The minimum and maximum values are (16/9)√3 and -(16/9)√3.
According to the statement
we have given that the function
First of all, since the constraint's graph is a circle, which is a closed loop, and f is continuous in [tex]R^2[/tex], there must exist a constrained global maximum and minimum value.
Lagrange Multipliers:
f(x,y) = xy^2
g(x,y) = x^2 + y^2
We want ∇f = λ∇g so we get the following system of equations
1. y^2 = 2λx
2. 2xy = 2λy
3. x^2 + y^2 = 4 ← The constraint.
Equation 2 implies that y = 0 or λ = x
We can ignore y = 0, since that will make f(x,y) = 0 and clearly f(x,y) takes on both positive and negative values subject to the constraint.
Plugging in the alternative, λ = x to equation 1 gives y^2 = 2x^2.
Plugging this into the constraint gives 3x^2 = 4 so that x^2 = 4/3 and y^2 = 8/3
Taking square roots gives
x = ±√(4/3) = ±(2/3)√3
y = ±√(8/3) = ±(2/3)√6
At the points < (2/3)√3 , ±(2/3)√6 >, f(x,y) = (16/9)√3 ← Maximum
At the points < -(2/3)√3 , ±(2/3)√6 >, f(x,y) = -(16/9)√3 ← Minimum
So, The minimum and maximum values are (16/9)√3 and -(16/9)√3.
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The lines below are perpendicular.
the slope of the solid line iS 4, what is the slope of the dashed line?
Answer:
[tex]\frac{-1}{4}[/tex]
Step-by-step explanation:
Perpendicular lines have opposite reciprocal slopes. This means you change the sign and flip the numerator and denominator.
If the original slope is 4, the perpendicular slope is [tex]\frac{-1}{4}[/tex]
A right pyramid with a square base has a base edge length of 24 feet and a slant height of 20 feet. a right pyramid with square base has a base edge length of 24 feet and slant height of 20 feet. what is the height of the pyramid?
The height of the pyramid is [tex]16ft.[/tex]
What does it mean by right pyramid?The centroid of the base sits just above the right pyramid. Oblique pyramids are non-right pyramids. A right pyramid is typically assumed to be a regular pyramid because it has a regular polygon base.
To find the height of the pyramid:
l=the slant height of the right pyramid.
b=the length side of the square base of the right pyramid.
h=the height of the right pyramid.
Pythagorean Theorem: [tex]l^{2} =h^{2} +(\frac{b}{2} )^{2}[/tex]
Solve:h
[tex]h^{2} =l^{2} +(\frac{b}{2} )^{2}[/tex]
[tex]h^{2} =20^{2} -12^{2}[/tex]
[tex]h^{2} =256[/tex]
[tex]h=16ft[/tex]
Therefore, The height of the pyramid is [tex]16ft.[/tex]
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Answer:
The height of the pyramid is 16ft
Step-by-step explanation:
Jessica and her friend found some money under the couch. they split the money evenly, each getting $16.32. how much money did they find?
There is two of them each getting $16.32
So take the $16.32 × 2=$32.64
Choose the best answer from the choices below: the perpendicular bisector of a chord ____.
The perpendicular bisector of a chord passes through the center of a circle.
What is a chord?In geometry, a chord is a straight line segment that connects two points on the circumference of a circle. It is the longest possible segment within the circle.
The perpendicular bisector of a chord is a line that passes through the midpoint of the chord, creating two equal halves, and is perpendicular to the chord.
Notably, for any chord in a circle, its perpendicular bisector always intersects at the center of the circle, making it a significant geometric property.
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A bag has marbles labeled ‘4’ and ‘7’ and a spinner has 3 options, What is the probability of drawing a number less than 8 or spinning a 4?
Answer:2
Step-by-step explanation:
im just guessing
The answers are :
P (Drawing a number < 8) = 1
P (Spinning a 4) = 0
There are only 2 numbers in the bag, 4 and 7, both of which are less than 8. Hence, as it is a sure event, the probability of the event is 1.
There are only 3 numbers on the spinner, 1, 2, and 3. As 4 is not on the spinner, it is an impossible event, hence the probability of the event is 0.
Darlene is building a rectangular fence. the length is 8 feet longer than the width. the perimeter of the rectangle is 76 feet. what is the length of the rectangle?
Answer:8 feet
Step-by-step explanation:
Yu-xi draws a triangle with two angles measuring and . what is the measure, in degrees, of the third angle? 78 79 101 102
Considering the definition of a right triangle and its properties, the value of the third angle of a triangle with two angles measuring 79 degrees and 23 degrees is 78 degrees.
Triangle definitionA triangle is a polygon that is determined by three line segments called sides, or by three non-aligned points called vertices.
