Answers
The extreme values of the function f(x, y, z) = x^2 + y^2 + z^2, x^2 + y^2 + z^2 + xy = 27 are minimum = 0 and maximum = 18
Calculating the extreme values of the functionGiven that
f(x, y, z) = x^2 + y^2 + z^2, x^2 + y^2 + z^2 + xy = 27
To find the extreme values of the function f(x, y, z) subject to the given constraint, we can use the method of Lagrange multipliers.
Defining the Lagrangian function L as follows:
L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z))
where g(x, y, z) is the constraint function and λ is the Lagrange multiplier.
In our case, we have:
f(x, y, z) = x^2 + y^2 + z^2
g(x, y, z) = x^2 + y^2 + z^2 + xy - 27
So, our Lagrangian function becomes:
L(x, y, z, λ) = x^2 + y^2 + z^2 - λ(x^2 + y^2 + z^2 + xy - 27)
To find the extreme values of f(x, y, z) subject to the constraint g(x, y, z) = 0, we need to solve the following system of equations:
∂L/∂x = 0
∂L/∂y = 0
∂L/∂z = 0
∂L/∂λ = 0
g(x, y, z) = 0
Taking the partial derivatives of L with respect to x, y, z, and λ, we get:
∂L/∂x = 2x - λ(2x + y) = 0
∂L/∂y = 2y - λ(2y + x) = 0
∂L/∂z = 2z - λz = 0
∂L/∂λ = x^2 + y^2 + z^2 + xy - 27 = 0
From the third equation, we get:
2z - λz = 0
z(2 - λ) = 0
So, either z = 0 or λ = 2.
If z = 0, then from the first two equations, we get:
2x - λy = 0
2y - λx = 0
Multiplying the first equation by y and the second equation by x, we get:
2xy - λy^2 = 0
2xy - λx^2 = 0
Subtracting the second equation from the first, we get:
λ(x^2 - y^2) = 0
Since λ cannot be zero, we have x^2 = y^2 and x = y
Similarly, if λ = 2, then from the first two equations, we get:
2x - 2y = 0
x = y
Substituting x = y into the constraint equation, we get:
2x^2 + x^2 = 27
x^2 = 9
x = ±3
So, the possible critical points are:
(3, 3, 0)
(-3, -3, 0)
(0, 0, 0)
Recall that
f(x, y, z) = x^2 + y^2 + z^2
So, we have
f(3, 3, 0) = 3^2 + 3^2 + 0^2 = 18
f(-3, -3, 0) = -3^2 + -3^2 + 0^2 = 18
f(0, 0, 0) = 0^2 + 0^2 + 0^2 = 0
Hence, the minimum is 0 and the maximum is 18
Read more about Lagrange multipliers at
https://brainly.com/question/10479355
#SPJ1
Related Questions
three cards are drawn from an ordinary 52-card deck without replacement (drawn cards are not placed back in the deck). find: (a) assuming sequence does not matter, how many different ways to draw three cards?
Answers
As per the concept of combination method, there are 22,100 different ways to draw three cards from a standard deck of 52 cards without replacement, where the order of the cards does not matter.
The number of ways to draw three cards from a deck of 52 cards without replacement can be calculated using combinations. A combination is a way of selecting a subset of objects from a larger set, where the order of selection does not matter. The number of combinations of r objects chosen from a set of n objects is given by the formula:
ⁿCₓ = n! / x!(n-x)!
Where n! (read as n factorial) is the product of all positive integers up to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
In our case, we want to find the number of combinations of three cards chosen from a set of 52 cards. Using the formula above, we have:
⁵²C₃ = 52! / 3!(52-3)!
= (52 x 51 x 50) / (3 x 2 x 1)
= 22,100
To know more about combination here
https://brainly.com/question/28998705
#SPJ4
evaluate the double integral. 8y2 da, d is the triangular region with vertices (0, 1), (1, 2), (4, 1) d
Answers
The value of the double integral is 128/27 - 32/9 + 8ln2/3. From triangular region with vertices (0, 1), (1, 2), (4, 1) d.
