To find the maximum and minimum values of the function f(x, y) = x + y + 4 on the circular disk x^2 + y^2 ≤ 1, evaluate the function at the boundary of the disk.
Explanation:To find the maximum and minimum values of the function f(x, y) = x + y + 4 on the circular disk x^2 + y^2 ≤ 1, we can use the method of level curves and gradients.
First, find the gradient of the function f(x, y) using partial derivatives.Next, find the critical points of the function by setting the gradient equal to zero and solving for x and y.Finally, evaluate the function at the critical points and the boundary of the circular disk to find the maximum and minimum values.In this case, since the function f(x, y) = x + y + 4 is linear, it does not have any critical points. Therefore, the maximum and minimum values of the function on the circular disk x^2 + y^2 ≤ 1 are obtained by evaluating the function at the boundary of the disk.
When x^2 + y^2 = 1, the function value is largest at the point (x, y) = (-1, 0), giving a maximum value of -1 + 0 + 4 = 3. The function value is smallest at the point (x, y) = (1, 0), giving a minimum value of 1 + 0 + 4 = 5.
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Which of these story problem describes the sum 19 + (-12)? Check all that apply. Show your work to justify your answer.
what is 16 divided by 7? and give me a real answer.
G find the point on the plane x + 4y + 2z =14 closest to the point (1,1,1)
Answer:
(4/3, 7/3, 5/3)
Step-by-step explanation:
The line perpendicular to the plane through the point will be ...
... (x, y, z) = (1, 1, 1) +t(1, 4, 2)
Using this expression for (x, y, z), we can find the point on the line where it intersects the plane. That point is the one of interest.
... (1 +t) +4(1 +4t) +2(1 +2t) = 14 . . . . . substituting for (x, y, z) in the equation of the plane
... 7 +21t = 14 . . . . . collect terms
... 21t = 7 . . . . . . . . subtract 7
... t = 7/21 = 1/3 . . . divide by 21
The point of interest is ...
... (x, y, z) = (1, 1, 1) + (1/3)(1, 4, 2) = (1 1/3, 2 1/3, 1 2/3)
Simplify.
16.3 – (–4.2) + 15.9
16.3 - (-4.2) +15.9 =
16.3 + 4.2 + 15.9 =
36.4
Which of the following numbers are divisible by 2? 74 , 296, 1110, 148, 370, 666, 9324, 111, 37, 555 , or 333?
It takes a boat 2 hours to travel 22 miles downstream and 4 hours to travel 20 miles upstream. What is the speed of the boat in still water?
True or false? If you took a true "if-then" statement, inserted a not in each clause l, and reversed the clauses, the statement would also be true
A pinewood derby car can be at most 2 3/4 inches wide. About how many pinwood derby cars can be displayed side-by-side on a shelf that is 31 1/2 inches long?
About 5 11/22 pinewood derby cars can be displayed side-by-side on a shelf that is 31 1/2 inches long.
Explanation:To find out how many pinewood derby cars can be displayed side-by-side on a shelf, we need to divide the length of the shelf by the width of each car. The shelf is 31 1/2 inches long, and each car can be at most 2 3/4 inches wide. So we divide 31 1/2 by 2 3/4:
Therefore, about 5 11/22 pinewood derby cars can be displayed side-by-side on the shelf.
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Please help me with all three! 25 Points!!
Use distributive property to rewrite each product below. Simplify your answer
a). 28x63
b). 17(59)
c). 458(15)
a. 28 * 63
28 (60) + 28 (3) or 20 (60) +20 (3) + 8 (60) + 8 (3)
1680 + 84=1764 1200 / 60 +480 +24=1764
b. 17(59)
17 (50) + 17 (9) or 10 (50) + 10 (9) + 7 (50) +7 (3)
850 + 153=1003 500 + 90 + 63 =1003
c.458 (15)
458 (10) +458 (5) or 10 (400) + 10 (50) + 10 (8) + 5 (400) +5 (50) + 5 (8)
4580 +2290 = 6870 400 + 500 + 80 + 2000 + 250 + 40 = 6870
My work is kinda confusing but hope it helped
Line s is the perpendicular bisector of JK. If line s intersects JK at point L, which of the following statements must be true? Check all that apply. A. Line s is perpendicular to JK B. JL = KL C. Line s is parallel to JK D. Point L is the midpoint of JK E. Line s intersects JK at a 180 angle
Line s, being the perpendicular bisector of JK, must be perpendicular to JK at point L, making JL equal to KL and meaning that point L is the midpoint of JK.
