Determine the magnitude of each vector: • Use "sqrt()" to denote ✓ • If you use a decimal approximation, you must be accurate to at least 3 decimal places. a. ||(6, 10) || = b. ||(-2, 10)|| = C. ||(12, -10)|| = d. ||(-8, -13)|| = 7 : <2,2> Dragū into one of the four quadrants, then give the magnitude and direction angle for ū. |17|| = O = You must complete this problem once for each quadrant: Quadrant Quadrant II Quadrant III Quadrant IV
Answers
a. ||(6, 10)|| = sqrt(6^2 + 10^2) = sqrt(136) ≈ 11.662
b. ||(-2, 10)|| = sqrt((-2)^2 + 10^2) = sqrt(104) ≈ 10.198
c. ||(12, -10)|| = sqrt(12^2 + (-10)^2) = sqrt(244) ≈ 15.620
d. ||(-8, -13)|| = sqrt((-8)^2 + (-13)^2) = sqrt(233) ≈ 15.264
For u = <2, 2>, regardless of which quadrant it is in:
||u|| = sqrt(2^2 + 2^2) = sqrt(8) ≈ 2.828
The direction angle (θ) can be found using the formula θ = tan^-1(y/x), where (x,y) is the vector.
Therefore, θ = tan^-1(2/2) = tan^-1(1) ≈ 45°
If we move u to Quadrant II by reflecting it across the y-axis, then its coordinates become (-2, 2).
||u|| = sqrt((-2)^2 + 2^2) = sqrt(8) ≈ 2.828
θ = tan^-1(2/-2) = tan^-1(-1) ≈ -45°
If we move u to Quadrant III by reflecting it across both axes, then its coordinates become (-2, -2).
||u|| = sqrt((-2)^2 + (-2)^2) = sqrt(8) ≈ 2.828
θ = tan^-1(-2/-2) = tan^-1(1) ≈ 45°
If we move u to Quadrant IV by reflecting it across the x-axis, then its coordinates become (2, -2).
||u|| = sqrt(2^2 + (-2)^2) = sqrt(8) ≈ 2.828
θ = tan^-1(-2/2) = tan^-1(-1) ≈ -45°
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Related Questions
An equilateral triangle with side lengths of 8.7 centimeters is shown. An apothem has a length of a and the radius has a length of 5 centimeters. The apothem and radius form a triangle with a base length of b.
Which statements about finding the area of the equilateral triangle are true? Select three options.
The apothem can be found using the Pythagorean theorem.
The apothem can be found using the tangent ratio.
The perimeter of the equilateral triangle is 15 cm.
The length of the apothem is approximately 2.5 cm.
The area of the equilateral triangle is approximately 65 cm2.
Answers
Options A,B and D. The correction about the area of the triangle are
The apothem can be found using the Pythagorean theorem.The apothem can be found using the tangent ratio.The length of the apothem is approximately 2.5 cm.How to solve for the length of the Apothem using the Pythagoras theoremFirstly this is an equilateral triangle
3 of the sides have the length of 8.7cm
b = half of 8.7
= 4.35cm
To get A we have
a² + b² = c²
a² + 4.35² = 5²
When we solve this out we would have
a² = 25 - 18.9225
a ≈ 2.5 cm.
Hence we have proved that the first option is correct.
For B,
In an equilateral triangles, each of the angles = 60 degrees
60/2 = 30 is the angle that is made from the base
using tangent ratiosa / 4.35 = tan 30
then a = 4.35 tan 30
This would also give us an approximate of 2.5
For D
The calculations we have done based on the statements in A and B shows that D is correct as 2.5.