To evaluate the double integral, we first need to set up the limits of integration. Since the region D is a triangle, we can use the following limits:
0 ≤ x ≤ 1
1 + x ≤ y ≤ 4 - x
The integral then becomes:
∫0^1 ∫1+x^4-x 8y^2 dy dx
Evaluating the integral with respect to y first, we get:
∫0^1 ∫1+x^4-x 8y^2 dy dx = ∫0^1 [(8/3)(y^3)]1+x^4-x dx
= ∫0^1 [(8/3)(1+x^3)^3 - (8/3)(1+x^2)^3] dx
= 128/27 - 32/9 + 8ln2/3
Learn more about integral: https://brainly.com/question/27419605
#SPJ11
5x5=25+1=26+6=
solve the eqation
Answers
Answer:
(5 x 5 = (25) + 1 = (26) + 6 = ?)
= 32Step-by-step explanation:
You're welcome.
Answer: 32
Step-by-step explanation:
5x5 is 25 +1=26 so 26+6=32
PLEASE HELP IF U DO GT BRAINLIEST A student attempted to generate an equivalent expression using the distributive property, as shown below: 7(y − 5) = y − 35 What was the mistake made? (1 point) a 7 was not multiplied by y b 7 was not multiplied by 5 c Incorrect sign was used for the first term d Incorrect sign was used for the second term
Answers
The mistake made by the student is (c) incorrect sign was used for the first term.
When using the distributive property, the expression 7(y-5) should be expanded as 7y - 35, not y - 35. The student incorrectly distributed the 7 to the second term only, while ignoring the first term, which led to an incorrect expression.
algrebra if8762 a rectangle has a perimeter of 18 cm. its length is 5 cm greater than its width. find the dimensions.
Answers
The perimeter of a rectangle is equal to twice the length plus twice the width. Therefore, 18 cm = 2(length + 5) + 2(width). This equation can be solved by subtracting 2(width) from both sides, resulting in 18 - 2(width) = 2(length + 5). Finally, dividing both sides by 2 gives us 9 - width = length + 5. Therefore, width = 4 cm and length = 9 cm.
In this problem, we were given the perimeter of a rectangle, and asked to find its dimensions. We used the formula for the perimeter of a rectangle to set up an equation and solve for the length and width. The width of the rectangle turned out to be 4 cm, and the length was 9 cm, with the length being 5 cm greater than the width.
Know more about dimensions here
https://brainly.com/question/28688567#
#SPJ11
which type of graph represents a system of linear equations that has muttiple common coordinate pairs as a solutions
Answers
The type of graph that represents a system of linear equations that has multiple common coordinate pairs as a solution is called a line of best fit.
A line of best fit is a line drawn through data points that are scattered on a graph in order to illustrate a relationship between the data.
This line helps identify patterns in the data, and when plotted, multiple points can exist on the same line, which in this case represents the multiple common coordinate pairs.
To create a line of best fit, you will need to calculate the slope and the intercept for the equation that models the data. Then, you can plot the points onto the graph and draw a line through them.
The resulting line will be the line of best fit, which represents the multiple common coordinate pairs that are solutions for the system of linear equations.
To know more about linear equations click on below link:
https://brainly.com/question/29739212#
#SPJ11
Angle r° = 2w°. What is the measure of angle r°?
A) 145°
B) 290°
C) 125°
D) 90°
Answers
Answer:
290
Step-by-step explanation:
Please help like ASAP now thxxx
Answers
Answer:
26
Step-by-step explanation:
True or False: Of the major economies in the world, Italy had the highest growth rate of real GDP per capita between 1982 and 2009
Answers
The given statement regarding the GDP of major economies of the world is False.
To determine the accuracy of this statement, we would need to gather data on the real GDP per capita growth rates of major economies between 1982 and 2009 and compare them. Without this data, we cannot definitively say whether the statement is true or false. However, based on historical trends and current economic data, it is unlikely that Italy had the highest growth rate of real GDP per capita during this time period.
