Looks like [tex]f(x,y)=x^2+y^2-x^2y+7[/tex].
[tex]f_x=2x-2xy=0\implies2x(1-y)=0\implies x=0\text{ or }y=1[/tex]
[tex]f_y=2y-x^2=0\implies2y=x^2[/tex]
If [tex]x=0[/tex], then [tex]y=0[/tex] - critical point at (0, 0).If [tex]y=1[/tex], then [tex]x=\pm\sqrt2[/tex] - two critical points at [tex](-\sqrt2,1)[/tex] and [tex](\sqrt2,1)[/tex]The latter two critical points occur outside of [tex]D[/tex] since [tex]|\pm\sqrt2|>1[/tex] so we ignore those points.
The Hessian matrix for this function is
[tex]H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}2-2y&-2x\\-2x&2\end{bmatrix}[/tex]
The value of its determinant at (0, 0) is [tex]\det H(0,0)=4>0[/tex], which means a minimum occurs at the point, and we have [tex]f(0,0)=7[/tex].
Now consider each boundary:
If [tex]x=1[/tex], then[tex]f(1,y)=8-y+y^2=\left(y-\dfrac12\right)^2+\dfrac{31}4[/tex]
which has 3 extreme values over the interval [tex]-1\le y\le1[/tex] of 31/4 = 7.75 at the point (1, 1/2); 8 at (1, 1); and 10 at (1, -1).
If [tex]x=-1[/tex], then[tex]f(-1,y)=8-y+y^2[/tex]
and we get the same extrema as in the previous case: 8 at (-1, 1), and 10 at (-1, -1).
If [tex]y=1[/tex], then[tex]f(x,1)=8[/tex]
which doesn't tell us about anything we don't already know (namely that 8 is an extreme value).
If [tex]y=-1[/tex], then[tex]f(x,-1)=2x^2+8[/tex]
which has 3 extreme values, but the previous cases already include them.
Hence [tex]f(x,y)[/tex] has absolute maxima of 10 at the points (1, -1) and (-1, -1) and an absolute minimum of 0 at (0, 0).
The function must first have its partial derivatives set to zero to find its critical points. Also, considering the given domain's boundaries with Lagrange multipliers can establish the function's maximum and minimum points.
Explanation:This function is a multivariable function and to find its extreme values within a specified domain you would use multivariable calculus methods. For the given function f(x, y) = x2 + y2 − x2y + 7, we first find its critical points by setting its partial derivatives equal to zero and then solve for x and y. Additionally, we need to consider the boundaries of the set {|x| ≤ 1, |y| ≤ 1} by using the method of Lagrange multipliers. The extreme values are obtained at the critical points within the domain, including the boundary.
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Gabriel is making a mixture of compost and soil to use for a special plant.He wants his final mix to be 2 parts compost to 6 parts potting soil.He wants to end up with 10 kilograms of mix.How many compost should Gabriel use?
Answer:
2.5 kg
Step-by-step explanation:
Parts of compost = 2 parts
Parts of potting soil = 6 parts
Total parts = 2 parts + 6 parts = 8 parts
from the above, we can see that 2 out of 8 parts of the total mix is compost,
i.e compost makes up [tex]\frac{2}{8}[/tex] of the total mix.
Given: total mix is 10 kg,
the amount of compost is then [tex]\frac{2}{8}[/tex] x 10 kg = 2.5 kg
Answer:Your answer is 2.5
Step-by-step explanation:
The soccer team collected $800 at a car wash fundraiser. They charged $5.00 for small vehicles and $10.00 for larger vehicles. The amount collected can be modeled by the equation. What is the equation?
Answer:
The equation is 5x + 10y = 800
Step-by-step explanation:
* Lets explain what is the equation and how can we make one
- The equation is a mathematical statement that two things are equal.
- It consists of two expressions, one on each side of an equal sign (=).