In other words, a triangle is a polygon made up of three sides, as well as three vertices and three interior angles.
One of the properties of any triangle is that the sum of the interior angles is equal to 180°.
Missing angle value in this caseIn this case, you know that:
One interior angle measures 79°.Another interior angle measures 23°.The sum of the interior angles is equal to 180°.So the value of the missing angle can be calculated by:
79° + 23° + missing angle = 180°
Solving:
102° + missing angle = 180°
missing angle= 180° - 102°
missing angle= 78°
Finally, the value of the third angle of a triangle with two angles measuring 79 degrees and 23 degrees is 78 degrees.
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Answer:
The answer is 78.
Step-by-step explanation:
Person above is right and I'm also taking the test!!
find the discriminant of the following quadratic equation
x^-5x+3=0
The discriminant of the given quadratic equation as in the task content can be evaluated by means of the formula; D = b²-4ac and it's value is; 13.
What is the discriminant of the quadratic equation as given in the task content?According to the task content, it follows that the quadratic equation whose discriminant is to be determined is; x²-5x+3=0.
By comparison with the standard form equation of a quadratic graph which goes thus; ax²+bx +c = 0, in which case, the determinant is given by the expression; b² - 4ac.
We can consequently evaluate the determinant of the quadratic equation in discuss as follows;
Determinant = b² -4ac = (-5)² - (4×1×3) = 25 - 12
Hence, the determinant in this case is; 13.
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The age of a father is three times that of his son. In ten year's time their ages will add up to 68 years. If the son is x years old today write down an algebraic expression for: the present age of the father and the son's age in ten year's time and the father's age in 10 year's time.
Answer:
46 and 22
Step-by-step explanation:
let the father b (f) and d son is (s)
so
f=3s.....(1)
(f+10) +(s+10)=68 .... (2)
substitute f=3s in equation (2)
we have
(3s+10)+(s+10)=68
4s+20=68
4s=68-20
4s=48
s=12 .......(3)
substitute (3) in (1)
f=3*12
which f=36
in 10 year time
f+10 =46
s+10=22
(x+3)^0 Rewrite the equation with positive exponents and simplify. x≠-3.
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Equation:}[/tex]
[tex]\mathsf{(x + 3)^0}[/tex]
[tex]\huge\textbf{Simplifying:}[/tex]
[tex]\mathsf{(x + 3)^0}[/tex]
[tex]\text{Anything raised to the 0 power it automatically equal to 1.}[/tex]
[tex]\huge\text{So, this mean that your answer should be: \boxed{\mathsf 1}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
2/13 = 0.153846 In the decimal above, 1 is the first digit in the repeating pattern. What is the 279th digit?
Answer:
F
Step-by-step explanation:
There are 6 digits in the repeating pattern.
279 / 6 = 46.5 remainder 3.
we care about the remainder as this is how many digits in. it is the 3rd digit in so is 3.
having trouble with this
Answer:
Step-by-step explanation:
13
i,dk how to do this pls help :)
Given the percentage taken off and the reduced price, the original price of the item is $142.
What was the original price of the item?Given the data in the question;
Percentage off the item = 20%Reduced price of the item = $113.60Original price of the item = ?The original price of the item is at 100%, if the 20% is taken off. The price is now at 80%.
Hence 80% of the original price will give the reduced price.
Let the original price be represented by x
80% × x = $113.60
We solve for x
80/100 × x = $113.60
80x/100 = $113.60
Cross multiply
80x = $11360
x = $11360/80
x = $142
Given the percentage taken off and the reduced price, the original price of the item is $142.
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The ratio is: 450g to 1kg
In it's simplest form
Answer:
9:20
Step-by-step explanation:
1kg is 1000g grams and in order to simify we need the units to be equal . 450:1000 would simplify to
45: 100 by divinding both the number by 10 then divide both number by 5 which would give 9: 20
hopefully this helped . if you need any more explanation pls ask
pls give brainliest if you like my answer :) thanks
Answer:
9: 20
Step-by-step explanation:
First, we have to convert both figures into the same unit.Here we can convert both into grams.As 450g is already given in grams, let us convert 1kg into grams.To convert kilograms into grams we have to multiply kilograms by 1000.Therefore,1kg × 1000 = 1000g
Now let us write those as a ratio.450 : 1000g
450 : 1000
To write the ratio in the simplest form we can divide both by 50.9: 20
As this cannot be simplified anymore, the simplest form of 450: 1000 is 9: 20.-3p+6p
help simplify
Step-by-step explanation:
-3p+6p is 3p
-3 + 6 = 3
So when you simplify the equation you get 3p
A function has a slope of 3, and one solution is given in the table. identify the missing outputs. x: 5 6 7 8 9 y: -50
The remaining value will increase by the value of the slope. Hence the remaining missing values are -47, -44, -41, -38 and -35
Slope and tablesThe slope of a line is the ratio of the rise to run of a line. It is also defined as the rate of change of coordinate y with respect to x. Mathematically;
slope = change in y/change in x
If the slope of the table given is 3, then using the coordinate points (5, -50)and (6, y)
Substitute
3 = y-(-50)/6-5
3 = y+50/1
y+50 = 3
y = -47
The remaining value will increase by the value of the slope. Hence the remaining missing values are -44, -41, -38 and -35
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Explanation to :an insurance agent is paid a salary of gh200 and a commission of 3% on all sales over gh2000 per month. if his total income in a particular month was gh530, what was the amount of his sales for that month
GH 13000 is the amount of his sales for that month.