Learn more about the Economy:
https://brainly.com/question/28210218
#SPJ4
If vector r = ❬4, 6❭ and s = ❬8, –3❭ then which is r – s?
Answers
Answer:
r - s = (4 - 8, 6 - (-3)) = (-4, 9)
mrs. stevens is an inpatient in your hospital. her prescriber just ordered hydroxyzine 10 mg every 8 hours. how many tablets of hydroxyzine 10 mg will you put in mrs. stevens' med cart drawer for a 24-hour fill?
Answers
We will need to place three tablets of hydroxyzine 10 mg in Mrs. Stevens' med cart drawer for a 24-hour supply because her prescriber has instructed her to take it every eight hours.
Mrs. Stevens will need to take 10 mg of hydroxyzine three times per day (every eight hours) to comply with the prescription. We will need to figure out how many tablets of hydroxyzine 10 mg she will need for those three daily doses because we need to fill the drawer for 24 hours.
We will require a total of 30 mg per day for three doses, each of which contains 10 mg. Since we are filling it for 24 hours, that means we will need
30 mg x 3 doses = 90 mgfor the entire day. There are 10 mg of hydroxyzine in each tablet., therefore,
90 mg / 10 mg per tablet = 9 tablets of hydroxyzine.However, as we are only using the drawer for three doses, we will require three 10 mg hydroxyzine tablets.
Learn more about Hydroxyzine:
https://brainly.com/question/29440151
#SPJ4
You work part-time at a diner that pays you $6. 75/hr plus tips. Tips average $3. 50/hr. Your deductions are FICA (7. 56%), Federal tax withholding (9. 6%), and state tax withholding (6. 7%). You work 13 hours per week. What is you gross pay per month including tips? $533. 00 $351. 00 $507. 00 $520. 0
Answers
Your gross pay per month including tips is $533.00. The answer is A) $533.00.
First, let's calculate your hourly wage including tips:
Hourly wage including tips = $6.75 + $3.50 = $10.25
Next, let's calculate your total earnings per week:
Total earnings per week = hourly wage including tips x hours worked per week
= $10.25 x 13 hours
= $133.25
Now let's calculate the deductions from your earnings:
FICA deduction = 7.56% x $133.25 = $10.08
Federal tax withholding = 9.6% x $133.25 = $12.80
State tax withholding = 6.7% x $133.25 = $8.92
Total deductions = $10.08 + $12.80 + $8.92 = $31.80
Finally, let's calculate your gross pay per month:
Gross pay per month = total earnings per week x 4 weeks - total deductions
= ($133.25 x 4) - $31.80
= $533.00
Therefore, your gross pay per month including tips is $533.00. The answer is A) $533.00.
To learn more about gross pay Click here:
brainly.com/question/13143081
#SPJ4
In the figure, lines a and b are parallel lines. Select all the statements that are true. Parallel lines A and B cut by a transversal. Line A and the transversal form angles 1, 2, 3, and an angle with measure 75 degrees. In the corresponding positions, line B and the transversal form an angle with measure x minus 15 degrees and angles 4, 5, and 6. m∠1 = m∠6 m∠2 = m∠6 m∠1 + m∠6 = 180° x = 90° x = 120°
Answers
Therefοre, the sοlutiοn οf the given prοblem οf angles cοmes οut tο be m∠1 + m∠6 = 180°.
What dοes an angle mean?The largest and smallest walls οf a skew are determined by the intersectiοn οf lines that create its ends. There is a chance that twο lines will cοme tοgether at a junctiοn. Anοther result οf twο οbjects interacting is an angle. They mοst clοsely resemble dihedral fοrms. Twο line beams can be arranged in different ways between their extremities tο fοrm a twο-dimensiοnal curve.
Here,
=> m∠1 + m∠2 = 180° (interior angles on the same side of the transversal and between the parallel lines)
=> m∠4 + m∠5 = 180° (interior angles on the same side of the transversal and between the parallel lines)
Other views shown in the figure include:
=> m∠3 = 75° (given)
=> m∠1 = m∠6 (corresponding angles)
Vertical angles: m2 = m6 x - 15° (alternate interior angles)
Angles corresponding to m4 are m5 (vertical angles)
So let's assess each assertion:
Since they are corresponding angles, m1 = m6 is correct.
