the three numbers whose sum is 21 and whose sum of squares is a minimum are 7, 7, and 7.
To find three numbers whose sum is 21 and whose sum of squares is a minimum, we can use a mathematical technique called optimization. Let's denote the three numbers as x, y, and z.
We need to minimize the sum of squares, which can be expressed as the function f(x, y, z) = x² + y² + z²
Given the constraint that the sum of the three numbers is 21, we have the equation x + y + z = 21.
To find the minimum value of f(x, y, z), we can use the method of Lagrange multipliers, which involves solving a system of equations.
First, let's define a Lagrange multiplier, λ, and set up the following equations:
1. ∂f/∂x = 2x + λ = 0
2. ∂f/∂y = 2y + λ = 0
3. ∂f/∂z = 2z + λ = 0
4. Constraint equation: x + y + z = 21
Solving equations 1, 2, and 3 for x, y, and z, respectively, we get:
x = -λ/2
y = -λ/2
z = -λ/2
Substituting these values into the constraint equation, we have:
-λ/2 - λ/2 - λ/2 = 21
-3λ/2 = 21
λ = -14
Substituting λ = -14 back into the expressions for x, y, and z, we get:
x = 7
y = 7
z = 7
Therefore, the three numbers whose sum is 21 and whose sum of squares is a minimum are 7, 7, and 7.
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Solve the following differential equation:
dV/dθ= V cotθ + V^3cosecθ
Comparing this with the left-hand side expression -V cosec^2θ - 3V^2cosec^2θcotθ, we can see that both sides are equal.
To prove the given equation, we will differentiate both sides with respect to θ and simplify the expression.
Differentiating V cotθ + V^3cosecθ with respect to θ:
d/dθ(V cotθ + V^3cosecθ) = d/dθ(V cotθ) + d/dθ(V^3cosecθ)
Applying the differentiation rules:
= V (-cosec^2θ) + 3V^2cosecθ(-cosecθcotθ)
= -V cosec^2θ - 3V^2cosec^2θcotθ
Now, we need to express the right-hand side of the original equation in terms of dV/dθ:
V cotθ + V^3cosecθ = V(cotθ + V^2cosecθ)
= V(cotθ + cotθV^2)
= Vcotθ(1 + V^2)
Taking the derivative of the right-hand side with respect to θ:
d/dθ(Vcotθ(1 + V^2)) = V(−cosec^2θ)(1 + V^2) + Vcotθ(0)
= -Vcosec^2θ(1 + V^2)
Comparing this with the left-hand side expression -V cosec^2θ - 3V^2cosec^2θcotθ, we can see that both sides are equal. Therefore, we have proved that dV/dθ = V cotθ + V^3cosecθ.
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I WILL GIVE U BRAINLYEST!! PLSS HELP!
The graph below represents the number of siblings each student in a class has.
what was the mean number of siblings?
The mean number of students in the class is 2.4 siblings.
What is the mean number of students?Mean is the average of a set of numbers. It is determined by adding the numbers together and dividing it by the total number
Mean = total sum of siblings / number of students
[(0 x 4) + ( 1 x 4) + ( 2 x 2) + ( 3 x 8) + ( 4 x 4) ] / ( 2 + 4 + 2 + 8 + 4)
= 48 / 20 = 2.4 siblings
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if there's a how and a guy picks her up who's getting abused
Use convolution notation with * and set up the integral to write the final answer of the following initial value ODE. There is no need to evaluate the integral. x" - 8x' + 12x = f(t)
The equation in integral form by taking the inverse Laplace transform: x(t) = L^-1((s^2 - 9s + 20)X(s)) + L^-1(F(s) + sx(0) + x'(0) - 8x(0)), where L^-1 represents the inverse Laplace transform.