* Lets solve the problem
- There are two types of vehicles, large one and small one
- The cost of washing the small vehicle is $5
- The cost of washing the larger vehicle is $10
- They collected from washing both is $800
- To make an equation for the amount collected let the number of the
small vehicles is x and the number of the large vehicles is y
∵ x is the number of the small vehicles were washed
∵ y is the number of the large vehicles were washed
∵ The cost of washing the small vehicle is $5
∵ The cost of washing the larger vehicle is $10
∴ The money collected from the small vehicles = 5 × x = 5x
∴ The money collected from the large vehicles = 10 × y = 10y
- find the total money collected from both
∴ The amount of money collected from both = 5x + 10y
∵ The amount of money collected is $800
- Equate the two expressions above
∴ 5x + 10y = 800
* The equation is 5x + 10y = 800
The vertices of ABC are (2,8), B (16,2), and C (6,2). the perimeter of ABC is units, and it’s area is square units
Answer:
Perimeters is 32.44 unit and area is 30 square unit.
Need help with this math question!!!!!!!!!
ANSWER
The vertex is (1,-3)
EXPLANATION
The given parabola has equation:
[tex] {y}^{2} + 6y + 8x + 1 = 0[/tex]
We group the variables to obtain:
[tex]{y}^{2} + 6y = - 8x - 1 [/tex]
We complete the square to get,
[tex]{y}^{2} + 6y + {3}^{2} = - 8x - 1 + {3}^{2} [/tex]
[tex] {(y+ 3)}^{2} = - 8x +8[/tex]
[tex] {(y+ 3)}^{2} = - 8(x -1)[/tex]
The vertex is of the parabola is (1,-3)
Answer:
(-1,-3)
Step-by-step explanation:
Using the following triangle, what is the tangent of angle B?
Answer:
tanB=b/a
Step-by-step explanation:
Need help with a math question
Answer:
x=7
Step-by-step explanation:
Equate the angles 7x-7 and 4x+14 and solve:
7x-7 = 4x+14
7x-4x = 14+7
3x = 21
x = 7
Answer:
x = 7
Step-by-step explanation:
Alternate interior angles are equal. Alternate interior angles look like these two angles.
7x - 7 = 4x + 14 Add 7 to both sides
7x - 7 + 7 = 4x + 14+7 Combine
7x = 4x + 21 Subract 4x from both sides
7x-4x = 4x - 4x + 21 Combine
3x = 21 Divide by 3
3x/3 = 21/3 Do the division
x = 7
A swimming pool measures 35 feet wide by 50 feet long. What is the ratio of the length to the perimeter in simplest form?
Answer:
i think it is 5:17
Step-by-step explanation:
length=50
perimeter=170
the highest factor of 50 and 170 is 10 so the answer will be 5:17
hopefully this made sense
two lengths of a triangle are show
I can't give you the correct answer but you can guess for it, cause I can give you the the scop of the length of KL.
So, as we know this;
The sum of the two sides of a triangle is greater than the third and the difference between the two sides is less than the third.
10+5 >= KL >= 10-5
15 >= KL >= 5
based on your option, the answer should be:
9 in
Andy timed himself throughout the school year to see how many math facts he could complete in 111 minute. Andy gets 222 minutes of extra computer time for every math fact he completes. How many extra minutes did he get in February?
Andy gets 222 minutes of extra computer time for every math fact he completes. To find out how many extra minutes he got in a specific month, multiply the number of math facts he completed in that month by 222.
Explanation:To find out how many extra minutes Andy got in February, we need to know how many math facts he completed in February. Let's say he completed x math facts in February. Since Andy gets 222 minutes of extra computer time for every math fact he completes, the total extra minutes he got in February can be calculated by multiplying the number of math facts he completed in February (x) by 222.
So, the total extra minutes Andy got in February is 222x.
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Find the area of a regular hexagon with an apothem 11.4 yards long and a side 13 yards long. Round your answer to the nearest tenth.