This is also accurate because the angles are vertical: m2 = m6.
M1 + M6 = 180°: Since they are linear pairs, this is correct.
=> x = 90°: According to the data in the illustration, this isn't always the case.
=> x = 120°: Based on the data shown in the illustration, this is also not always the case.
Thus, the following assertions are true:
=> m∠1 = m∠6
=> m∠2 = m∠6
=> m∠1 + m∠6 = 180°
To know more about angles visit:
brainly.com/question/14569348
#SPJ1
Could someone please help me I don’t understand this
Answers
Solution:
DC= 1 / 3
• The slope of DC is the same as AB
solve the system 2 -1 5 -2 with -2 -1 . give your solution in real form. , . an ellipse with counterclockwise orientation 1. describe the trajectory.
Answers
The solution of the system 2 -1 5 with -2 -1 in real form is (x, y) = (7/4, 7/2).
To solve the system with the given coefficients, first, let's rewrite it as a linear system of equations:
2x - y = 5
-2x - y = -2
Now, let's solve this system step-by-step:
Step 1: Solve for y in the first equation:
y = 2x - 5
Step 2: Substitute the expression for y from Step 1 into the second equation:
-2x - (2x - 5) = -2
Step 3: Simplify and solve for x:
-2x - 2x + 5 = -2
-4x = -7
x = 7/4
Step 4: Substitute the value of x back into the expression for y:
y = 2(7/4) - 5
y = 7/2
The solution in real form is (x, y) = (7/4, 7/2).
Regarding the ellipse with counterclockwise orientation, the trajectory can be described as a smooth, closed curve in the shape of an ellipse. The ellipse will have a major and minor axis, with the points on the ellipse moving continuously in a counterclockwise direction along the curve.
To learn more about trajectory: https://brainly.com/question/13244761
#SPJ11
using a single screen window, create the following four distinct drawings, each occupying one quadrant of the window. 1. a logarithmic spiral with k
Answers
Divide your single screen window into four equal quadrants. You can do this by drawing two perpendicular lines that intersect in the center of the window.
How to create the four distinct drawings in a single screen window?To create the following four distinct drawings in a single screen window, with each occupying one quadrant of the window, follow these steps:
. Divide your single screen window into four equal quadrants. You can do this by drawing two perpendicular lines that intersect in the center of the window.
. In the first quadrant, draw a logarithmic spiral with a given value for k. To do this, start at the center of the quadrant and draw the spiral outwards, increasing the distance between the spiral lines as you move away from the center, according to the formula r = a * e^(k * theta), where r is the distance from the center, a is a constant, e is the base of the natural logarithm, k is the given value, and theta is the angle.
. In the second quadrant, create a distinct drawing that is different from the logarithmic spiral. This can be any other type of curve, shape, or pattern that you would like.
. In the third quadrant, create another distinct drawing that is different from the first two. Again, this can be any curve, shape, or pattern that you choose.
. In the fourth quadrant, create a final distinct drawing that is different from the first three. Make sure to choose a different curve, shape, or pattern for this quadrant as well.
By following these steps, you will have created four distinct drawings in a single screen window, each occupying one quadrant of the window, with the first quadrant featuring a logarithmic spiral with the specified value for k.
Learn more about logarithmic spiral
brainly.com/question/30645990
#SPJ11
use your z-score table. what percentage of scores (i.e., of a gaussian distribution) is between -1.96 and 1.96 z-scores? you don't have to put the % sign.
Answers
Approximately 95% of scores lie between -1.96 and 1.96 z-scores. This is because the area under the standard normal distribution curve between these two values is approximately 0.95.
