.76 dollars per dozen, and the video games were on sale for a special price—12 for 156 dollars. What was the total cost of Toby’s order, in dollars?
Answers
7 × 2 gross (1 gross = 144 units)
7 cents x 288 = 2016 cents = $20.16
Total cost of teddy bear:
59.76 x (60÷12) (1 dozen = 12 units)
59.76 x 5 = $298.80
Total cost of video games:
156 x 5 = $780.00
Total spent:
780 + 298.80 + 20.16 = $1098.96
last week, several gas stations in a neighborhood all charged the same price for a gallon of gas.The table below shows how much gas prices have changed from last week to this week.
a. Order the numbers in the table from least to greatest.
b. Which gas station has the cheapest gas this week?
c. Critical Thinking. Which gas station changed their price the least this week?
Gas and Samson. Star
Go. Gas. Gas
-6.6. 5.8. -6 3/4
Corner store. tip top
27/5. shop
-5 5/8
Answers
–6 3/4 = –6.75
27/5 = 5.4
–5 5/8 = –5.625
Now that the numbers are –6.6, 5.8, –6.75, 5.4 and –5.625.
a) A negative number is always less than a positive number. In addition, a negative with a larger magnitude would be less. Like –4 < –3. So the order from least to greatest is –6.75 < –6.6 < –5.625 < 5.4 < 5.8. In other words, –6 3/4 < –6.6 < –5 5/8 < 27/5 < 5.8.
b) A negative change means that the gas price decreased. The gas station that decreased the gas price the most is the Star gas since the –6 3/4 is the least number in the list.
c) The changes in prices can be compared in relation to each other based on their magnitude, which is the value without the sign. For example, the magnitude of –4 is 4. The magnitudes of the changes in prices are 6.6, 5.8, 6.75, 5.4 and 5.625. So the least change is 5.4, that is, 27/5. So the Corner Store changed the price the least.
Without the correctly formatted data, we cannot give direct answers. However, the answers would come from ordering the data from least to greatest, identifying the least positive or most negative change as the cheapest gas, and the smallest change as the least change in price.
Explanation:Unfortunately, the question you provided is a bit unclear because the table is not formatted correctly. However, it's clear that you're asked to analyze a set of data, which refers to changes in gas prices at several gas stations. Usually, to order the numbers from least to greatest, you would list them in ascending numerical order, taking into account whether they are positive or negative, with negative numbers being less than positive numbers. The gas station with the cheapest gas this week would be the one with the least positive change or the greatest negative change. The gas station that changed their price the least would be the gas station with the smallest change in price, whether positive or negative. However, without the correct formatting of data, I am unable to provide specific answers. It's always important to check your data for clarity in order to perform accurate mathematical analyses. These principles of data analysis and number order are crucial skills in your mathematical education.
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Suppose five construction companies have the ability to build a factory overseas to produce a manufactured good. The marginal cost of building a factory for each construction company is shown in the table below: Producer Marginal Cost Company 1 $1,000,000 Company 2 $1,250,000 Company 3 $1,300,000 Company 4 $1,350,000 Company 5 $1,500,000 If the market price of an overseas factory is $ 1 470 000 , what is the surplus for these five companies?
Answers
The surplus for the five companies is amounting to $830,000
What is the surplus for the five companies?Producer Surplus is the difference between the price at which a producer is willing to spend and the price which he is getting from the market for his product.
The minimum price at which a producer is willing to spend is the Marginal Cost of producing that product.Therefore, Producer surplus is Market Price- Marginal CostProducer Marginal Cost Market Price Producer Surpus Company 1 1,000,000 1,470,000 (1,470,000-1,000,000)=470,000
Company 2 1,250,000 1,470,000 (1,470,000-1,250,000)=220,000
Company 3 1,300,000 1,470,000 (1,470,000-1,300,000)=170,000
Company 4 1,350,000 1,470,000 (1,470,000-1,350,000)=120,000
Company 5 1,500,000 1,470,000 (1,470,000-1,500,000)=-30,000
Total producer surplus= 470,000 + 220,000 + 170,000 + -30,000 = 830,000
The surplus for the five companies is amounting to $830,000
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Final answer:
The surplus for Companies 1, 2, and 3 is $470,000, $220,000, and $170,000 respectively, as their marginal costs are less than the market price of $1,470,000. Companies 4 and 5 do not have a surplus since their marginal costs are above the market price. The total surplus is $860,000.
Explanation:
The surplus for the five construction companies based on their marginal costs and the market price of an overseas factory can be calculated as the difference between the market price and the marginal cost for each company. Only the companies that have a marginal cost lower than the market price will experience a surplus. Given that the market price is $1,470,000:
Company 1 has a surplus of $470,000 ($1,470,000 - $1,000,000).