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does anyone know this
Answers
Answer:
Answer below
Step-by-step explanation:
The coordinates of vertex in triangle 1 - A = 4, 2 B= 6, 8 C= 8, 4
The coordinates of vertex in triangle 2- A= -4, 2 B= -6, 8 C= -8, 4
To do this you write the coordinates of the vertex ( corners of the the triangle )
HOPE IT HELPED
Answer:
Shape #1:
A (4,2)
B (6,8)
C (8,5)
Shape #2:
A' (-4, 2)
B' (-6, 8)
C' (-8, 5)
Hope this helps :)
I have no idea what's going on
Answers
Answer:
\(3B-4C=\begin{pmatrix}-8&-23\\ 1&-15\end{pmatrix}\)
Step-by-step explanation:
Let
\(B=\begin{pmatrix}8&-5\\ -1&3\end{pmatrix}\)
and
\(C=\begin{pmatrix}8&2\\ \:-1&6\end{pmatrix}\:\)
Finding 3B-4C
\(\:3B-4C\:=\:3\begin{pmatrix}8&-5\\ -1&3\end{pmatrix}\:-4\begin{pmatrix}8&2\\ \:-1&6\end{pmatrix}\)
first solving
\(3\begin{pmatrix}8&-5\\ -1&3\end{pmatrix}\)
Scalar multiplication: Multiply each of the matrix elements by a scalar
\(3\begin{pmatrix}8&-5\\ \:\:-1&3\end{pmatrix}=\begin{pmatrix}3\cdot \:\:8&3\left(-5\right)\\ \:3\left(-1\right)&3\cdot \:\:3\end{pmatrix}\)
simplify each element
\(=\begin{pmatrix}24&-15\\ -3&9\end{pmatrix}\)
now solving
\(4\begin{pmatrix}8&2\\ \:\:-1&6\end{pmatrix}\)
Scalar multiplication: Multiply each of the matrix elements by a scalar
\(4\begin{pmatrix}8&2\\ \:\:-1&6\end{pmatrix}=\begin{pmatrix}4\cdot \:\:8&4\cdot \:\:2\\ \:4\left(-1\right)&4\cdot \:\:6\end{pmatrix}\)
simplify each element
\(=\begin{pmatrix}32&8\\ -4&24\end{pmatrix}\)
now combining the results
\(\:3B-4C\:=\:3\begin{pmatrix}8&-5\\ -1&3\end{pmatrix}\:-4\begin{pmatrix}8&2\\ \:-1&6\end{pmatrix}\)
\(=\begin{pmatrix}24&-15\\ -3&9\end{pmatrix}-\begin{pmatrix}32&8\\ -4&24\end{pmatrix}\)
subtract the elements in the matching positions
\(=\begin{pmatrix}24-32&\left(-15\right)-8\\ \left(-3\right)-\left(-4\right)&9-24\end{pmatrix}\)
simplifying the elements
\(=\begin{pmatrix}-8&-23\\ 1&-15\end{pmatrix}\)
Therefore,
\(3B-4C=\begin{pmatrix}-8&-23\\ 1&-15\end{pmatrix}\)
HELP ASAP!!! please, I need help
Answers
Your answer would be be subtract 3
hope it helps
Hope that
Helped u
Which number line shows the points −125 and –2.8?
Answers
is -20/2 irrational or rational
Answers
Answer:
rational
Step-by-step explanation:
a rational number can be expressed in the form
\(\frac{a}{b}\) where a and b are integers
\(\frac{-20}{2}\) is in this form and is therefore rational
Answer:
rational
Step-by-step explanation:
Rational Numbers :
It is expressed in the ratio, where both numerator and denominator are the whole numbers
Which property explains the step shown in solving the equation?5(x − 2) = 12x − 39 ⇒ 5x − 10 = 12x − 39
Answers
Hi there!
»»————- ★ ————-««
I believe your answer is:
Distribution property
»»————- ★ ————-««
Here’s why:
⸻⸻⸻⸻
From the first step to the next step, the 5 gets distributed into the 'x-2'. The distribution property is getting used.
⸻⸻⸻⸻
\(5(x-2) = 12x -39\\\\\rightarrow5(x-2)\leftarrow\\\\\left \{ {{5*x=5x} \atop {5*-2=-10}} \right. \\\\\rightarrow5(x-2)=\boxed{5x - 10}\\\\\\5(x - 2) = 12x - 39\\\\\rightarrow\boxed{5x-10}=12x-39\)
⸻⸻⸻⸻
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
The five values for a data set are: minimum = 0 lower quartile = 2 median = 3. 5 upper quartile = 5 maximum = 10 Bruno created the box plot using the five values. What error did he make? The right whisker should go from 3. 5 to 10. The left whisker should go from 0 to 2. The box should go from 2 to 3. 5. The box should go from 3. 5 to 5
Answers
The five values for a data set are: minimum = 0 lower quartile = 2 median = 3. 5 upper quartile = 5 maximum = 10 Bruno created the box plot using the five values. Bruno made error. The left whisker should go from 0 to 2.
About quartileQuartiles is a type of quartile that divides data into four parts with approximately the same number. The first quartile or lower quartile (Q1) is the middle value between the smallest value and the median of the data group. The first quartile is a marker that the data in that quartile is 25% below the data group.