To calculate this, first calculate the area under the standard normal distribution curve between -1.96 and 1.96. This can be done by using a Z-score table, which provides the area under the curve to the left of a given value. To calculate the area between -1.96 and 1.96, subtract the area to the left of -1.96 from the area to the left of 1.96. This is approximately 0.95.
Since the total area under the curve is 1, this implies that approximately 95% of scores lie between -1.96 and 1.96 z-scores. This is commonly known as the "95% Rule," and it is a useful way to quickly estimate the amount of scores that are within a certain range.
It is important to note that this calculation is only an approximation, as the exact value depends on the actual values that are used. Therefore, it is important to use a Z-score table with more accurate values if more precision is needed.
See more about normal distribution at: https://brainly.com/question/14243195
#SPJ11
THIS IS URGENT AM I CORRECT PLEASE HALP!!!!!!
Answers
The total time that it took the fireman to be ready and aboard the firetruck can be obtained by summing all of the time spent. When summed, the total figure will be 32 seconds.
How much time did it take?The total time taken to board the firetruck can be calculated by summing all the time given in seconds. We start by getting the time it took the firefighter to reach the fire pole which is 9.5 seconds.
Next is the time it took him to slide down the pole which is 3.2 seconds. Next is 13.6 seconds for sliding down the pole and 5.7 seconds for climbing aboard. The sum of all these is 32 seconds.
Learn more about summation here:
https://brainly.com/question/542712
#SPJ1
Find the length of the third side
Answers
Answer:
2 radical 11
Step-by-step explanation:
hyp= short (2)
x= √11 (2)
x= 2√11
Name and draw a quadrilateral that is NOT a rectangle or a rhombus. Is there another shae you couldve drawn? Explain
Answers
One example of a quadrilateral that is not a rectangle or a rhombus is a trapezoid. A trapezoid has one pair of opposite sides that are parallel, but the other pair of sides are not. (the figure is attached is attached below)
Yes, there are many other shapes that could have been drawn, such as a kite, parallelogram, or irregular quadrilateral. The possibilities are endless, as long as the shape has four sides.
A trapezoid is a quadrilateral with one pair of opposite sides parallel, but the other pair of sides are not. It can be visualized as a shape that resembles a ladder. Unlike a rectangle or rhombus, a trapezoid does not have equal side lengths or equal angles. It can be used in various mathematical contexts, such as finding the area or perimeter of a geometric figure. Additionally, there are many other shapes that could have been drawn instead of a trapezoid, as long as they have four sides. The shapes could have different angles and side lengths, leading to different properties and applications.
Learn more about quadrilateral here: brainly.com/question/29934440
#SPJ4
a bank pin is a string of seven digits, each digit 0-9. five of the digits {0, 2, 4, 6, 8} are even and five of the digits {1, 3, 5, 7, 9} are odd. how many pins are there with at least one odd digit?
Answers
three and seven tenths times five
in a sampling distribution of means 1. the distribution will approximate the normal distribution (bell curve). 2. the mean of the sampling dsitribution will equal the mean of the population 3. the standard deviation of the sampling distribution is called the standard error. this is based on the
Answers
In a sampling distribution of means, the distribution will approximate the normal distribution, the mean of the sampling distribution will equal the mean of the population and the standard deviation of the sampling distribution is based on Central Limit Theorem, option A.
The central limit theorem (CLT) of probability theory states that, under the assumption that all samples are of equal size and regardless of the population's actual distribution shape, the distribution of a sample variable approaches a normal distribution (i.e., a "bell curve") as the sample size increases.
In other words, the central limit theorem (CLT) is a statistical assumption that, given a sufficiently enough sample size from a population with a limited degree of variance, the mean of all sampled variables from the same population will be roughly equal to the mean of the entire population. According to the law of large numbers, these samples also resemble a normal distribution, with their variances almost equaling the variation of the population as the sample size increases.
Learn more about Central Limit Theorem:
https://brainly.com/question/18403552
#SPJ4
Complete question:
In a sampling distribution of means... 1. the distribution will approximate the normal distribution (bell curve). 2. the mean of the sampling distribution will equal the mean of the population 3. the standard deviation of the sampling distribution is called the standard error. This is based on the...