Company 2 has a surplus of $220,000 ($1,470,000 - $1,250,000).
Company 3 has a surplus of $170,000 ($1,470,000 - $1,300,000).
Companies 4 and 5 do not have a surplus because their marginal costs are higher than the market price. The total surplus for the three companies with a surplus is $860,000 ($470,000 + $220,000 + $170,000).
At an interest rate of 8% compounded annually, how long will it take to double the following investments?
$50
$500
$1700
Answers
recall your logarithm rules for an exponential,
[tex]\bf \textit{Logarithm of exponentials}\\\\ log_{{ a}}\left( x^{{ b}} \right)\implies {{ b}}\cdot log_{{ a}}(x)\\\\ -------------------------------\\\\ \qquad \textit{Compound Interest Earned Amount} \\\\ [/tex]
[tex]\bf A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\$100\\ P=\textit{original amount deposited}\to &\$50\\ r=rate\to 8\%\to \frac{8}{100}\to &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annnually, thus once} \end{array}\to &1\\ t=years \end{cases} \\\\\\ 100=50\left(1+\frac{0.08}{1}\right)^{1\cdot t}\implies 100=50(1.08)^t \\\\\\ \cfrac{100}{50}=1.08^t\implies 2=1.08^t\implies log(2)=log(1.08^t) \\\\\\ [/tex]
[tex]\bf log(2)=t\cdot log(1.08)\implies \cfrac{log(2)}{log(1.08)}=t\implies 9.0065\approx t\\\\ -------------------------------\\\\ [/tex]
now, for the second amount, if the Principal is 500, the accumulated amount is 1000 when doubled,
[tex]\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\$1000\\ P=\textit{original amount deposited}\to &\$500\\ r=rate\to 8\%\to \frac{8}{100}\to &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annnually, thus once} \end{array}\to &1\\ t=years \end{cases} \\\\\\ 1000=500\left(1+\frac{0.08}{1}\right)^{1\cdot t}\implies 1000=500(1.08)^t \\\\\\ [/tex]
[tex]\bf \cfrac{1000}{500}=1.08^t\implies 2=1.08^t\implies log(2)=log(1.08^t) \\\\\\ log(2)=t\cdot log(1.08)\implies \cfrac{log(2)}{log(1.08)}=t\implies 9.0065\approx t\\\\ -------------------------------[/tex]
now, for the last, Principal is 1700, amount is then 3400,
[tex]\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\$3400\\ P=\textit{original amount deposited}\to &\$1700\\ r=rate\to 8\%\to \frac{8}{100}\to &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annnually, thus once} \end{array}\to &1\\ t=years \end{cases}[/tex]
[tex]\bf 3400=1700\left(1+\frac{0.08}{1}\right)^{1\cdot t}\implies 3400=1700(1.08)^t \\\\\\ \cfrac{3400}{1700}=1.08^t\implies 2=1.08^t\implies log(2)=log(1.08^t) \\\\\\ log(2)=t\cdot log(1.08)\implies \cfrac{log(2)}{log(1.08)}=t\implies 9.0065\approx t[/tex]
Randy school requires that there are two adult chaperones for every 18 students when the students go on a field trip to the museum if there are 99 students going to the museum how many adult shop around our need it
Answers
2 adults to 18 students is as x adults to 99 students.
2/18 = x/99
18x = 2 * 99
18x = 198
x = 11
Answer: 11 adults
Some luggage pieces have wheels and a handle so that the luggage can be pulled along the ground. Suppose the length from the bottom of the bag to the place on the floor perpendicular to the hand on the bag is 14 inches, and the length of the bag with its handle is 19 inches. At which angle made by the bag and the floor would it be comfortable to roll the bag?
Answers
The trigonometric ratio cosine, relates the angle with those lengths:
cos(angle) = adjacent-leg / hypotenuse = 14 / 19
=> angle = arc cosine (14/19) = 42.5°
Answer: 42.5°
If an article costs a store $7.45 and if it sells the article for $9.95, what percentage (to the nearest tenth) markup is it using?
Answers
subtract to find the difference:
9.95 - 7.45 = 2.50
divide 2.50 by cost price to get mark up percentage
2.50 / 7.45= 0.3355
= 33.6 percent to the nearest tenth
Answer:
Markup Percentage is 33.56 %
Step-by-step explanation:
Given: Cost of an article = $ 7.45
Selling price of the article = $ 9.95.
To find: Markup percentage or profit percentage
Clearly Selling price is greater we have profit in this case.