The second quartile (Q2) is the median data which marks 50% of the data (dividing the data in half). The third or upper quartile (Q3) is the middle value between the median and the highest value of the data set. The third quartile is a marker that the data in that quartile is 75% below the data group. Quartiles are a form of an ordered statistic because to determine quartiles, data needs to be sorted from smallest to largest value first.
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Helpppppp pleaseeee !!!!
Answers
Find both the vector equation and the parametric equations of the line through (0,0,4) in the direction of the vector v=−4,2,0​, where t=0 corresponds to the given point. Question content area bottom Part 1 The vector equation is x,y,z
Answers
The parametric equations of the line are
**x(t) = -4t**
**y(t) = 2t**
**z(t) = 4**
The vector equation of the line through (0,0,4) in the direction of the vector **v=−4,2,0** is:
r(t) = (0, 0, 4) + t(-4, 2, 0)
In this equation, **r(t)** represents the position vector of any point on the line, **t** is a parameter that determines the position of a point along the line, and **(0, 0, 4)** is the initial point of the line.
To obtain the parametric equations of the line, we can express the coordinates **x**, **y**, and **z** as functions of the parameter **t**.
For the **x-coordinate**, we have:
**x(t) = 0 - 4t**
For the **y-coordinate**, we have:
**y(t) = 0 + 2t**
For the **z-coordinate**, we have:
**z(t) = 4 + 0t = 4**
These equations describe the position of any point on the line in terms of the parameter **t**. Thus, the parametric equations of the line are:
**x(t) = -4t**
**y(t) = 2t**
**z(t) = 4**
By substituting different values of **t**, we can determine the corresponding coordinates of points along the line.
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adding and subtracting functions
g(x) = 4x + 5 f (x) = −3x^2 − 3x Find g(x) + f (x)
Answers
Answer:
- 3x² + x + 5
Step-by-step explanation:
g(x) + f(x)
= 4x + 5 - 3x² - 3x ← collect like terms
= - 3x² + x + 5
lapeer flour mills purchased new equipment and made the following expenditures: purchase price $ 64,000 sales tax 5,450 shipment of equipment 890 insurance on the equipment for the first year 590 installation of equipment 1,780
Answers
The journal entry for the following transaction will be:
Equipment A/c....Dr. 73120
Prepaid Insurance A/c....Dr. 590
Cash A/c....Cr. 3260
Accounts payable A/c....Cr. 70450
Any cost that has been incurred by the company to make an asset ready to use, is also included in its purchasing price. So, any transportation, legal, or maintenance fee is also included in the asset cost.
The total cost of the new machine = Purchase price + Sales tax + Shipment of machine + Installation of machine
Total cost of new machine = 65,000 + 5,450 + 890 + 1,780
The total cost of the new machine = $73,120
Journal entry for the same:
Equipment A/c....Dr. 73120
Prepaid Insurance A/c....Dr. 590
Cash A/c....Cr. 3260
Accounts payable A/c....Cr. 70450
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Ryan's friends have organized a treasure hunt. He is using the map below to find the clues. The position of the last clue, which
leads to the treasure, is located at point (-2,4).
6
+
2
2
-2
He figures that the treasure is 3 units either to the east or south of the location of the last clue. Which of these could be the
location of the treasure?
OA. point (1, 4) or point (-2, 1)
OB. point (-5, 4) or point (-2, 1)
OC. point (1, 1) or point (-5,7)
OD. point (1, 4) or point (-2,7)
Answers
Using translation concepts, it is found that the location of the treasure could be given by these following points:
A. point (1, 4) or point (-2, 1)
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
We consider that he is at point (-2,4), hence:
A shift east 3 units leads to point (1,4).A shift south 3 units leads to point (-2,1).Hence option A is correct.
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Which one is greater 25 fluid ounces or 1 pint?
Answers
Answer:
25 Ounces =
1.5625 Pints
or
25 Ounces =
25 Pints
16
Step-by-step explanation:
Answer:
Step-by-step explanation:
1 pint is bigger because 16 fluid ounces would equal to 1 pint so 1 pint is greater
Check the image for the question.
Answers
The mathematical model that best fits the given data is option : y = 82.3 - 8.275x.
To construct a scatterplot and determine the mathematical model that best fits the given data, we will plot the points and analyze the trend.
Given Data:
x: 4, 8, 12, 16, 20
y: 40, 20, -11, -37, -97
Let's plot these points on a scatterplot:
(x, y) points:
(4, 40)
(8, 20)
(12, -11)
(16, -37)
(20, -97)
After plotting the points, we can analyze the trend in the data.