Central Limit Theorem
Standardization of Transformation
Z-Score Distribution Table
Relative Frequency of Probability
A cylinder has radius 1.6 meters. Its volume is 95 cubic meters. Find its height to the nearest tenth of a meter. Use 3.14 as approximation for
Answers
The height of the given cylinder with radius 1.6 meters is 11.81 meters.
Explain about the volume of cylinder?A cylinder is a solid made up of two congruent circles in parallel planes, respective interiors, and all the parallel line segments with ends on the circular regions that are perpendicular to the took the shape the centers of both circles.
A three-dimensional solid's volume is the amount of room it occupies up. When calculating the volume, make sure that each of the quantities belong to the same unit.
Radius r = 1.6 meters.
Volume of cylinder V = 95 cubic meters
Formulae for volume of cylinder = πr²h
Height = ?
V = πr²h
95 = 3.14 * 1.6² * h
h = 95 /(3.14 * 1.6²)
h = 11.81
Thus, the height of the given cylinder with radius 1.6 meters is 11.81 meters.
Know more about the volume of cylinder
https://brainly.com/question/23935577
#SPJ1
The box plot represents the number of tickets sold for a school dance.
A horizontal line labeled Number of Tickets sold that starts at 7, with tick marks every one unit up to 32. The graph is titled Tickets Sold for A Dance. The box extends from 15 to 21 on the number line. A line in the box is at 19. The lines outside the box end at 8 and 31.
Which of the following is the appropriate measure of variability for the data, and what is its value?
The IQR is the best measure of variability, and it equals 13.
The IQR is the best measure of variability, and it equals 6.
The range is the best measure of variability, and it equals 13.
The range is the best measure of variability, and it equals 6.
Answers
The correct statement regarding the best measure of the variability is given as follows: The IQR is the best measure of variability, and it = 6. The Interquartile Range (IQR of a data-set. As it not affected by outliers, it is the best measure of variability of a data-set. The quartiles for this problem are given as follows: First quartile: Q1 = 15. Third quartile Q3 = 21. Hence the IQR is obtained as follows: IQR = 21 - 15 IQR = 6
Answer: The IQR is the best measure of variability, and it equals 6.
IQR is the best way to measure the variability in this question and its also equal to six. I took the test just now and got it right i cant really explain anything else.
What is the equation of this graph?
A.) y=-x²-2x-8
B.) y=x²-2x-8
C.) y=3x²-12x-8
D.) y=-3x²-12x-8
Answers
The equation of the quadratic function that passes through these points is:
[tex]y = -3x^2 - 12x - 8.[/tex]
What is quadratic equation?
A quadratic equation is a polynomial equation of the second degree, which means it contains one variable with an exponent of 2. The general form of a quadratic equation is:
[tex]ax^2 + bx + c = 0[/tex]
Using polynomial interpolation with these 5 points, we can find a quadratic polynomial that passes through all the points. Let the equation of the polynomial be [tex]y = ax^2 + bx + c[/tex].
Using the point (0, -8), we get:
[tex]-8 = a(0)^2 + b(0) + c[/tex]
c = -8
Using the point (-1, 1), we get:
[tex]1 = a(-1)^2 + b(-1) - 8[/tex]
a - b = 9
Using the point (-3, 1), we get:
[tex]1 = a(-3)^2 + b(-3) - 8[/tex]
9a - 3b = 27
Using the point (-4, -8), we get:
[tex]-8 = a(-4)^2 + b(-4) - 8[/tex]
16a - 4b = 0
Using the point (-2, 4), we get:
[tex]4 = a(-2)^2 + b(-2) - 8[/tex]
4a - 2b = 12
Solving this system of equations, we get:
a = -3
b = -12
c = -8
Therefore, the equation of the quadratic function that passes through these points is:
[tex]y = -3x^2 - 12x - 8.[/tex]
To learn more about quadratic equation visit:
https://brainly.com/question/1214333
#SPJ1
During a catered lunch, an average of 4 cups of tea are poured per minute. The lunch will last 2 hours. How many gallons of tea should the caterer bring if there are 16 cups in one gallon? _____ gallons
Answers
Answer:30 gallons
Step-by-step explanation:
Answer:
30 gallons of tea
Step-by-step explanation:
Step 1:
First, before we do anything, let's convert 2 hours into minutes. There are 60 minutes in an hour, so we would multiply 60 and 2 to get 120 minutes.