Explanation:If line s is the perpendicular bisector of segment JK and intersects it at point L, we can determine several truths about the geometric relationship between these elements:
A. Line s is perpendicular to JK: Yes, this is true because by definition, a perpendicular bisector not only cuts a line segment into two equal parts but also does so at a 90° angle.B. JL = KL: This statement must also be true. Since line s is a bisector, it divides JK into two equal parts, hence JL (the length from J to L) is equal to KL (the length from K to L).D. Point L is the midpoint of JK: True again, as the point where the perpendicular bisector intersects the line segment is known as the midpoint, implying that point L is equidistant from both J and K.Statements C and E are incorrect. Line s cannot be parallel to JK as it intersects it, and a 180° angle refers to a straight line, not the angle formed by a perpendicular intersection.
Please help..click on picture...I rrally need help
Find the tax on a dinette set with a cash price of $200 that has been reduced by 20% if the tax rate is 6.5%.
2.60
10.40
11.70
13.00
Answer:
The correct option is B.
Step-by-step explanation:
The initial cash price is $200.
The cash price reduced by 20%.
20% of $200 is
[tex]200\times \frac{20}{100}=40[/tex]
The cash reduced by $40.
The new cash price is
[tex]\$200-\$40=\$160[/tex]
The tax rate is 6.5%.
6.5% of remaining cash is
[tex]160\times \frac{6.5}{100}=10.40[/tex]
Therefore option B is correct.
Liliana takes 4 breaths per 10 seconds during yoga. At this rate, about how many breaths would Liliana Take in 2 minutes of yoga?
A machine is made up of 3 components: an upper part, a middle part, and a lower part. the machine is then assembled. 5 percent of the upper parts are defective, 4 percent of the middle parts are defective, and 1 percent of the lower parts are defective. what is the probability that a machine is not defective
Final answer:
The probability that a machine with three components (upper, middle, and lower parts) is not defective is found by multiplying the non-defective probabilities of each part. Calculating this gives a probability of 90.24% that the machine is non-defective.
Explanation:
The question asks what is the probability that a machine with three components (upper, middle, and lower parts) is not defective. To calculate this, we first note the probability of each part being non-defective:
Upper part non-defective probability = 95% (since 5% are defective, 100%-5% = 95%)
Middle part non-defective probability = 96% (since 4% are defective, 100%-4% = 96%)
Lower part non-defective probability = 99% (since 1% are defective, 100%-1% = 99%)
Since the machine is not defective only if all three parts are non-defective, we multiply these probabilities together:
Probability(machine is non-defective) = Probability(upper is non-defective) * Probability(middle is non-defective) * Probability(lower is non-defective)
Probability(machine is non-defective) = 0.95 * 0.96 * 0.99 = 0.9024 or 90.24%
Frank's Factories has a staff of sales people like Anthony, who receives a monthly salary of $2,750.00 plus an incremental commission based on the table below. If he sells $27,300.00, what is his total gross pay for the month?
The total gross pay for the month is $3,930.80 if Frank's Factories has a staff of salespeople like Anthony, who receives a monthly salary of $2,750.00.
What is the percentage?It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
It is given that:
Frank's Factories has a staff of salespeople like Anthony, who receives a monthly salary of $2,750.00 plus an incremental commission based on the table below. If he sells $27,300.00,
Fixed monthly salary = $2,750.00
First commission = 4.1% of 15,000 = $615.00
Second commission = 4.6% of (27300 - 15000) = $565.00
Total gross pay for the month = $2,750.00 + $615.00 + $565.80
= $3,930.80
Thus, the total gross pay for the month is $3,930.80 if Frank's Factories has a staff of salespeople like Anthony, who receives a monthly salary of $2,750.00.
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
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To calculate Anthony's total gross pay for the month, consider his base salary and incremental commission based on sales, resulting in a total gross pay of $4,934.00.
To calculate Anthony's total gross pay for the month, we need to consider his base salary and incremental commission based on sales.
Base salary: $2,750.00
Commission on $27,300.00 sales: Using the incremental commission table, the commission percentage for sales between $27,194-$31,479 is 8%. Thus, commission = $27,300.00 * 8% = $2,184.00
Total gross pay = Base salary + Commission = $2,750.00 + $2,184.00 = $4,934.00
Find the radius of a circle with the given circumference.
15 ft.
r =
A.30 ft.
B.30 ft.
C.7.5 ft.
D.7.5 ft.
Answer: [tex]\dfrac{7.5}{\pi}\ ft[/tex]
Step-by-step explanation:
We know that , the circumference of a circle is given by :-
[tex]C=2\pi r[/tex] , where r is the radius of the circle.