By observing the scatterplot, it is clear that a linear or quadratic model would not fit the data well since the points do not follow a straight line or a parabolic curve.
Next, we can try the logarithmic, exponential, and power models to see if any of them fit the data better.
Using a calculator or computer to perform the regression analysis, we find that the regression equation with the highest R² value is:
y = 82.3 - 8.275x
This equation corresponds to option : y = 82.3 - 8.275x
The regression equation represents a linear model that best fits the given data. The R² value indicates how well the model explains the variability in the data, and in this case, the linear model provides the highest R² value compared to the other models.
It's important to note that the specific values of the coefficients in the regression equation may vary slightly depending on the calculator or software used for the analysis. However, the overall best-fit model remains the same.
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According to a recent report, a sample of 360 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is =σ5.1 pounds.Construct a 99.5% confidence interval for the mean weight of all one-year-old baby boys in the United States. Round the answer to at least one decimal place.
Answers
The 99.5% confidence interval for the mean weight of all one-year-old baby boys in the United States is (24.8, 26.2) pounds.
We are given a sample of 360 one-year-old baby boys in the United States with a mean weight of 25.5 pounds and a population standard deviation of σ=5.1 pounds. We need to construct a 99.5% confidence interval for the mean weight of all one-year-old baby boys in the United States.
To construct the confidence interval, we can use the formula:
Confidence interval = sample mean ± (z-score)(standard error)
where the z-score is based on the confidence level and the standard error is calculated as:
standard error = σ/√n
Plugging in the given values, we get:
standard error = 5.1/√360 = 0.27 pounds
Since we want a 99.5% confidence interval, the corresponding z-score is 2.807. Therefore, the confidence interval is:
Confidence interval = 25.5 ± (2.807)(0.27) = (24.8, 26.2) pounds
We can be 99.5% confident that the mean weight of all one-year-old baby boys in the United States falls within the range of 24.8 to 26.2 pounds.
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An animal reserve has 24,000 elks. The population increases by 12% a year. How long until the population is equal to 96,000,000
Answers
Answer:33,325,000
Step-by-step explanation:95,976,000÷2.88=33,325,000
asapppppp
Find the median of the data: 22, 16, 14, 26, 32 and 30
Answers
Step-by-step explanation:
14 ,16,22,26,30,32
Have a great day
Answer:
\(\boxed {\boxed {\sf 24}}\)
Step-by-step explanation:
We are asked to find the median of the following data set:
22, 16, 14, 26, 32, 30The median is the middle number of the data set. It separates the higher half of the set from the lower half. We put the data set in order from least to greatest and find the middle number.
14, 16, 22, 26, 30, 32We eliminate a number from each side of the data set until we find the middle number.
16, 22, 26, 3022, 26However, the data set had 6 numbers, which is an even number. There isn't one middle number, so we must find the average of the middle 2 numbers. Add the numbers and divide by 2.
22+ 26 = 48 48 / 22= 24The median of the data set is 24.
If the equation of a line is: Y= 200 -3X This line is
downward sloping
upward sloping
vertical
horizontal
Answers
Step-by-step explanation:
y = 200 - 3 *x has slope = -3 this is downward sloping from L to R
Answer:downward sloping
Step-by-step explanation:
negative slope
A real estate agent receives a 3% commission on the sale price of a home.
How much does the real estate agent receive as a commission if the house sells for $232,000?
O $69,600.00
$6.960.00
$696.00
$69.60
Answers
multiply the percent by the money
One particular hotel in the downtown area costs \$90$90dollar sign, 90 a night. An additional 10\, percent tax is added to the original cost of the hotel room. There is also a one-time \$12$12dollar sign, 12 parking charge, and a family can expect to spend \$30$30dollar sign, 30 on tips during their stay. How many nights can a family spend at the hotel without exceeding their budget of \$600$600dollar sign, 600
Answers
Answer:
5
Step-by-step explanation:
total cost of hotel stay = cost per night + tax + parking charge + tips
tax = 10% x 90
= 0.1 x 90 = 9
let x = number of stays
90x + 9x + 12 + 30 ≤ 600
combine like terms
99x ≤ 558
x ≤ 5.6
if they stay 5 nights, they would spend 537
if they stay 6 nights, they would spend 636
Answer:
5
Step-by-step explanation:
Given the following functions, find g(f(x)). f(x) = 2x + 1 g(x) = x2 + 6
Answers
Answer:
Step-by-step explanation:
g(x) = x^2 + 6
f(x^2 + 6) = 2(x^2 + 6) + 1 = 2x^2 + 12 + 1 = 2x^2 + 13
Which of these strategies would eliminate a variable in the system of equations?