To express the given initial value ordinary differential equation (ODE) using convolution notation with *, we can rewrite it as:
(x'' - 8x' + 12x) = f(t)
Taking the Laplace transform of both sides, we have:
s^2X(s) - sx(0) - x'(0) - 8(sX(s) - x(0)) + 12X(s) = F(s)
where X(s) is the Laplace transform of x(t), x'(0) is the initial condition of the derivative of x(t), x(0) is the initial condition of x(t), and F(s) is the Laplace transform of f(t).
Simplifying the equation, we get:
(s^2 - 8s + 12)X(s) - sx(0) - x'(0) + 8x(0) = F(s)
Now, let's express the left-hand side of the equation using convolution notation. Using the property that the Laplace transform of the derivative of a function f(t) is sF(s) - f(0), we can rewrite the equation as:
((s^2 - 8s + 12) - s + 8)X(s) = F(s) + sx(0) + x'(0) - 8x(0)
Simplifying further, we have:
(s^2 - 9s + 20)X(s) = F(s) + sx(0) + x'(0) - 8x(0)
Finally, we can express the equation in integral form by taking the inverse Laplace transform:
x(t) = L^-1((s^2 - 9s + 20)X(s)) + L^-1(F(s) + sx(0) + x'(0) - 8x(0))
where L^-1 represents the inverse Laplace transform.
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what is the slope and y intercept of -5x+3y=-9
Answer:
slope = 5/3
y-intercept = (0,-3)
Step-by-step explanation:
Make the equation in slope-intercept form:
-5x+3y=-9
add 5x on both sides
3y=5x-9
divide by 3 on both sides
y=5/3x-3
slope = 5/3
y-intercept = (0,-3)
During basketball practice, you make 8 free throws out of 20 shots taken. Using
experimental probability, how many free throws would you score out of 50
shots taken?
Answer:
if you make 4 free throws for every 10 shots taken you would make 20 out of 50 free throws
Step-by-step explanation:
HELP ME PLEASE! its very hard and i need it for tonight!!
Answer:
288 cm
Step-by-step explanation:
The formula to solve this polygon is: area = 1/2 x perimeter x apothem
To find the perimeter all you have to is find the sum of all sides together: 12+12+6+6+6+6= 48
Then the Apothem is, the "segment that joins the polygon's center to the midpoint of any side that is perpendicular to that side." To find this you find the point from the middle to the edge which is 12cm.
Then you plug in this information into the polygon. 1/2 x 48 x 12= 288!
The polygon is 288 cm!
Hope this helps! :) Please let me know if it does, and mark as brainliest!
Can someone help me find this please and thank you:)
Answer:
The perimeter to the question is 30.
Step-by-step explanation:
.
[tex]3 + 3 + 3 + 3 + 9 + 9 \\ 6 + 6 + 18 \\ 12 + 18 \\ 30[/tex]
Solve the proportion
17/4 = y/6
Answer:
This is your answer ☺️☺️
Nash uses his credit card with a 17.8% APR, compounded monthly, to pay for a cruise totaling $1,789.17. He can pay $650 per month on the card. What will the total cost of this purchase be?
Answer: 17.8% + $1.789.17 + $650 = 2439.348 or 2439
Step-by-step explanation: if you read the word problem it has a key word the key word is total.
Nash uses his credit card with a 17.8% APR, compounded monthly, to pay for a cruise totaling $1,789.17. He can pay $650 per month on the card. What will the total cost of this purchase be?
A 30-ft cable is stretched from the top of an antenna to an anchor point 12-ft from the base of the
antenna. How tall is the antenna? Round to the nearest tenth of a foot.
compute the accumulated value of $8600at 6.45% after 8
months (simple interest)
The accumulated value of $8600 at 6.45% after 8 months (simple interest) is $8971.90.
To compute the accumulated value of $8600at 6.45% after 8 months (simple interest), we need to use the formula for simple interest, which is given by:
I = P × r × t
Where, I is the interest earned, P is the principal amount, r is the interest rate, and t is the time in years.
Here, we have t in months, so we need to convert it into years by dividing by 12.
So, t = 8/12 = 2/3 years.