Formula for radius :
[tex]r=\dfrac{C}{2\pi}[/tex] , where C is the circumference of the circle.
Given : Circumference of circle = 15 ft.
Then , the radius of a circle will be :-
[tex]r=\dfrac{15}{2\pi}=\dfrac{7.5}{\pi}\ ft[/tex]
Hence, the radius of a circle= [tex]\dfrac{7.5}{\pi}\ ft[/tex]
which number line represents the solutions to |-2x|=4?
Please help! First rewrite 17/18 and 11/12 so that they have a common denominator then use <, =, or > to order 17/18 and 11/12
A heavy truck and a light car are traveling at the same speed on the same roadway. if the coefficients of static friction between their tires and the road are the same, which vehicle will be able to stop in the shortest distance? assume both have the same braking force.
Help ASAP please and thank you
One evening, 2/3 of the students in Rick's class watched television.
Mr. Obama would like his daughter to attend Harvard in 6 years when she finishes high school. Mr. Obama would like to invest enough money in a certificate of deposit (CD) now to pay for his daughter’s college expenses. If he estimates that he will need $250 thousand in 5 years, how much should he invest now in a CD that has a rate of 2.8% compounded monthly?
Opening up the monthly cell phone bill can be alarming due to the uncertainty of the bill. suppose the amount of minutes used per month is distributed normally with a mean of 700 minutes and a standard deviation of 120 minutes. what is the probability that more than 940 minutes were used?
Answer:
the required probability = 0.25
Step-by-step explanation:
Mean = 700 minutes , Standard Deviation = 120 minutes
To find : the probability that more than 940 minutes were used.
Solution : First find the z-score for 940 minutes and then find the area under the curve to the right of associated standard deviations from the mean.
[tex]z-score=\frac{\mu_2-\mu_1}{\sigma}\\\\\mu_1=700,\mu_2=940,\sigma=120\\\\\implies z-score=\frac{940-700}{120}\\\\\implies z-score=2[/tex]
Now, calculate the area under the curve to the right of 2 standard deviations from the mean.
Take confidence level to be 95%.
The area to the right of 2 standard deviations is 0.5 that is half the area under the standard normal curve.
[tex]\implies Area < \frac{0.95}{2}\\\\\implies Area<0.475[/tex]
Now, to find the probability that more than 940 minutes were used = 0.5 - 0.475 = 0.25
Hence, the required probability = 0.25
How do you graph a circle x^2 + y^2=25 and the line is given y=2
The formula for the circumference C of a circle, in terms of the radius, is C=2πr, where r stands for the radius. Rearrange the quantities in this formula to give a new formula for the radius of a circle in terms of the circumference.
A. r= 2C/3.14
B. r=3.14/2C
C. r=C/2(3.14)
D. r= 1/2(3.14)C
To get the radius of a circle in terms of circumference, divide the formula for the circumference by 2π, thus r=C/(2π). The correct option is C: r=C/2(3.14), considering 3.14 is the approximate value of π.
Explanation:The formula for the circumference of a circle is given by C=2πr. We are asked to rearrange this formula to make the radius (r) the subject of the formula. To achieve this, we divide both sides of the equation by 2π.
So, dividing both sides of the equation by 2π gives r=C/(2π).
Looking at your options, Option C: r=C/2(3.14), is correct if we consider 3.14 to be the value of π. Therefore, the radius of a circle in terms of the circumference is given by r=C/(2π).
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Shakira went bowling with her friends. She paid $3 to rent shoes and then $4.75 for each game of bowling. If she spent a total of $21, then how many games did Shakira bowl?
which of the following numbers does not belong here? 64, 16, 36, 32, 8, 4
Which of these statements is true?
A) 4 x 5 – 7 + 9 = 28
B) 4 x 5 – (7 + 9) = 22
C) 4 x 5 – (7 + 9) = 4
D) 4 x 5 – 7 + 9 = 44
Please answer ASAP: Determine which of the following situations can be modeled by the ratio 7/2. Select all situations that apply
For every twenty-eight cats, there are 4 dogs
Sara runs an average of fourteen miles every four days.
An online clothing store charges $7.50 to ship ten clothing items
A recipe calls for 3 cups of flour for every 1 cup of sugar.
Bottled water is on sale this week for a rate of $1.50 for two bottles
the only one that applies is:
Sara runs 14 miles every 4 days = 7 miles every 2 days = 7/2
How many numbers in the data set are within 1 standard deviation of the mean?
a.13
b.14
c.15
d.16