2x- 6y=6
6x - 4y = 2
Choose all answers that apply: more than 1
Multiply the bottom equation by 3 then subtract the bottom equation from the top equation
Multiply the bottom equation by -3/2 then add the equations.
Multiply the top equation by-3. then add the equations
Answers
Answer:
Multiply the bottom equation by -3/2 then add the equations.
Multiply the top equation by-3. then add the equations
Step-by-step explanation:
Given the simultaneous equation
2x- 6y=6 ... 1
6x - 4y = 2 ... 2
To eliminate a variable, we have to make the coefficient of one of the variable to be the same.
Multiply equastion 1 by -3
-6x+18y= -18
6x - 4y = 2
Add the result:
-6x + 6x + 18y-4y = -18+2
18y-4y = -18+2
14y = -18
y = -9/7
Another way is to Multiply the bottom equation by -3/2 then add the equations.
Multiplying equation 2 by -3/2 will give;
6x(-3/2) - 4y(-3/2) = 2(-3/2)
-9x + 6y = -3
Add to equation 1;
2x- 6y=6
-9x + 2x + 0 = -3+6
-7x = 3
x = -3/7
Hence the correct two options are;
Multiply the bottom equation by -3/2 then add the equations.
Multiply the top equation by-3. then add the equations
WILL GIVE BRAINLIEST *NO LINKS*
Answers
Answer:46%
Step-by-step explanation:
Answer:
23/50 or 46%
Step-by-step explanation:
I just added red and purple out of the total(red+purple/total=11+12/50=23/50)
then I multiplied the fraction by 2 to get out of 100 for the percentage(23/50*2=46/100=46%)
HELP!! For questions 9-12 evaluate each function for the given value
Answers
Answer:
9). h² - 7h + 11
10). 2a² + 14a + 16
11). 3
12). 2h + 4x - 3
Step-by-step explanation:
9). If j(x) = x² + x - 1
j(h - 4) = (h - 4)² +(h - 4) - 1
= h² - 8h + 16 + h - 4 - 1
= h² - 7h + 11
10). If f(x) = 2x² + 2x - 8
f(a + 3) = 2(a + 3)² + 2(a + 3) - 8
= 2(a² + 6a + 9) + 2a + 6 - 8
= 2a² + 12a + 18 + 2a - 2
= 2a² + 14a + 16
11). If f(x) = 3x - 1
\(\frac{f(x+h)-f(x)}{h}=\frac{3(x+h)-1-(3x-1)}{h}\)
\(=\frac{3x+3h-1-3x+1}{h}\)
= 3
12). If, f(t) = 2t² - 3t + 7
\(\frac{f(x+h)-f(x)}{h}=\frac{2(x+h)^{2}-3(x+h)+7-(2x^2-3x+7)}{h}\)
\(=\frac{2(x^2+h^2+2hx)-3x-3h+7-2x^2+3x-7}{h}\)
\(=\frac{2x^2+2h^2+4hx-3x-3h+7-2x^2+3x-7}{h}\)
\(=\frac{2h^2+4hx-3h}{h}\)
= 2h + 4x - 3
A boy saved over $5. His father gave him $2. The boy now had $y altogether. Write an inequality in y
Answers
Answer:
$7 total
Step-by-step explanation:
$5 + $2 = $7
A line in the coordinate plane has a slope of 4, and a distance of 1 unit from the origin. Find the area of the triangle determined by the line and the coordinate axesA line in the coordinate plane has a slope of m=4, and a distance of 1 unit from the origin. => that is a point (1,0)
so, the equation of this line will be:
y-y1=m(x-x1)...plug in given data
y-0=4 (x-1)
y=4x-4
now we know y intercept is at (0,-4)Find the area of the triangle determined by the line and the coordinate axes
the triangle determined by the line and the coordinate axes is right triangle, and its legs are base and altitude
base is 1 unit long, altitude is 4units long
Answers
The area of the triangle according to the information given in the question is 17/8.
What is meant by triangle?A triangle is a three-sided polygon that is also known as the trigon.
Angles in a triangle are formed by connecting the three sides end to end at a point.
Consider the following right-angled triangle, with vertices (0,-4), (0.0), and (1,0).
Its legs are 4 and 1 meters long, and its hypotenuse is 42+12=17 meters long.
The height "h" (altitude) of this triangle drawn to the hypotenuse is easily determined.