Profit amount = 9.95 - 7.45 = $ 2.50
Profit percentage = [tex]\frac{2.50}{7.45}\times100=33.5570469799=33.56[/tex] %
Therefore, Markup Percentage is 33.56 %
Find an equation of the normal line to the parabola y = x2 â 9x + 7 that is parallel to the line x â 7y = 2.
Answers
I'm taking the liberty of correcting your input: "Find an equation of the normal line to the parabola y = x2 â 9x + 7 that is parallel to the line x â 7y = 2." may be:
Find an equation of the normal line to the parabola y = x^2 - 9x + 7 that is parallel to the line x - 7y = 2.
How do you find the slope of a normal line to the graph of a given quadratic? Differentiate y = x^2 - 9x + 7: dy/dx = 2x - 9
You can see that this slope varies with x. You have not named an x-value here, so I will have to stop at dy/dx = 2x - 9.
The slope of a line perpendicular to the tangent line is the negative reciprocal of the slope of the tangent line, or
-1
------- which obviously depends upon the x value you choose or are given.
(2x)
Now focus on the given line x - 7y = 2. Solving for y, 7y =x-2. The slope of this line is (1/7). Thus,
-1 1
--------- = ---------- Cross-multiplying, -7 = 2x; x = -3.5
2x 7
So it appears we DO have the x-value at the point of tangency on the curve.
Find the associated y-value by subst. x = -3.5 into the eq'n of the parabola.
Call it Y.
Then the eq'n of the normal line to the parabola that is parallel to x - 7y = 2
is y-Y = (1/7)(x-[-3.5])
The normal line to the parabola y = x^2 - 9x + 7 that is parallel to the line x - 7y = 2 is y = -7x + 65, which was determined by finding the normal slope and point of tangency using the derivative.
Explanation:In mathematics, the normal line to a curve at a particular point is the line that is perpendicular to the tangent of the curve at that point. First, it's important to establish that any line parallel to x - 7y = 2 would have the same slope, which in this case is 1/7. To find the normal line, we need a line that is perpendicular to this -- hence, it would have a slope of -7.
Now, to find the point of tangency, you must take the derivative of y = x^2 - 9x + 7, making it 2x - 9. Set this equal to -7 (the slope of the normal line) and solve for x, which comes out to be 8. Substituting x = 8 back into our parabola equation gives us y = -9. Therefore, our point of tangency is (8, -9).
Finally, using the point-slope form of a line formula, we can find our equation of the normal line, which is y - (-9) = -7*(x - 8), simplifying to y = -7x + 65.
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how do i calculate an estimated regression line with the information
n=12, ∑x=66, ∑y=588, ∑xy=2244, ∑x2=396
Answers
y = ax + b
where
a = [(Σy)(Σx²) - (Σx)(Σxy)]/D
b = [n(Σxy) - (Σx)(Σy)]/D
D = n(Σx²) - (Σx)²
Therefore,
D = 12(396) - 66² = 396
a = [588*396 - 66*2244]/396 = 214
b = [12*2244 - 66*588]/396 = -30
Answer:
The regression line is y = 214x - 30
Find the length and width of a rectangle when the width is 4ft. Shorter than the length. The perimeter of the rectangle is greater than 72ft.
Answers
width= 34
Each class at briarwood elementary collected at least 54 cans of food lduring the food drive.If tbere are 29 clases in the school what was the least numbet of cans collected?
Answers
For each class → x ≥ 54
Full School → x ≥ 54(29)
If we simplify the second inequality, we get this:
x ≥ 1566
The answer to your query is x ≥ 1566. Hope this helps and have a fantastic day!
Cameron can display his 12 models cars
Answers
If the equation were y=40x+70.What would the x-intercept be? Plz...Help!?!
Answers
[tex]( - \frac{7}{4} .0)[/tex]
the period is a comma
If you put all the flowers together end to end, what would be the total length of all the flowers?
Answers
1/8 + 3/4 + 1 + 5/8 + 3/4 + 7/8 + 1 =
13/8 + 6/4 + 2 =
13/8 + 12/8 + 16/8 =
41/8 =
5 1/8 inches <==
Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, two green ones, four white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.] She does not have all the green ones.
Answers
Explanation of calculating the probability of choosing 3 green marbles out of 10 from a bag with 50 marbles.
Explanation:
The probability of choosing exactly 3 green marbles if a total of 10 marbles are selected from a bag filled with 50 marbles, 15 of which are green, can be calculated as follows:
Calculate the total number of ways to choose 3 green marbles out of 15: C(15, 3).Calculate the total number of ways to choose 7 non-green marbles out of 35: C(35, 7).Find the total number of ways to choose 10 marbles out of 50: C(50, 10).Calculate the probability by dividing the product of the above two calculations by the total ways to choose 10 marbles.What is the value of r, the part of the job that Marina can complete in 1 hour? 0.1 0.4 0.5 0.6
Answers
if in two hours she and Marina can be done, then Marina will do 1(the whole job)-2/10 (twice as much as she can do in an hour)=0.8 of the job. So if Marina does 0.8 of the job in two hours, each hour she will do half of it: 0.4 - and this is the correct answer
There are 365,493 blue pens in a warehouse and 549384 black pens how many pens are there in all
Answers
365,493+549,384
Your answer would be 366,042.384
I hope this helps!
Benny wants to put a rectangular fence in his yard, and he wants the length of the fence to be 5 less feet than twice its width. He also found that every foot of fencing will cost $3.16. The total cost of the fence is C = 3.16(6w - 10), where w is the width of the fence. What does 6w - 10 represent?
Answers
What is the probability that Eric rolls an even number or draws an ace from a standard deck of cards?
Answers
It is logic that the actions are roll a die and draw a card.
The two events are indepent, so the combined probability is the sum of the individual probabiliites.
Probability = number of positive events / number of possible events
Probability of rolling an even number = number of faces with even numbers / total number of numbers.
Probability of rolling an even number = 3 / 6 = 1/2
Probability of drawing an ace = number of aces in the deck of cards / number of cards
Probability of drawing an ace = 4 / 52 = 1 / 13
Combined probability = 1/2 + 1/13 = (13 + 2) / 26 = 15 / 26
Answer: option D. 15 / 26
If 4f(x)+f(5-x)=x2
What is f(x)?
Answers
Substitute x with 5-x, then the above equation becomes:
[tex]4f(5-x)+f(5-(5-x))=(5-x)^2[/tex], that is
[tex]4f(5-x)+f(x)=(5-x)^2[/tex]
So, we have the following system of equations:
i) [tex]4f(x)+f(5-x)=x^2[/tex]
ii) [tex]4f(5-x)+f(x)=(5-x)^2[/tex]
multiply the first equation by -4, so that we eliminate f(5-x)'s
i) [tex]-16f(x)-4f(5-x)=-4x^2[/tex]
ii) [tex]4f(5-x)+f(x)=(5-x)^2[/tex]
adding the 2 equations side by side we have:
[tex]-15f(x)=-4x^2+(5-x)^2[/tex]
expanding the binomial, and collecting same terms we have:
[tex]-15f(x)=-4x^2+(25-10x+x^2)[/tex]
[tex]-15f(x)=-3x^2-10x+25[/tex]
dividing by -5:
[tex]3f(x)=\frac{3}{5}x^2+2x-5[/tex]
dividing by 3:
[tex]\displaystyle{f(x)= \frac{1}{5}x^2+ \frac{2}{3}x-\frac{5}{3} [/tex]
Answer: [tex]\displaystyle{f(x)= \frac{1}{5}x^2+ \frac{2}{3}x-\frac{5}{3} [/tex]
Find the minimum and maximum values of the function subject to the given constraint. (if an answer does not exist, enter dne.) f(x, y) = x2y + x + y, xy = 5
Answers
[tex]L(x,y,\lambd)=x^2y+x+y+\lambda(xy-5)[/tex]
[tex]L_x=2xy+1+\lambda y=0[/tex]
[tex]L_y=x^2+1+\lambda x=0[/tex]
[tex]L_\lambda=xy-5=0[/tex]
[tex]\underbrace{10}_{2xy}+1+\lambda y=0\implies \lambda=-\dfrac{11}y[/tex]
[tex]xy=5\implies y=\dfrac5x\implies\lambda=-\dfrac{11}5x[/tex]
[tex]\impliesx^2+1+\left(-\dfrac{11}5x\right)x=0\implies x^2=\dfrac56\implies x=\pm\sqrt{\dfrac56}[/tex]
[tex]xy=5\implies y=\pm\sqrt{30}[/tex]
At these points, we get local minima of [tex]f\left(\pm\sqrt{\dfrac56},\pm\sqrt{30}\right)=\pm2\sqrt{30}[/tex].
- - -
Another way to do this is to make [tex]f(x,y)[/tex] a function independent of [tex]y[/tex], which is made possible by the constraint.
[tex]xy=5\implies y=\dfrac5x[/tex]
[tex]\implies f(x,y)=f\left(x,\dfrac5x\right)=F(x)=6x+\dfrac5x[/tex]
[tex]\implies F'(x)=6-\dfrac5{x^2}=0\implies x=\pm\sqrt{\dfrac56}[/tex]
and so on.