Now, substituting the given values, we get:
I = 8600 × 6.45/100 × 2/3 = $371.90
Therefore, the accumulated value of $8600 at 6.45% after 8 months (simple interest) is $8600 + $371.90 = $8971.90.
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HELPPPP
Diane draws an obtuse, isosceles triangle with two of the angles measuring 25° each.
What is the measure of the obtuse angle in her triangle?
Answer:
130⁰. unsure. angle sum, 25+25+x =180, so x=180-25-25
hi please help i’ll give brainliest
Answer:
Choice D
Step-by-step explanation:
I believe that it is Choice D. Due to the fact that the moons orbit around Earth is inclined around 5 degrees, not allowing an eclipse to happen every New or Full Moon. Most of the time, the Sun, Earth, and Moon don't line up precisely enough to cause an eclipse.
Loretta is rolling an unfair 6 sided die with a single number between 1 and 6 on each face. She has a 70% chance of rolling a four. She is playing a game with a friend and knows that if she rolls a four on three of her next five rolls she will lose the game. She wants to determine the probability that she rolls a four on three of her next five rolls.
Which simulation design has an appropriate device and a correct trial?
Complete question is;
Loretta is rolling an unfair 6 sided die with a single number between 1 and 6 on each face. She has a 70% chance of rolling a four. She is playing a game with a friend and knows that if she rolls a four on three of her next five rolls she will lose the game. She wants to determine the probability that she rolls a four on three of her next five rolls.
Which simulation design has an appropriate device and a correct trial?
A) Using a fair coin let heads represent rolling a four and tails represent not rolling a four. Flip the coin five times.
B) Using a table of random digits select a digit between 0 and 9. Let 0-6 represent rolling a four and 7-9 represent not rolling a four. Select five digits.
C) Roll a fair die with a single digit between 1 and 6 on each face. Let four represent rolling a four and 1-3 and 5 and 6 represent not rolling a four. Roll the die five times.
D) Using a table of random digits select a digit between 1 and 6, ignoring digits 0, 7, 8, and 9. Let 4 represent rolling a four and 1-3 and 5 and 6 represent not rolling a four Select five digits.
Answer:
B) Using a table of random digits, select a digit between 0 and 9. Let 0-6 represent rolling a four and 7-9 represent not rolling a four. Select five digits.
Step-by-step explanation:
Since she knows that if she rolls a four on three of her next five rolls she will lose the game, then the best simulation that she will roll a four on three of the next five rolls will be option B because it uses a table of random digits and doesn't ignore any number but is well ordered with 0-6 representing a four and 7-9 not rolling a four.
Tea costs £1.40 and coffee
costs 65p how much will 5
coffees and 2 teas cost?
Answer:
£6.05
Step-by-step explanation:
Multiply the number of coffees by the cost of coffee:
5 × 65p
325p
Multiply the number of teas by the cost of tea:
2 × £1.40
£2.80
Add the two answers together:
325p + £2.80
Convert p to £
£3.25 + £2.80
£6.05
Find the sample standard deviation. 12, 12, 12, 15, 18,
18, 18
show work pls
Answer:
Sample standard deviation = 3
Step-by-step explanation:
We can use the following steps to calculate the sample standard deviation for the given data set:
Step 1: First, calculate the mean of the data set by adding all the values and dividing by the total number of values:
(12 + 12 + 12 + 15 + 18 + 18 + 18) / 7 = 15.
Thus, the mean is 15.
Step 2: Next, subtract the mean from each value in the data set to get the deviation of each value from the mean.
(12 - 15) = -3
We can simply write -3 thrice since 12 appears thrice in the data set.(15 - 15) = 0
(18 - 15) = 3
Similarly, we can write 3 thrice since 18 appears thrice in the data set.Thus, the deviations are -3, -3, -3 0, 3, 3, and 3.
Step 3: Square each deviation:
(-3)^2 = 9
We can write 9 thrice since -3 appears as a deviation thrice.0^2 = 0
(3)^2 = 9
We can again write 9 thrice since 3 appears as a deviation thrice.Thus, the squares of all our deviations are 9, 9, 9, 0, 9, 9, and 9.