Based on the area, we get the equation (41)/2=(1/2)17h h=4/17 h=0.970143.
As we can see, it is not a single unit.
It means that the triangle's legs should be 17/4 as long as the original triangle's legs (1 unit and 4 units).
As a result, the area of the seeking triangle is (1/2)(17/4)(417/4) = (1/2)(17/4) =17/8
As a result, the area of the triangle in the question is 17/8.
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Find the volume of this object.Use 3 for a.Volume of a CylinderV=Tr2hVolume of a Sphere4 cmV=-Tr36 cm8 cmV ~ [?]cm3
Answers
The figure given in the question is a composite figure, meaning that it comprises two different figures
The volume of the composite figure can be found as follow
The two figures are:
Cylinder and sphere.
To solve this, we will first find the area of a cylinder
\(\begin{gathered} \text{Area of a cylinder is given by:} \\ V_{\text{cylinder}}=\pi r^2h \\ \text{where} \\ \pi=3 \\ r=4 \\ h=6 \end{gathered}\)So, we will have
\(\begin{gathered} V_{\text{cylinder}}=3\times4^2\times6 \\ V_{\text{cylinder}}=288\operatorname{cm}^3 \end{gathered}\)Then, we will find the volume of the sphere
\(\begin{gathered} V_{\text{sphere}}=\frac{4}{3}\pi r^3 \\ \text{where} \\ \pi=3 \\ r=4 \end{gathered}\)Thus, the volume of the sphere will be
\(\begin{gathered} V_{\text{sphere}}=\frac{4}{3}\times3\times4^3 \\ V_{\text{sphere}}=256\operatorname{cm}^3 \end{gathered}\)Thus, the total volume will be
\(288+256=544\operatorname{cm}^3\)The volume is:
\(544\operatorname{cm}^3\)The answer is 544cm³
I need help pls im confuzzled
Answers
Heart = 1
Water = 2
star = 4
Step-by-step explanation:
Heart x water = water, so heart is probably 1.
Water x water=star and water + water= star so water is 2. Star is 4
Answer:
heart= 1
tear= 2
star=4
leaf=2
diamond=3
circle=6
Step-by-step explanation:
if heart x tear equals tear one of those numbers has to be one.
so heart equals 1
tear + tear= star and tear x tear= star
only number that work are 2 since 2+2=4 and 2x2=4
leaf +leaf +leaf = circle
so plug in 2 for the leaves. 2+2+2=6
2x ____=6 plug in 3 so diamond is 3
3+3=6 so those are the correct choices.
Calculate all extrema (except for marginal values) of the function f(x,y)=x2+xy+x under the constraint g(x,y)=x+y−4=0. (It is not required to show that the found extrema are feasible.)
Answers
To find the extrema of the function f(x, y) = x^2 + xy + x under the constraint g(x, y) = x + y - 4 = 0, we can use the method of Lagrange multipliers. Let's proceed step by step:
1. Define the Lagrangian function L(x, y, λ) as:
L(x, y, λ) = f(x, y) + λ * g(x, y)
L(x, y, λ) = x^2 + xy + x + λ * (x + y - 4)
2. Compute the partial derivatives of L(x, y, λ) with respect to x, y, and λ:
∂L/∂x = 2x + y + 1 + λ
∂L/∂y = x + λ
∂L/∂λ = x + y - 4
3. Set the partial derivatives equal to zero and solve the resulting system of equations:
2x + y + 1 + λ = 0 ...(1)
x + λ = 0 ...(2)
x + y - 4 = 0 ...(3)
From equation (2), we have x = -λ. Substituting this into equations (1) and (3), we get:
-2λ + y + 1 + λ = 0 ...(4)
-λ + y - 4 = 0 ...(5)
4. Solve equations (4) and (5) simultaneously:
Subtracting equation (5) from equation (4), we get:
-λ + 5 = 0
λ = 5
Substituting λ = 5 into equation (5), we get:
-5 + y - 4 = 0
y = 9
Substituting y = 9 into equation (2), we get:
x + 5 = 0
x = -5
5. The extrema of f(x, y) = x^2 + xy + x under the constraint g(x, y) = x + y - 4 = 0 are obtained at the point (x, y) = (-5, 9).
Note: It's important to note that finding the extrema using the method of Lagrange multipliers gives the critical points of the function under the given constraint. To determine if these points are maximum, minimum, or saddle points, further analysis is required (such as using the second derivative test or checking the Hessian matrix). However, since feasibility is not required in this case, we stop at finding the critical point.
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