Answer:
Step-by-step explanation:
area=6×11.4×13÷2=444.6 yd²
Final answer:
The area of a regular hexagon with an apothem of 11.4 yards and a side of 13 yards is approximately 444.6 square yards when rounded to the nearest tenth.
Explanation:
To find the area of a regular hexagon with a given apothem and side length, you can use the formula for the area of a regular polygon:
Area = (1/2) × Perimeter × Apothem
In this case, the perimeter of the hexagon can be calculated by multiplying the length of one side by 6 (since a hexagon has 6 sides). Therefore, the Perimeter = 13 yards × 6 = 78 yards.
Now, plug the values into the area formula:
Area = (1/2) × 78 yards × 11.4 yards
Area = (1/2) × 889.2 square yards
Area = 444.6 square yards
So, the area of the regular hexagon is approximately 444.6 square yards, when rounded to the nearest tenth.
Rosana is tracking how many customers she can serve in a morning. She listed the number of customers served per hour in the following table:
Hour (x) Number of Customers f(x)
1 3
2 5
3 7
Determine if these data represent a linear function or an exponential function, and give the common difference or ratio.
A. This is a linear function because there is a common difference of 2.
B. This is an exponential function because there is a common ratio of 2.
C. This is a linear function because there is a common difference of 3.
D. This is an exponential function because there is a common ratio of 3.
Answer:
A. This is a linear function because there is a common difference of 2
Step-by-step explanation:
Differences are ...
5 - 3 = 27 - 5 = 2The differences of 2 are common, so this is an arithmetic function.
Option: A is the correct answer.
A. This is a linear function because there is a common difference of 2.
Step-by-step explanation:Linear function--
A function is said to be linear if the rate of change is constant.
i.e. the function is increasing by a fixed constant.
Here the table is given by:
Hour (x) Number of Customers f(x)
1 3
2 5
3 7
From the table of values we observe that as the value of increasing by 1 the value of y is increasing by 2.
i.e. the rate of change is constant i.e. 2.
Also, the common difference is 2.
since,
5-3=2
and 7-5=2
Hence, the table represents a linear function.
HELP ME!!!!! ......
Using point-slope form, write the equation of the line that passes through the point (-4, 12) and has a slope of -3/4
Answer:
[tex]y-12=-\frac{3}{4}(x+4)[/tex]
Step-by-step explanation:
we know that
The equation of the line into slope point form is equal to
[tex]y-y1=m(x-x1)[/tex]
In this problem we have
[tex](x1,y1)=(-4,12)[/tex]
[tex]m=-\frac{3}{4}[/tex]
substitute
[tex]y-12=-\frac{3}{4}(x+4)[/tex]
Find the area of the kite QRST.
For this case we have that the area of the kite is formed by the ara of two triangles with the same base of[tex]9 + 9 = 18[/tex] meters and one with height 9 meters and another with height 6 meters.
So:
[tex]A = \frac {1} {2} b * h + \frac {1} {2} b * h\\A = \frac {1} {2} 18 * 9 + \frac {1} {2} 18 * 6\\A = \frac {1} {2} 162+ \frac {1} {2} 108\\A = 81 + 54\\A = 135 \ m ^ 2[/tex]
ANswer:
Option B
Answer:
135 m^2
Step-by-step explanation:
Gradpoint approved, good luck guys.
A woman has 14 different shirts: 10 white shirts and 4 red shirts. If she randomly chooses 2 shirts to take with her on vacation, then what is the probability that she will choose two white shirts? Show your answer in fraction and percent, round to the nearest whole percent.
Answer:
1/5; 20%
Step-by-step explanation:
If she is only choosing 2 shirts, and they both have to be white, the probability of choosing those 2 white shirts out of 10 white shirts is 2/10 or, equivalently, 1/5. In percent form, this is 20%
The probability of a woman randomly choosing two white shirts from a collection of 14 shirts (10 white and 4 red) is approximately 49%, derived through the process of combinations calculation in the field of probability.