Step 2:
Since we know that 4 cups of tea are poured every minute, we must multiply 4 by 120 because it is poured every 1 minute, and 4 × 120 = 480.
Step 3:
We also know that there are 16 cups of tea in one gallon, so we would divide 480 by 16 because we are trying to get how many gallons there are. 480 ÷ 16 = 30, so there are 30 gallons of tea poured in two hours.
how many ways are there to put 23 identical objects into 7 distinct containers so that no container is empty?
Answers
This problem is a classic example of applying the stars and bars combinatorial method.
To solve this problem, we need to distribute 23 identical objects into 7 distinct containers so that none of the containers is empty. Let's start by assuming that each container has at least one object. This means we have distributed 7 objects, leaving 16 objects to be distributed.
We can now use the stars and bars method to distribute these 16 objects. Imagine placing all 16 objects in a line, and placing 6 dividers between them to separate them into 7 groups (one for each container). Each group will then receive the objects that are to the left of the divider.
For example, if we have the sequence OO|OOO|O|OOO|OO|OO|O, this corresponds to 2 objects in the first container, 3 objects in the second container, 1 object in the third container, and so on.
The number of ways to place the 6 dividers in the sequence of 16 objects is the same as the number of ways to choose 6 positions out of 16 to place the dividers. This can be computed using the formula for combinations:
$${16 \choose 6} = \frac{16!}{6!(16-6)!} = 8008$$
Therefore, there are 8008 ways to distribute 23 identical objects into 7 distinct containers so that no container is empty.
For more details about combinatorial click here:
https://brainly.com/question/30574889#
#SPJ11
Triangle TUV, with vertices T(-8,2), U(-2,8), and V(-9,9), is drawn inside a rectangle, as shown below.
Answers
The area of triangle TUV with vertices T(-8,2), U(-2,8), and V(-9,9) are 13.4 units.
What is triangle?A triangle is a closed two-dimensional plane figure that has three sides, three angles, and three vertices. The sum of the angles of a triangle is always 180 degrees. Triangles can be classified based on the length of their sides and the size of their angles. Some common types of triangles include equilateral, isosceles, scalene, acute, obtuse, and right triangles. Triangles are a fundamental concept in geometry and are used in many areas of mathematics and science.
Here,
To find the area of triangle TUV, we can use the formula:
Area = 1/2 * base * height
We can choose any two sides of the triangle as the base and the corresponding height. Let's choose TU as the base and the perpendicular distance from V to TU as the height.
First, let's find the length of TU:
TU = √[(8 - 2)² + (-2 - (-8))²]
= √[6² + 6²]
= 6√(2)
Next, let's find the slope of TU:
mTU = (8 - 2) / (-2 - (-8))
= -3/2
The line perpendicular to TU passing through V has a slope equal to the negative reciprocal of mTU:
mVQ = 2/3
The equation of the line passing through V and perpendicular to TU is:
y - 9 = (2/3)(x + 9)
Solving for x and y at the point where this line intersects TU, we get:
y = (2/3)x + 19
(2/3)x + 19 = -3x/2 + 7
x = -8/7
y = 94/21
The perpendicular distance from V to TU is the absolute value of y - 8:
|94/21 - 8| = 2/21
So, the area of triangle TUV is:
Area = 1/2 * TU * (2/21)
= (1/21)√(2)
To find the area of rectangle QRS, we need to find the length and width. We can use the distance formula to find the length QR and the width QS:
QR = √[(9 - (-8))² + (9 - 2)²]
= √[289]
= 17
QS = √[(9 - (-9))² + (2 - 2)²]
= √[324]
= 18
So, the area of rectangle QRS is:
Area = QR * QS
= 17 * 18
= 306
Area of triangle QRS = Area of rectangle QRS - Area of triangle TUV
= 306 - (1/21)√(2)
≈ 13.4 units
So, the answer is (C) 13.