Step 4: Add up all the squared deviations:
9 + 9 + 9 + 0 + 9 + 9 + 9 = 54.
Thus, the sum of the squared deviations is 54.
Step 5: Find the variance by dividing the sum of squared deviations by one less than the number of values in the data set:
54 / (7 - 1) = 9
54 / 6 = 9
Thus, the variance is 9
Step 6: Finally, find the sample standard deviation by taking the square root of the variance:
√9 = 3
So, the sample standard deviation for this data set is 3.
Please help me if you want brianleist! :(
Answer:
2ft
Step-by-step explanation:
16^3 ft
4 * 2=8
16/8 =2
Answer:
2 ft
Step-by-step explanation:
V = lwh
16 = 4(2)w
16 = 8w
w = 2 ft
The scatter plot shows the amount of time Adam spent studying and his test scores, Predict Adam's test score when he studies for 6 hours
100
90
80
Test Score
70
O.
60
0
0 1 2 3 4 5
Time Studying (hours)
When he studies for 6 hours, Adam should score about
Answer:
I think 100
Step-by-step explanation:
find the value of the variable for each polygon
The value of variable h is,
⇒ h = 18
We have to given that,
A heptagon with angles are shown in figure.
Here, All the angles are,
⇒ 6h°, 132°, 146°, 146° , (6h + 10)° , (6h + 10)° , 146°, and 132°
We know that,
Sum of all the angles in a heptagon is, 900°
Hence, We get;
⇒ 6h° + 132° + 146° + 146° + (6h + 10)° + (6h + 10)° + 146° + 132° = 900°
Combine like terms,
⇒ 18h + 576 = 900
⇒ 18h = 900 - 576
⇒ 18h = 324
⇒ h = 18
Thus, The value of variable h is,
⇒ h = 18
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what is the quadratic equation for this graph? will make brainiest if also tell me how to put into vertex from
Answer:
Step-by-step explanation: The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y -axis. The coefficients a,b, and c in the equation y=ax2+bx+c y = a x 2 + b x + c control various facets of what the parabola looks like when graphed.
Hope you like it.
determine the normal and shear stresses at point d that act perpendicular and parallel, respectively, to the grains. the grains at this point make an angle of 35 ∘ with the horizontal as shown.
To determine the normal and shear stresses at point D, which act perpendicular and parallel to the grains, respectively, we require additional information such as the applied forces or loadings at that point.
The given question mentions the need to determine the normal and shear stresses at point D, which act perpendicular and parallel to the grains, respectively. However, to accurately calculate these stresses, we need more information about the system under consideration. Key details include the applied forces or loadings on the system, the material properties, and the specific orientation and arrangement of the grains at point D.
Normal stress, also known as axial stress, refers to the force per unit area acting perpendicular to the surface. Shear stress, on the other hand, represents the force per unit area acting parallel to the surface. The magnitudes and directions of these stresses depend on the applied forces, the geometry of the system, and the material properties.
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SAT Scores the national average for mathematics on a standardized test in 2011 was 518. Suppose that the distribution of scores was approximately bell- shaped and that the standard deviation was approximately 48. Round your answers to at least one decimal place as needed.
The national average for mathematics on a standardized test in 2011 was 518, with a standard deviation of approximately 48.
In statistics, the bell-shaped distribution is known as the normal distribution or the Gaussian distribution. It is characterized by its symmetry and the majority of the data falling within a certain range.
The national average score of 518 represents the central tendency of the distribution. This means that a large number of students scored around this average score.
The standard deviation of approximately 48 measures the variability or spread of the scores. It indicates how much the scores deviate from the average. In a normal distribution, about 68% of the data falls within one standard deviation of the mean.
By knowing the average score and the standard deviation, we can determine the proportion of students who scored above or below a certain score, as well as calculate percentiles and compare individual scores to the national average.