Explanation:The question relates to the field of probability in mathematics. To calculate her probability of selecting two white shirts, we first need to understand the number of total possible outcomes when she is selecting 2 out of the 14 shirts. This is represented by the combination formula C(n, r) = n! / r!(n-r)!, where n is the total number of items to choose from and r is the number of items to choose. Using this combination formula, we find that there are C(14, 2) = 91 total possible combinations of shirts she could end up with.
Next, we need to find the number of combinations that involve her picking 2 white shirts. As there are 10 white shirts, this is represented by C(10, 2) = 45. So, the probability of her taking two white shirts is then the number of desired outcomes divided by the total number of outcomes, or 45 / 91 which simplifies to approximately 0.4945 or 49% when rounded to the nearest whole percent.
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Evaluate the following expression for m= 8 and p= -12
M^2-|p|
76
52
-52
-76
Answer:
52
Step-by-step explanation:
Put the numbers in the expression and do the arithmetic.
(-8)² - |-12| = 64 -12 = 52
A hot dog vendor has determined that the number of hot dogs he sells per day is inversely proportional to the price he charges. The vendor wants to decide if increasing his price by 55 cents will drive away too many customers. On average, he sells 200 hot dogs a day at a price of $3.85 per hot dog. How many hot dogs can he expect to sell if the price is increased by 55 cents? Round your answer to the nearest hot dog.
Answer:
175 hot dogs
Step-by-step explanation:
The new price will be $3.85 + 0.55 = $4.40. Since the price has increased by a factor of 4.40/3.85 = 8/7, the number of hot dogs sold, which is inversely proportional, will be ...
(7/8)·200 = 175 . . . hot dogs sold at the higher price
The vendor can expect to sell approximately 233 hot dogs if the price is increased by 55 cents.
Explanation:To determine how many hot dogs the vendor can expect to sell if the price is increased by 55 cents, we can use the inverse proportionality relationship between the number of hot dogs sold and the price charged.
First, let's set up a proportion with the initial price and number of hot dogs sold:
$3.85 / 200 = (new price + $0.55) / x
Next, we can cross multiply and solve for x:
x = (200 * ($3.85 + $0.55)) / $3.85
Calculating this expression gives us a value of x ≈ 233.77.
Since the number of hot dogs sold must be a whole number, we round down to the nearest whole number, giving us an estimated value of 233 hot dogs.
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Someone help me out please
For this case we have to define trigonometric relations of rectangular triangles that, the cosine of an angle is given by the leg adjacent to the angle on the hypotenuse of the triangle. That is, according to the data of the figure we have:
[tex]Cos (45) = \frac {6} {h}[/tex]
Where:
h: It's the hypotenuse
So:
[tex]\frac {\sqrt {2}} {2} = \frac {6} {h}[/tex]
We cleared h:
[tex]h \sqrt {2} = 6 * 2\\h \sqrt {2} = 12\\h = \frac {12} {\sqrt {2}}[/tex]
We rationalize:
[tex]h = \frac {12 \sqrt {2}} {(\sqrt {2}) ^ 2}\\h = \frac {12 \sqrt {2}} {2}\\h = 6 \sqrt {2}[/tex]
Answer:
Option C
ANSWER
[tex] 6\sqrt{2} \: units[/tex]
EXPLANATION
This is an isosceles right triangle.
The two legs of an isosceles right triangle are equal, hence each of them is 6 units each.
Let h be the hypotenuse, then from the Pythagoras Theorem,
[tex] {h}^{2} = {6}^{2} + {6}^{2} [/tex]
[tex]{h}^{2} = {6}^{2} \times 2[/tex]
Take square root of both sides;
[tex]{h} = \sqrt{ {6}^{2} \times 2 } [/tex]
[tex]{h}= \sqrt{ {6}^{2} } \times \sqrt{2} [/tex]
[tex]{h} = 6\sqrt{2} \: units[/tex]
The correct answer is C
The number –1,000 can be used to indicate a(n) A. increase in inventory of 1,000 units. B. harvest of 1,000 bushels of wheat. C. withdrawal of $1,000 from a checking account. D. receipt of $1,000.