how to find sales of the month?
total income = salary + commission
total income = GH530 (given)
salary = GH200 (given)
Commission is 3% on the sales above Gh2000 (given)
Let sales be x
total income = salary + commission
530 = 200+3%(x-2000)
530-200=0.03(x-2000)
330/.03=x-2000
11000=x-2000
x= 13000
total sales = GH13000
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Allison is saving money to buy a new phone that costs $600 by selling her graphic designs. She is using an app to manage her sales, but it keeps a fraction of each sale. Her net pay is modeled by the function P(x) = x2 + 18x – 160, where x represents the number of sales. How many sales does Allison to make to earn $600?
Answer:
20
Step-by-step explanation:
x^2+18x-160
If you put this into your calculator or desmos, you will get a parabola. You want to know the value of x when y=600. You can look at the table for the graph or scan to where y = 600. The value of x is 20 at that point.
To check you can plug 20 in for x. When you do that, y = 600.
help me with this question
Answer: A) 23,00
Step-by-step explanation:
A=23,00
B=8,600
C=12,00
D=3,600
Expand 10a(2a-7) pls its very urgent
Answer:
[tex]20a^2-70a[/tex]
Step-by-step explanation:
1) Expand by distributing terms.
[tex]10\times2a+10a\times-7[/tex]
2) Take out the constants.
[tex](10\times2)aa+10a\times-7[/tex]
3) Simplify [tex]10\times2[/tex] to [tex]20.[/tex]
[tex]20aa+10a\times7[/tex]
4) Use Product Rule:[tex]x^ax^b=x^{a+b}[/tex].
[tex]20a^2+10a\times-7[/tex]
5) Simplify [tex]10a\times-7[/tex] to [tex]-70a.[/tex]
[tex]20a^2-70a[/tex]
Thank you,
Eddie
Answer:
20a² - 70a
Step-by-step explanation:
(10a)(2a-7)
= 10a*2a + 10a*-7
= 20a² - 70a
Solve for n.
7+2n = 11
Answer:
2
Step-by-step explanation:
7+2n = 11
2n = 11-7
2n = 4
n = 4/2
therefore n = 2
A system of equations is shown:
4x = -3y + 17
3x - 4y = -6
What is the solution to this system of equations? (5 points)
(3,2)
(-3,-2)
(-2,-3)
(2,3)
Answer: (2;3).
Step-by-step explanation:
[tex]\displaystyle\\\left \{ {{4x=-3y+17} \atop {3x-4y=-6}} \right. \ \ \ \ \left \{ {{4x+3y=17\ |*4} \atop {3x-4y=-6\ |*3}} \right. \ \ \ \ \left \{ {{16x+12y=68} \atop {9x-12y=-18}} \right. \\Let's\ sum \ up\ these\ equations:\\25x=50\ |:25\\x=2.\\4*2=-3y+17\\8+3y=17\\3y=9\ |:3\\y=3.[/tex]
Barrett works at an ice cream shop. the function f(x) represents the amount of money in dollars barrett earns per gallon of ice cream, where x is the number of gallons of ice cream he makes. the function g(x) represents the number of gallons of ice cream barrett makes per hour, where x is the number of hours he works. f(x) = 2x2 4 g(x) = the square root of three times x cubed find f(g(x)). f of g of x equals the square root of the quantity 6 times x to the fifth power plus 4 dollars over hour f of g of x equals the square root of the quantity 6 times x to the fifth power plus 4 gallons over hour f(g(x)) = 6x3 4 gallons over hour f(g(x)) = 6x3 4 dollars over hour
Amount of money earned per number of hours of work = [tex]f(g(x)) = 6x^{3} + 4[/tex]
What is a composite function?A composite function is generally a function that is written inside another function. Composition of a function is done by substituting one function into another function. For example, f [g (x)] is the composite function of f (x) and g (x). The composite function f [g (x)] is read as “f of g of x”.To evaluate a composite function f(g(x)) at some x = a, first compute g(a) by substituting x = a in the function g(x). Then substitute g(a) into the function f(x) by substituting x = g(a). In the same way, we can calculate g(f(a)) as well.