Understanding the characteristics of the distribution, such as the average and standard deviation, helps in interpreting and analyzing the scores, making meaningful comparisons, and identifying students' performance relative to the national average.
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HELP ASAP I WILL GIVE BRAINLIEST IF I CAN
Answer: add 3
Step-by-step explanation:
I am the number that is 5,000 greater than the smallest number u can make using six of the digits what number am I?——————————————
Only can use 1 8 3 4 9 6 2 7 I think..
12,351,789
ignore this ------
12346789
+ 5000
----------------
12351789
Which of the following equations could be the line of best fit?
A. x = 100
B. y = 100
C. y = x + 100
D. y = x - 100
The real answer is B
I don’t understand this
Answer:
It is plugging it the 36pye
Step-by-step explanation:
h = 36pye
what it wants you to do is plug that into all the numbers
so it be
2 x 36pye (you get it for the next one)
for the fraction ones it would
be divding 36pye by the number on the bottom
Fill in the blank using a term found in the vocabulary list. The ratio of the value of a subtotal to the value of the total is called _________________.
Answer:
Relative frequency
Step-by-step explanation:
Given that a is in Quadrant 2 and cos(a) = give an exact answer for the following: a sin(20) b. cos(2a) c. tan(20) = 2. Given that B is in Quadrant 4 and sin(B) = give an exact answer for the following: a sin(25) = b.cos(2B) c. tan(28) . Decimal approximations are not allowed for this problem, • Enter your answer in exact form. • Use "sqrt()" to represent.
a) In Quadrant 2, a sin(20) is equal to -sin(20).
b) In Quadrant 2, cos(2a) is equal to -cos(2a).
c) In Quadrant 2, tan(20) is equal to -tan(20).
In Quadrant 2, the angle 'a' is between 90 degrees and 180 degrees, or π/2 and π radians. Knowing that cos(a) is a negative value, we can determine the exact values for the given trigonometric expressions.
a) a sin(20) = -sin(20):
The sine function (sin) is positive in Quadrant 1 and negative in Quadrant 2. Therefore, the value of a sin(20) in Quadrant 2 is equal to the negative of sin(20). This means that the answer for a sin(20) is -sin(20).
b) cos(2a) = -cos(2a):
The cosine function (cos) is negative in Quadrant 2. Since we are given that a is in Quadrant 2, the angle 2a will also be in Quadrant 2. Therefore, cos(2a) in Quadrant 2 is equal to the negative of cos(2a). Thus, the answer for cos(2a) is -cos(2a).
c) tan(20) = -tan(20):
The tangent function (tan) is negative in Quadrant 2. Hence, the tangent of any angle in Quadrant 2 will be equal to the negative of its positive counterpart. Consequently, the answer for tan(20) in Quadrant 2 is -tan(20).
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Alice is using a rope to pull a wagon. He exerts a force of 2 newtons at an angle of 48° from the floor.
Part A: Suppose the direction of the force changes from a 48° angle with the floor to a 70° angle with the floor. Determine the effect on the horizontal and vertical components of the force. Give forces to the nearest hundredth of a unit.
Part B: What implications does this have for pulling the wagon? Explain.
Answer:
Part A - i. 0.68 N ii. 1.88 N
Part B - The wagon will move slowly while being raised up off the ground.
Step-by-step explanation:
Part A: Suppose the direction of the force changes from a 48° angle with the floor to a 70° angle with the floor. Determine the effect on the horizontal and vertical components of the force. Give forces to the nearest hundredth of a unit.
Since the force of 2 N is exerted at this new angle of 70°,
i. the horizontal component u = 2cos70° = 0.684 N ≅ 0.68 N and its
ii. vertical component, v = 2sin70° = 1.879 N ≅ 1.88 N
Part B: What implications does this have for pulling the wagon? Explain.
Since we have a horizontal component of 0.68 N and a vertical component of 1.88 N, most of the force is used in raising the wagon off of the ground with less of it moving it forward. So, the wagon will move slowly while being raised up off the ground.