To know more about triangle,
https://brainly.com/question/28600396
#SPJ1
Complete question:
Triangle TUV, with vertices T(-8,2), U(-2,8), and V(-9,9), is drawn inside a rectangle. What is the area, in square units, of the triangle TUV?
A. 7
B. 10
C. 13
D. 18
Answer these for points
Answers
6. f(x)g(x)+h(x) = 20x3-4x. This answer is arrived at by using the rules of algebra and the values of the given functions.
7. f(x)g(x) = -15x2 - 18x - 15
What are polynomials?Polynomials are algebraic expressions consisting of variables and coefficients that are combined through addition, subtraction, multiplication and division operations. For example, the polynomial 4x² + 3x - 5 can be written as the sum of 4x², 3x and -5.
6. In order to calculate f(x) g(x)+h(x), it is necessary to first multiply f(x) and g(x). Since f(x)=4x and g(x) = 5x2-4, it follows that f(x)g(x)=20x3-4x. Next, it is necessary to add h(x) to this product. Since h(x) = 9x, it follows that f(x)g(x)+h(x) = 20x3-4x+9x = 20x3-4x. Since each coefficient is correctly represented in the answer, it follows that the correct answer is 20x3-4x.
Multiplying f(x) and g(x) gives the product of 20x3-4x, and adding h(x) to this product gives the sum of 20x3-4x. As each coefficient is correctly represented in the answer, the correct answer is 20x3-4x.
7. The product of two polynomials is found by multiplying each term of one polynomial by each term of the other polynomial. This is known as the FOIL method (First, Outer, Inner, Last).
The first term is the product of the first terms of each polynomial, which is 3x x 4x = 12x2.
The second term is the product of the outer terms, which is 3x x -5x = -15x2.
The third term is the product of the inner terms, which is 5 x -5x = -25x.
Finally, the fourth term is the product of the last terms of each polynomial, which is 5 x -3 = -15.
We can now combine the terms to get the product of the two polynomials, which is:
f(x)g(x) = -15x2 - 18x - 15.
Therefore, the correct coefficient for each term is -15x2 for the first term, -18x for the second term, and -15 for the third term.
For more questions related to algebra
https://brainly.com/question/28036476
#SPJ1
11. Hannah recorded the number of miles she jogged. She started jogging 3 41 miles. Every week she jogged an additional
1 21 miles. Select all true statements. The y-intercept is the rate of change. The y-intercept is the starting value. The slope is 121. The slope is 3 41. The y-intercept is 1 21. The y-intercept is 3 41
Answers
Using coordinate geometry,
The y-intercept is the starting value (true).The slope is 1/2 or 0.5, not 121 or 3/41 (false).The y-intercept is 3/41 is also false as it contradicts the first true statement, therefore, the y-intercept is the starting value and the y-intercept of 3/41 is also true.The problem provides two pieces of information about Hannah's jogging routine: she started with 3 41 miles and added 1/21 miles every week. We can use this information to create a linear equation that represents the number of miles Hannah jogged as a function of the number of weeks she has been jogging.
To create this equation, we first identify the starting value, which is 3 41 miles. This is also the y-intercept of the line. Next, we determine the slope of the line, which is the rate at which the number of miles increases each week. The slope is equal to the change in y (miles) divided by the change in x (weeks), which in this case is (1/21 - 3/41)/1 = -2/1 = -2. Therefore, the slope of the line is -2.
Putting these two pieces of information together, we get the equation y = -2x + 3 41, where y is the number of miles jogging and x is the number of weeks of jogging. This is a linear equation in slope-intercept form, where the slope is -2 and the y-intercept is 3 41.
Learn more about the co-ordinate geometry at
https://brainly.com/question/21094061
#SPJ4