Answer:
C. withdrawl of $1000 from a checking account
Step-by-step explanation:
When you withdraw money from a checking account, you lose the amount of money which you withdraw. In this situation, you are withdrawing $1000, which means you have -1000 dollars in your checking account
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A farmer wants to plant peas and carrots on no more than 200 acres of his farm. If x represents the number of acres of peas and y represents the number of acres of carrots for solution (x, y), then which is a viable solution?
A.(−50, 160)
B.(80, 160)
C.(75, −200)
D.(60, 135)
Answer:
Option D.(60, 135)
Step-by-step explanation:
Let
x -----> the number of acres of peas
y ----> the number of acres of carrots
we know that
The inequality that represent the problem is equal to
[tex]x+y\leq 200[/tex]
so
Verify each case
case A) (-50,160)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]-50+160\leq 200[/tex]
[tex]110\leq 200[/tex] ----> is true
The point is a solution for the inequality, but is not a viable solution because the number of acres can not be a negative number
case B) (80,160)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]80+160\leq 200[/tex]
[tex]240\leq 200[/tex] ----> is not true
therefore
The point is not a solution
case C) (75,-200)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]75-200\leq 200[/tex]
[tex]-125\leq 200[/tex] ----> is true
The point is a solution for the inequality, but is not a viable solution because the number of acres can not be a negative number
case D) (60,135)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]60+135\leq 200[/tex]
[tex]195\leq 200[/tex] ----> is true
therefore
The point is a viable solution
Kedar is comparing the costs of phone plans. For phone plan A, the cost is $15.00 to connect and then $0.02 per minute. For phone plan B, the cost is $4.69 to connect and then $0.08 per minute. For what usage does plan A cost the same as plan B? (Hint: Find total minutes and then convert to hours and minutes.)
3 h 2 min
2 h 42 min
3 h 12 min
2 h 52 min
Answer:
The answer is (D): 2 h 52 min
Step-by-step explanation:
write these equations separately as functions:
A(x)=15+0.02m
B(x)=4.69+0.08m
Since we want to find when they are equal, set them equal to each other:
15+0.02m=4.69+0.08m
simplifying this equation gives
10.31+0.02m=0.08m
subtract 0.02m from both sides
10.31=0.06m
divide both sides by 0.06
m=171.833 etc approximately equals 172 minutes
172 min/60=
2 hours and 52 min
Hope this helps!
Answer:
2 h 52 min
Step-by-step explanation:
Let m represent the number of minutes that will make the plans have equal cost. Equating the two plan costs, we have ...
15.00 + 0.02m = 4.69 + 0.08m
10.31 = 0.06 m . . . . . subtract 4.69+0.02m
10.31/0.06 = m = 171.8333... ≈ 172 . . . . minutes at equal cost
There are 60 minutes in an hour, so ...
172 minutes = 120 minutes + 52 minutes = 2 hours + 52 minutes
Plan A will cost the same as Plan B for 2 h 52 min of usage.
Pia printed two maps of a walking trail. The length of the trail on the first map is 8 cm. The length of the trail on the second map is 6 cm.
(a) 1 cm on the first map represents 2 km on the actual trail. What is the scale factor from the map to the actual trail? What is the length of the actual trail?
(b) A landmark on the first map is a triangle with side lengths of 3 mm, 4 mm, and 5 mm. What is the scale factor from the first map to the second map? What are the side lengths of the landmark on the second map? Show your work.
Answer:
Two trail maps:
Trail on the first map is 8 cm
Trail on the second map is 6 cm
Scale on first map is 1 cm : 2 km
A. What is the scale factor from the map to the actual trail?
For the first map, the scale factor is 1 cm: 2km. Therefore the actual trail is 8 centimeters * 2 kilometers = 16 km.
