The extreme values of a function are the minimum and the maximum values of the function.
The extreme values are: -8 and 131.91
The given parameters are:
[tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]
[tex]\mathbf{x^2 + y^2 \le 81}[/tex]
Find the gradient of F(x,y)
[tex]\mathbf{f_x(x) = 2x + 4}[/tex]
[tex]\mathbf{f_y(y) = 2y - 4}[/tex]
Set to 0, to solve for x and y
[tex]\mathbf{2x + 4 = 0}[/tex]
[tex]\mathbf{2x = -4}[/tex]
[tex]\mathbf{x = -2}[/tex]
[tex]\mathbf{2y - 4 = 0}[/tex]
[tex]\mathbf{2y = 4}[/tex]
[tex]\mathbf{y = 2}[/tex]
So, the critical point is:
[tex]\mathbf{(x,y) = (-2,2)}[/tex]
Also, we have: [tex]\mathbf{x^2 + y^2 \le 81}[/tex]
Calculate the gradients
[tex]\mathbf{g_x = 2x}[/tex]
[tex]\mathbf{g_y = 2y}[/tex]
Equate the gradients as follows:
[tex]\mathbf{f_x = \lambda \cdot g_x}[/tex]
[tex]\mathbf{f_y = \lambda \cdot g_y}[/tex]
So, we have:
[tex]\mathbf{2x + 4 = \lambda \cdot 2x}[/tex]
[tex]\mathbf{2y - 4 = \lambda \cdot 2y}[/tex]
Make [tex]\mathbf{\lambda}[/tex] the subject, in the above equations
[tex]\mathbf{\lambda = \frac{x + 2}{x}}[/tex]
[tex]\mathbf{\lambda = \frac{y - 2}{y}}[/tex]
Equate the above equations, so we have:
[tex]\mathbf{\frac{x + 2}{x} = \frac{y - 2}{y} }[/tex]
Cross multiply
[tex]\mathbf{xy + 2y = xy - 2x}[/tex]
Subtract xy from both sides
[tex]\mathbf{2y =- 2x}[/tex]
Divide both sides by 2
[tex]\mathbf{y =- x}[/tex]
Substitute [tex]\mathbf{y =- x}[/tex] in [tex]\mathbf{x^2 + y^2 = 81}[/tex]
[tex]\mathbf{x^2 + (-x)^2 = 81}[/tex]
[tex]\mathbf{x^2 + x^2 = 81}[/tex]
[tex]\mathbf{2x^2 = 81}[/tex]
Divide both sides by 2
[tex]\mathbf{x^2 = \frac{81}{2}}[/tex]
Take square roots
[tex]\mathbf{x = \±\frac{9}{\sqrt2}}[/tex]
Rationalize
[tex]\mathbf{x = \±\frac{9\sqrt2}{2}}[/tex]
Recall that: [tex]\mathbf{y =- x}[/tex]
So, we have:
[tex]\mathbf{y = \±\frac{9\sqrt2}{2}}[/tex]
The ordered pairs are:
[tex]\mathbf{(x,y) = \{(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]
Hence, the points are:
[tex]\mathbf{(x,y) = \{(-2,2),(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]
Substitute these values in [tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]
[tex]\mathbf{F(2,2) = (-2)^2 + 2^2 + 4(-2) - 4(2) =-8}[/tex]
[tex]\mathbf{F(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}) = (\frac{9\sqrt2}{2})^2 + (-\frac{9\sqrt2}{2})^2 + 4(\frac{9\sqrt2}{2}) - 4(-\frac{9\sqrt2}{2}) =131.91}[/tex]
[tex]\mathbf{F(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2}) = (-\frac{9\sqrt2}{2})^2 + (\frac{9\sqrt2}{2})^2 + 4(-\frac{9\sqrt2}{2}) - 4(\frac{9\sqrt2}{2}) =30.09}[/tex]
Hence, the extreme values of the function are: -8 and 131.91
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Find equations for the horizontal and vertical tangent lines to the curves
You are debating about whether to buy a new computer for $800.00 or a refurbished computer with the same equipment for $640.00. If you are paid $10.00/hour and your deductions are FICA (7.65%), federal tax withholding (12%), and state tax withholding (8%), how many hours do you have to work to pay for the new computer?
A) 111 hours
B) 89 hours
C) 98 hours
D) 87 hours
First calculate all the deduction
FICA = $10 per hour x 0.0765 = $0.77 per hr
Fed tax = $10 per hr x 0.12 = $1.2 per hr
State tax = $10 per hr x 0.08 = $0.8 per hr
Remaining pay = 10 – 0.77 – 1.2 – 0.8 = $7.23 per hour
Time required = $640 / $7.23 per hr
Time required = 89 hrs
After calculating deductions from the gross wage of $10.00, the net income per hour is $7.235. To buy the new computer for $800, we divide the cost by the net wage, resulting in approximately 110.58 hours. Therefore, the student would need to work about 111 hours to afford it.
To calculate the number of hours needed to work to pay for the new computer, we must first determine the net income per hour after all the deductions. With a gross wage of $10.00 per hour, the deductions would be:
FICA: 7.65% of $10.00 = $0.765Federal Tax: 12% of $10.00 = $1.20State Tax: 8% of $10.00 = $0.80This totals $2.765 in deductions per hour. Subtracting this from the gross wage gives a net income per hour:
$10.00 - $2.765 = $7.235.
To afford the new computer costing $800, we divide the cost of the computer by the net income per hour to find the total number of hours needed:
$800.00 / $7.235 ≈ 110.58 hours.
Since we cannot work a fraction of an hour, we will need to round up to the nearest whole number, which suggests that the student would need to work approximately 111 hours to afford the new computer.
The shortest side of a right triangle measures 11m. The lengths of the other two sides are consecutive integers. Find the lengths of the other two sides.
What is the least possible sum of 24
There are four large groups of people, each with 1000 members. any two of these groups have 100 members in common. any three of these groups have 10 members in common. and there is 1 person in all four groups. all together, how many people are in these groups? (scheinerman 116) scheinerman, edward
a. mathematics: a discrete introduction, 3rd edition. cengage learning, 20120305. vitalbook file.
Final answer:
Using the principle of Inclusion-Exclusion, and taking the common members in two, three, and four groups into account, we find that there are 3439 unique members across the four groups.
Explanation:
To solve this problem, we need to use the principle of Inclusion-Exclusion. Let's denote the four groups as A, B, C, and D. According to the question, each group has 1000 members (n(A) = n(B) = n(C) = n(D) = 1000).
Any two groups have 100 members in common. That means, for instance, n(A ∩ B) = n(A ∩ C) = n(A ∩ D) = n(B ∩ C) = n(B ∩ D) = n(C ∩ D) = 100.
Any three groups have 10 members in common, which gives us n(A ∩ B ∩ C) = n(A ∩ B ∩ D) = n(A ∩ C ∩ D) = n(B ∩ C ∩ D) = 10.
There is 1 person in all four groups, so n(A ∩ B ∩ C ∩ D) = 1.
The total number of unique members in all groups, denoted as n(A ∩ B ∩ C ∩ D), is given by:
n(A ∩ B ∩ C ∩ D) = n(A) + n(B) + n(C) + n(D) - [sum of the number of common members between every two groups] + [sum of the number of common members between every three groups] - [number of members common to all four groups].
Substituting the values:
n(A ∩ B ∩ C ∩ D) = 4(1000) - [6(100)] + [4(10)] - 1 = 4000 - 600 + 40 - 1 = 3439.
Therefore, there are 3439 unique members across all groups.
In one month Miranda earned $800 at her part-time job, and $70 was withheld for federal income tax. Suppose she earns $1200 next month. How much will be withheld for federal income tax?
the temperature of a pot of water is shown. the temperature of the water changed -2.5 degree F per minute. What was the temperature after 20 minutes?
Atif has a z score of 1.02 from a converted iq score. from a z table, the percentage between his score and the mean is 34.61%. what percent of people have iqs at or lower than atif's score?
There are currently 3 students signed up for a trip. The van can transport only 7 students.
Which graph shows all the possible values for the extra number of students that need to sign up so that more than one van is needed to transport them?
Number line with closed circle on 4 and shading to the right.
Number line with closed circle on 4 and shading to the left.
Number line with open circle on 4 and shading to the right.
Number line with open circle on 4 and shading to the left.
Answer:
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Step-by-step explanation:
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Suzanne ran 3 3/4 miles in the morning and 4 1/2 miles in the afternoon. How many kilometers did she run altogether? Assume 1 mile = 1.6 kilometers.
Answer:- Total distance traveled by Suzanne in kilometers=13.2 kilometers.
Explanation:-
Distance traveled by Suzanne in the morning=[tex]3\frac{3}{4}\text{ miles}[/tex]
Distance traveled by Suzanne in the evening=[tex]4\frac{1}{2}\text{ miles}[/tex]
Total distance traveled by Suzanne=[tex][3\frac{3}{4}+4\frac{1}{2}\text{ miles}[/tex]
[tex]=\frac{15}{4}+\frac{9}{2}\text{ miles}[/tex]
[tex]=\frac{15+18}{4}\text{ miles}[/tex]
[tex]=\frac{33}{4}\text{ miles}[/tex]
[tex]=8.25\text{ miles}[/tex]
Total distance traveled by Suzanne in kilometers=[tex]8.25\times1.6\text{ kilometers}=13.2\text{ kilometers}[/tex]
Total distance traveled by Suzanne in kilometers[tex]=13.2\text{ kilometers}[/tex]
Suppose that work hours in new zombie are 200 in year 1 and productivity is $18 per hour worked. instructions: in part a, enter your answer as a whole number. in part b, round your answer to 2 decimal places.
a. what is new zombie's real gdp
A certain recipe calls for 2 3/4 cups of sugar.if the recipe is to be doubled, how much sugar should be used
Each newborn baby has a probability of approximately 0.49 of being female and 0.51 of being male. for a family with four children, let x = the number of children who are girls.
I believe this problem has 3 questions:
a. Explain why the three conditions are satisfied for X to have the binomial distribution.
b. Identify n and p for the binomial distribution.
c. Find the probability that the family has two girls and two boys.
Answers:
a. First because there are only 2 possible outcomes for
each birth: male or female. Hence a binomial distribution.
Second, because the probability of giving out a girl is
constant: 0.49 for each birth.
Third, the probability of a giving out a girl does not depend
on whether or not there is already a boy or a girl in the family.
b. The n is the total number of children, so n = 4
While the p is the success of being a girls, so P = 0.49
c. We use the binomial probability equation:
P (X) = nCx * p^x * q^(n-x)
P(X=2) = 4!/(2!2!) * 0.49^2 * 0.51^2 = 0.3747
Final answer:
Inheritance of sex chromosomes in babies is determined by the father, leading to a 50:50 chance of a male or female child. The likelihood of a newborn being female is around 0.49.
Explanation:
Sex Inheritance: The sex of a baby is determined by the father, who has a 50 percent chance of passing on a Y chromosome resulting in a male child. If the baby inherits two X chromosomes, it will be female. This 50:50 chance applies to each baby. For a family with four children, let x = the number of girls, the probability of a child being female is approximately 0.49.
Please help me on 3 and 4 please thanks ❤️
Solve for x.
5x - 1 = 26
x = 27/5
x = 5
x = -5
F(x, y, z) = xe^(5yz) find the gradient of f
How many grams are in 1.40 moles of sodium chloride (NaCl)?
The molar mass of NaCl is 58.44 g / mol. So we can calculate the mass by simply multiplying the number of moles and molar mass. That is:
mass = 1.40 moles * 58.44 g / mol
mass = 81.82 grams
Final answer:
To find the mass in grams of 1.40 moles of sodium chloride, multiply the number of moles by the molar mass of NaCl, which is 58.44 g/mol, resulting in 81.816 grams.
Explanation:
To calculate the mass of 1.40 moles of sodium chloride (NaCl), we will use the molar mass of NaCl as a conversion factor. The molar mass of NaCl is 58.44 grams per mole. Here's how you can do the conversion:
Find the molar mass of NaCl: 22.99 g/mol (Na) + 35.45 g/mol (Cl) = 58.44 g/mol (NaCl).
Multiply the number of moles of NaCl by the molar mass to get the mass in grams: 1.40 moles × 58.44 g/mol = 81.816 grams of NaCl.
So, 1.40 moles of sodium chloride weigh 81.816 grams.
A company makes 3 types of cable. cable a requires 3 black, 3 white, and 2 red wires. b requires 1 black, 2 white, and 1 red wire. c requires 2 black, 1 white, and 2 red wire. they used 120 black, 110 white and 100 red wires. how many of each cable were made? setup the system of equations to solve this problem by selecting the coefficients and constants for each colored wires.
To determine the number of each cable produced, a system of equations is set up with variables representing the cable types and coefficients corresponding to the number of wires used per cable. The system consists of three equations based on the colors of wires and the totals used.
To solve this problem, we can set up a system of equations where the unknowns are the number of each type of cable produced. Let's represent the number of type A cables as x, type B cables as y, and type C cables as z. Considering the number of wires used for each type of cable and the total used, we form the following equations:
3x + y + 2z = 120 (black wires)3x + 2y + z = 110 (white wires)2x + y + 2z = 100 (red wires)To find the number of each cable made, we need to solve this system of equations simultaneously. The coefficients represent the number of each color wire needed for each cable type, and the constants (120, 110, 100) represent the total number of wires used.
how many factors are in the expression x(1+y)^z
a. 3
b.2
c.0
d.1
There are two factors: ( x ) and (1+y). So option (b) is correct.
To find the factors in the expression ( x(1+y)^z ), we break it down into its components.
1. ( x ) - This is a factor because it's multiplied by the rest of the expression.
2. (1+y) - This is also a factor because it's part of the expression being raised to the power of \( z \).
3. ( z ) - The exponent does not affect the number of factors, as it's simply the power to which the expression is raised.
So, there are two factors: ( x ) and (1+y).
Detailed calculation:
( x ) is multiplied by the entire expression, so it's a factor.
(1+y) is a factor because it's part of the expression.
( z ) is an exponent and doesn't count as a separate factor.
Thus, the answer is (b) 2.
What is the quotient of d-2 / d4-6d3+d+17=
A. d3− 4d2 + 9d + 35 B. d3− 8d2 + 17d –17 C. d3− 8d2 + 16d − 31 R 79 D. d3– 4d2– 8d –15 R–13
Answer:
Step-by-step explanation:
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Geraldo wants to start a group bike ride on Sunday mornings for students at his school. He asked 30 students in the lunchroom how many miles they ride their bikes each month. He obtained the following results. 8, 3, 0, 1, 2, 3, 6, 3, 0, 4, 13, 1, 3, 12, 15, 2, 1, 3, 2, 2, 0, 3, 4, 2, 6, 7, 12, 8, 4, 0 The fraction x/30 shows the ratio of the number of students who ride their bikes more than 10 miles each month to the number of students surveyed. What is the value of x?
Answer:
x = 4 is the answer.
Step-by-step explanation:
List of rides in a month of 30 students have been given in the question.
We have to find the value of x if fraction x/30 shows the ratio of the number of students who ride their bikes more than 10 miles each month to the number of students surveyed.
Therefore x/30 = 4/30
x = 4×30/30 = 4
So x = 4 is the answer.
Consider the function below.
x -1 0 1 2
f(x) -2 3 8 13
Which of the following functions could be the inverse of function f?
x -2 3 8 13
q(x) -1 0 1 2
x -2 -3 -8 -13
r(x) 1 0 -1 -2
x -1 0 1 2
p(x) 2 -3 -8 -13
x 1 0 -1 -2
s(x) -2 3 8 13
If an item is 1/3 off, to calculate the final price simply divide the original price by 3.
true or false
A condo in Culver City, California, listed for $1.4 million with 20% down and financing at 5% for 30 years. What's the monthly payment?
Answer:
Step-by-step explanation:
$6,014.40
on monday your bank account was balanced -$12.58 because you didn't realize this, you wrote a check for -$30.72 for groceries what is the new balance in your checking account?
The new balance after I bought groceries worth of 30.72 dollars is -43.3 dollars.
What are he types of sign distributions?Sign distributions are of four types they are:- negative distributed to negative results in positive and positive distributed positive results to positive. Negative distributed to positive or vice versa results in negative.
According to the given question on Monday I had -12.58 dollars.
I bought some groceries worth -30.72 dollars which will also be deducted from my bank account.
∴ The final balance I have is ( -12.58 - 30.72) dollars which is,
= -43.3 dollars.
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Solve the system
4x+2y=7 y=5x
A. (1,5)
B. (3,15)
C. (2,10)
D. (0.5,2.5)
(1 point) consider the function f(t)=⎧⎩⎨⎪⎪⎪⎪0,−5,−6,6,t<00≤t<11≤t<7t≥7;f(t)={0,t<0−5,0≤t<1−6,1≤t<76,t≥7; 1. write the function in terms of unit step function
Final answer:
The function f(t) can be represented in terms of the unit step function as f(t) = -5u(t) - u(t-1) + 6u(t-7), with each term turning on at specific breakpoints.
Explanation:
To write the function f(t) in terms of the unit step function, you can think of f(t) as a piecewise function that changes values at various breakpoints. The unit step function, denoted as u(t), is defined to be 0 when t < 0 and 1 when t ≥ 0. For the given function f(t), we can write it using the unit step function as follows:
f(t) = -5u(t) -u(t-1) + 6u(t-7)
This representation uses the properties of the unit step function to 'turn on' different function values at specified points. Each term corresponds to a segment of the piecewise function, with the coefficients representing the value of f(t) within each segment.
Which group of 3 cards would allow Carlos to win with 21 points?
A. 6 of clubs, 7 of spades, 8 of diamonds
B. Ace of hearts, Ace of diamonds, 10 of hearts
C. Jack of clubs, 7 of clubs, 4 of spades
D. 8 of clubs, 9 of hearts, 9 of spades
At a pet store, Davina counted 12 parrots out of 20 birds. What is an equivalent ratio of parrots to birds in the pet store?
Lucy is thinking of a number.The number is greater than two hundred twenty- five. her number is less than 2 hundred 2 tens 7 ones. what is lucy's number?