Required calculation -
Amount of money (the earning) per unit x = [tex]f(x) = 2x^{2} + 4[/tex]
Number of gallons of ice cream that Barrett makes per hour, where x is the number of hours he works = [tex]g(x) = \sqrt{3x^{3}}[/tex]
we are to find the composite function [tex]f(g(x))[/tex]
Substituting g(x) into the x of f(x), we find:
[tex]f(g(x)) = 2(\sqrt{3x^{3} } )^{2} + 4\\\\= 2(3x^{3} ) + 4\\= 6 x^{3} + 4[/tex]
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Answer:
f(g(x)) = 6x3 + 4 dollars over hour
Step-by-step explanation:
Rod waxes two thirds of his car with one fourth bottle of car wax. What fraction of the bottle of car wax will Rod need to wax his entire car?
The total volume of a tree increases 6% each year. what will its volume be after 7 years if its volume is 5 cubic meters now?
Answer:
7.52 m^3
Step-by-step explanation:
This is like compound interest in a banking calculation
Future volume = 5 (1 + .06)^7 = 7.52 m^3
Which two of the following expressions can be simlified by using the Quotient of Powers Property?
Answer:
[tex]\frac{15x^{4}y^{3}}{5x^{3}} \ \ \text{and }\ -10x^{5}\div 2x^{3}y[/tex]
Step-by-step explanation:
the Quotient of Powers Property :
let a be a number where a ≠ 0
[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]
……………………
[tex]\frac{15x^{4}y^{3}}{5x^{3}}[/tex]
[tex]= \frac{3 \times 5x^{3}\times x \times y^{3}}{5x^{3}}[/tex]
[tex]= \frac{3 \times x \times y^{3}}{1}[/tex]
[tex]= 3 x y^{3}[/tex]
=============
[tex]-10x^{5}\div 2x^{3}y[/tex]
[tex]= [-5 \times (2x^{3}) \times x^2] \div [(2x^{3}) \times y][/tex]
[tex]= -5 \times x^2 \div y[/tex]
[tex]= -5 x^2 \div y[/tex]
when the product of 15 and 40 is diveded by the sum of 15 and 45 what is the quotient
Answer:
The answer is 10Step-by-step explanation:
First, you have to multiply. 15x4=60. All you need to do is add a zero to that product. You will get 600. Product means the answer to a multiplication problem. That is why you would have to multiply to get your answer. Your answer to that part is 600. 15*40= 600. Then, you would do 45+15= 60. 600/60= 10. Your answer is 10.
The area of a circle of radius 10 units is equal to the surface area of a sphere of radius 5 units.
A. True
B.False
Explanation:
The area of the circle with radius 10 is...
A = pi*r^2
A = pi*10^2
A = 100pi
The surface area of the sphere with radius 5 is...
SA = 4pi*r^2
SA = 4pi*5^2
SA = 100pi
Both results are the same, so the claim is true. We can think of surface area as the amount of wrapping paper needed to cover the sphere.
If you are playing a game using this spinner, what is the probability of not
landing on yellow or purple?
blue
green
yellow
orange
Your answer should be in the form of a fraction. Type it like this: 15/52.
Answer:
3/4
Step-by-step explanation:
Fraction of total area of the circle each color occupies in the spinner.
purple = 1/8
red = 1/4
orange = 1/8
yellow = 1/8
green = 1/4
blue = 1/8
Landing on yellow or purple: 1/8 + 1/8 = 1/4
Not landing on yellow or purple = 1 - 1/4 = 3/4
A scale measured a 4.5-pound brick as weighing 5.3 pounds. which measurement is more accurate but less precise than 5.3 pounds? 4.98 pounds 5 pounds 5.52 pounds 6 pounds
The correct option is B.
5 pound
The number (5.3) after Decimal is <5 so it round off to 5.
What is the property of rounding off numbers to one decimal place?It is the same to round a number to one decimal place as to the nearest tenths. At this instance, it is known which numeral is in the hundredths position. When the number in the hundredths place is higher than or equal to 5, the tenths digit is raised by one unit.
How do you round off decimals examples?A typical rule of thumb is to glance at the digit immediately to the right of the place value you want to round to and make your choice. For instance, rounding 5.1837 to the closest hundredth would result in 5.18 (because 3<5), whereas rounding 5.184 to the nearest thousandth would result in 5.184 (because 7>5).
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I understand that the question your are looking for is:
A scale measured a 4.5-pound brick as weighing 5.3 pounds. which measurement is more accurate but less precise than 5.3 pounds?
A. 4.98 pounds
B. 5 pounds
C. 5.52 pounds
D. 6 pounds