The scale factor of the second map is 16 km / 6 cm = 2.67 km : 1 cm
B. The length of the actual trail is 16 kilometers.
Answer:
Given:
Two trail maps:
Trail on the first map = 8 cm
Trail on the second map = 6 cm
Scale on first map = 1 cm : 2 km
A) What is the scale factor from the map to the actual trail?
For the first map, the scale factor is 1 cm: 2km. Therefore the actual trail is 8 centimeters * 2 kilometers = 16 km.
The scale factor of the second map is 16 km / 6 cm = 2.67 km : 1 cm
B) The length of the actual trail is 16 kilometers.
Step-by-step explanation:
A box has a base of 13 inches by 12 inches and a height of 30 inches. What is length of the interior diagonal of the box? Round to the nearest hundredth
The diagonal of a box is found by the formula:
Diagonal = √(length^2 + width^2 + height^2)
Diagonal = √(13^2 + 12^2 + 30^2)
Diagonal = √(169 + 144 + 900)
Diagonal = √1213
Diagonal = 34.828
Rounded to nearest hundredth = 34.83 inches.
Answer:
34.83 in
Step-by-step explanation:
See attached
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Sum of Arithmetic Series (Sigma Notation)
Find the numerical answer to the summation given below.
(Image Shown Below)
[tex]\displaystyle\sum_{n=2}^{91}(3n+8)=\sum_{n=1}^{91}(3n+8)-a_1=(*)\\\\S_n=\dfrac{a_1+a_n}{2}\cdot n\\a_1=3\cdot1+8=11\\n=91\\a_n=3n+8\\d=a_n-a_{n-1}\\a_2=3\cdot2+8=14\\d=14-11=3\\a_{91}=11+(91-1)\cdot 3=11+90\cdot3=281\\\\S_{91}=\dfrac{11+281}{2}\cdot91=146\cdot91=13286\\\\(*)=13286-11=\boxed{13275}[/tex]
Genet multiplied a 3-digit number by 1002 and got AB007C, where A, B, and C stand for digits. What was Genet's original 3-digit number?
Answer:
539
Step-by-step explanation:
The 007C requires the least significant two digits be in the range 35-39. In order for the 10-thousands digit to be zero, the sum of 1000 times the least digit of Genet's number and 2 times the hundreds digit must result in a sum with no thousands. About the only way to do that is to make the least digit 9 and the hundreds digit 5.
Then you have ...
539 × 1002 = 540078 . . . . . ABC =548
HELP!!!!!! Can someone help me write a system of linear inequalities to represent the graph? I'm having trouble since the one line is on the y-axis.
Answer:
x ≤ 0y ≤ 3/4x -5Step-by-step explanation:
The equation for the y-axis is x = 0. Your shading is to the left of that, so the inequality will be x ≤ 0.
The blue line has a rise of 3 for a run of 4, so its slope is 3/4. It intercepts the y-axis at y=-5, so the slope-intercept equation for that line is y = 3/4x -5. Since the shading is below that solid line, the inequality is ...
y ≤ 3/4x -5
9x 2 - 18x - 7 ÷ (3x + 1)
Answer:
The quotient is: 3x-7
The remainder is: 0
Step-by-step explanation:
We need to divide 9x^2 - 18x - 7 ÷ (3x + 1)
The Division is shown in the figure attached.
The quotient is: 3x-7
The remainder is: 0
This week, 300 tickets were sold to the school play. This is 120 percent of the number of tickets sold last week. How many tickets were sold last week for the school play?
Answer:
250 tickets
Step-by-step explanation:
300/1.2 or 120%
= 250 tickets
Answer:
250 tickets
Step-by-step explanation:
question is in the picture
Answer:
Step-by-step explanation:
The area of a triangle is:
A=05*b*h
b= base = x
h = height = x-12
A = (1/2)(x)(x-12)
Fill in the blank. if necessary, use the slash mark ( / ) for a fraction bar.if cos = , then tan = _____.
Answer:
5/4
Step-by-step explanation: