The given function f(x,y) = x^2 + y^2 + xy has a maximum value of 3/4 at (x,y) = (1/2,1/2) and a minimum value of -1/4 at (x,y) = (-1/2,1/2) inside the disk x^2 + y^2 < 1. There are no critical points inside the disk.
To find the critical points, we need to take partial derivatives of the function with respect to x and y and solve the resulting equations simultaneously.
fx = 2x + y = 0
fy = 2y + x = 0
Solving these equations, we get the critical point at (x,y) = (-1/2,-1/2) outside the disk. Hence, we do not consider it further.
Next, we need to find the boundary points of the disk, which is the circle x^2 + y^2 = 1. We can parameterize this circle as x = cos(t) and y = sin(t), where t ranges from 0 to 2π.
Substituting these values in the given function, we get:
f(cos(t), sin(t)) = cos^2(t) + sin^2(t) + cos(t)sin(t)
= 1/2 + 1/2sin(2t)
Now, we need to find the maximum and minimum values of this function. Since sin(2t) ranges from -1 to 1, the maximum value of the function is 3/4 when sin(2t) = 1, i.e., when t = π/4 or 5π/4. At these points, x = cos(π/4) = 1/2 and y = sin(π/4) = 1/2.
Similarly, the minimum value of the function is -1/4 when sin(2t) = -1, i.e., when t = 3π/4 or 7π/4. At these points, x = cos(3π/4) = -1/2 and y = sin(3π/4) = 1/2.
Therefore, the function has a maximum value of 3/4 at (x,y) = (1/2,1/2) and a minimum value of -1/4 at (x,y) = (-1/2,1/2) inside the disk x^2 + y^2 < 1. There are no critical points inside the disk.
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Kristin made a scatter plot to represent the relationship between the temperature and the number of bottles of water sold at her concession stand at a soccer tournament. Which is a description of the location of the point Kristin will add to represent the 13 bottles of water that were sold when the temperature was 39 degrees Fahrenheit? Select one:
O Top left of the scatter plot
OBottom right of the scatter plot
O Top right of the scatter plot
OBottom left of the scatter plotâ
The location of the point Kristin will add to represent the 13 bottles of water sold at 39 degrees Fahrenheit is the bottom left of the scatter plot.
A scatter plot represents the relationship between two variables. In this case, the temperature (independent variable) is plotted along the x-axis, while the number of bottles of water sold (dependent variable) is plotted along the y-axis. As the temperature increases, it is expected that more bottles of water would be sold.
The bottom left area of the scatter plot is where lower values of both temperature and the number of bottles sold would be found. Since 39 degrees Fahrenheit is relatively low and 13 bottles of water is a lower quantity, the point representing this data will be in the bottom left quadrant of the scatter plot.
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Complete question:
Kristin made a scatter plot to represent the relationship between the temperature and the number of bottles of water sold at her concession stand at a soccer tournament. Which is a description of the location of the point Kristin will add to represent the 13 bottles of water that were sold when the temperature was 39 degrees Fahrenheit? Select one:
O Top left of the scatter plot
O Bottom right of the scatter plot
O Top right of the scatter plot
O Bottom left of the scatter plotâ
Edwin fills 15 test tubes with a solution. each test tube contains 150 milliliters of solution.
how many liters of solution in all is there in the test tubes?
2.25 l
22.5 l
225 l
2,250 l
2.25 liters of solution in all is there in the test tubes. So, the correct option is 2.25 l.
The total volume of solution in the 15 test tubes can be calculated by multiplying the volume of one test tube by the number of test tubes:
Total volume = 15 test tubes × 150 milliliters/test tube
Total volume = 2,250 milliliters
However, the question asks for the answer in liters, so we need to convert milliliters to liters by dividing by 1,000:
Total volume = 2,250 milliliters ÷ 1,000
Total volume = 2.25 liters
Therefore, there are 2.25 liters of solution in all the test tubes.
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he figure below is a net for a rectangular prism. Side a = 62 centimeters, side b = 21 centimeters, and side c = 16 centimeters. What is the surface area of this figure?
The surface area of the rectangular prism is 4960 cm².
The rectangular prism can be divided into six rectangular faces, with opposite faces having the same area. To find the surface area, we need to calculate the area of each face and add them up.
The net shows three rectangles with dimensions of 62 cm x 21 cm, 62 cm x 16 cm, and 21 cm x 16 cm.
Therefore, the surface area of the rectangular prism is:
Area of the first rectangle = 62 cm x 21 cm = 1302 cm²
Area of the second rectangle = 62 cm x 16 cm = 992 cm²
Area of the third rectangle = 21 cm x 16 cm = 336 cm²
Total surface area = 2(Area of first rectangle) + 2(Area of second rectangle) + 2(Area of third rectangle)
= 2(1302) + 2(992) + 2(336)
= 2604 + 1984 + 672
= 4960 cm²
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to find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage of the disease, are selected. five mice receive the treatment and 4 do not. survival times, in years, from the time the experiment commenced are as follows: treatment 2.1 5.3 1.4 4.6 0.9 no treatment 1.9 0.5 2.8 3.1 at the 0.05 level of significance, can the serum be said to be effective? assume the two populations to be normally distributed with equal variances.
The serum be said to be effective can't be concluded, since the test statistic is less than the critical value, we fail to reject the null hypothesis.
Let [tex]n_A[/tex] denotes the number of mice which receiving treatment. Therefore,
[tex]n_A[/tex] = 5,
Let [tex]n_B[/tex] denotes the number of mice which do not receive treatment. Therefore, [tex]n_B[/tex] = 4
Survival times for the mice receiving the treatment are: 2.1; 5.3; 1.4; 4.6; 0.9
Survival times for the mice not receiving the treatment are: 1.9; 0.5; 2.8; 3.1
Let [tex]x_A[/tex] be the mean of survival time for the mice receiving the treatment and [tex]x_B[/tex] be the mean of survival time for the mice not receiving the treatment.
We have: [tex]x_A[/tex] = 2.86
[tex]x_B[/tex] = 2.075
Standard deviation be:
[tex]S_A=\sqrt{\frac{\sum (x_a-x_A)^2}{n_A-1} }[/tex]
[tex]=\sqrt{\frac{[(2.1-2.86)^2+(5.3-2.86)^2+(1.4-2.86)^2+(4.6-2.86)^2+(0.9-2.86)^2]}{4} }[/tex]
= 1.971
[tex]S_B=\sqrt{\frac{\sum (x_b-x_B)^2}{n_B-1} }[/tex]
[tex]=\sqrt{\frac{[(1.9-2.08)^2+(0.5-2.08)^2+(2.8-2.08)^2+(3.1-2.08)^2]}{3} }[/tex]
= 1.167
[tex]\mu_A[/tex] and [tex]\mu_B[/tex] are population means for the groups receiving the treatment and not receiving the treatment respectively.
Level of significance is α = 0.05
If P-value is less then 0.05, we will reject [tex]H_o[/tex]
The test statistic is,
[tex]t=\frac{(x_A-x_B)-(\mu_A-\mu_B)}{s_p\sqrt{\frac{1}{n1} +\frac{1}{n2} } }[/tex]
[tex]=\frac{2.86-2.07)-(0)}{1.674388\sqrt{\frac{1}{5} +\frac{1}4} } }[/tex]
= 0.79/1.123
t = 0.70
Degrees of freedom is,
[tex]d_f=n_A+n_B-2[/tex]
= 5 + 4 - 2
= 7.
According to the value in the table, the test's critical value is 1.895.
We are unable to reject the null hypothesis since the test statistic is less than the threshold value.
We thus cannot draw the conclusion that the serum is working.
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Find the next term in each sequence.
Question 1:
35, 29, 23, 17, ?.
Question 2:
1, 2, 5, 10, ?.
Please Include an Explanation of how to solve problems like this!
Thanks a ton!
1. For the sequence : 35, 29, 23, 17, ?; the next term is 11
2. For the sequence : 1, 2, 5, 10, ?; the next term is 17
Calculating the term in a sequenceFrom the question, we are to calculate the next term in each of the given sequence
From the given sequence,
35, 29, 23, 17, ?.
To determine the next term, we will determine the common difference
Common difference = Second term - First term
Common difference = 29 - 35
Common difference = -6
Thus,
To determine the next term, we will add the common difference to the last term
That is,
17 + - 6 = 17 - 6
= 11
The next term is 11
For the sequence 1, 2, 5, 10, ?.
Common difference = successive odd numbers
To get the second term, we will add to the first term the first natural odd number
To get the third term, we will add to the second term the second natural odd number
And so on.
In the given sequence, we are to determine the 5th term
Thus,
We will add to the fourth term, the fourth natural odd number
That is,
10 + 7 = 17
Hence, the next term is 17
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Find the work done by the force field F(x,y) = x^2i – ryj in moving a particle along the F semicircle y = Sqrt(4 – x^2) from P(2,0) to Q(-2,0) and then back along the line segment from Q to P.
The work done by the force field F along the semicircle and the line segment is 32/3.
The work done by a force field F along a curve C from point A to point B is given by the line integral:
W = ∫ F dot dr
where dot represents the dot product and dr is the differential displacement vector along the curve C.
Let's divide the curve C into two parts: the semicircle from P to Q, denoted by C1, and the line segment from Q to P, denoted by C2.
For C1, the curve can be parameterized as x = 2cos(t) and y = 2sin(t) for t in [0, pi]. The differential displacement vector is then given by:
dr = (-2sin(t) dt)i + (2cos(t) dt)j
The force field F(x,y) = x^2i - ryj, so we have:
F(x,y) = (2cos^2(t))i - (2rsin(t))j
The dot product F dot dr is then:
F dot dr = (2cos^2(t))(-2sin(t) dt) + (2rsin(t))(2cos(t) dt)
= -4cos^2(t)sin(t) dt + 4rcos(t)sin(t) dt
= 4sin(t)cos(t)(r - cos(t)) dt
Therefore, the work done along C1 is:
W1 = ∫ C1 F dot dr
= ∫[0, pi] 4sin(t)cos(t)(r - cos(t)) dt
This integral can be evaluated using the substitution u = cos(t), du = -sin(t) dt:
W1 = -∫[1, -1] 4u(r - u) du
= 4r∫[1, -1] u du - 4∫[1, -1] u^2 du
= 0
Hence, the work done along C1 is 0.
For C2, the curve is simply the line segment from Q(-2,0) to P(2,0), which is parallel to the x-axis. Therefore, the differential displacement vector is given by:
dr = dx i
where i is the unit vector in the x-direction. The force field is the same as before, F(x,y) = x^2i - ryj. Along C2, y = 0, so the force field reduces to:
F(x,y) = x^2i
The dot product F dot dr is then:
F dot dr = x^2 dx
Therefore, the work done along C2 is:
W2 = ∫ C2 F dot dr
= ∫[-2, 2] x^2 dx
= 32/3
Hence, the work done along C2 is 32/3.
The total work done along the curve C is the sum of the work done along C1 and C2:
W = W1 + W2 = 0 + 32/3 = 32/3
Therefore, the work done by the force field F along the semicircle and the line segment is 32/3.
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Suppose $40,000 is deposited into an account paying 2. 5% interest, compounded continuously.
How much money is in the account after eight years if no withdrawals or additional deposits are made?
The formula for calculating the amount of money in an account with continuous compounding is:
[tex]A = Pe^{(rt)}[/tex]
where A is the amount of money in the account, P is the principal (initial deposit), e is the mathematical constant e (approximately equal to 2.71828), r is the interest rate (expressed as a decimal), and t is the time (in years).
Plugging in the given values, we get:
A =[tex]40000 * e^{(0.025 * 8)[/tex]
Using a calculator, we find that [tex]e^{(0.025 * 8)[/tex] is approximately 1.2214, so:
A = 40000 * 1.2214 = $48,856.12
Therefore, the amount of money in the account after eight years with continuous compounding is $48,856.12.
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What is the x intercept of f(x)= 2x^2+5x+3
Answer: x intercepts = (-1.5,0) and (-1,0)
Step-by-step explanation: Graphed it in desmos :)
Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. Domestic Traveler Spending in the U.S., 1987-1999 Spending (dollars in billions) A graph titled Domestic Traveler Spending in the U S from 1987 to 1999 has year on the x-axis, and spending (dollars in billions) on the y-axis, from 225 to 450 in increments of 25. Year Source: The World Almanac, 2003 a. positive correlation; as time passes, spending increases. b. no correlation c. positive correlation; as time passes, spending decreases. d. negative correlation; as time passes, spending decreases.
There is a positive correlation and as such as time passes, spending increases.
Checking the correlation of the graphThe descriptions of the graph from the question are given as
Year (x - axis): 1987 to 1999Spending (y - axis, dollars in billions) 225 to 450 in increments of 25.From the above statements, we can make the following summary
As the year increase, the spending also increase
The above summary is about the correlation of the graph
And it means that there is a positive correlation and as such as time passes, spending increases.
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Helppppppppppppppppp
if you increase your reading speed so that each page takes you 30 seconds less than it did before, and you begin reading 20 minutes per day, how many 200 page books can you now read in a year
We can read 91 books in a year if you increase your reading speed so that each page takes you 30 seconds less than it did before.
How is the number of books calculated?Now If each page now takes 30 seconds less to read than before,
then you will save 30*200 = 6000 seconds (or 100 minutes) on each book.
So, the time it will take you to read a 200-page book will be
20 minutes - 100 minutes = -80 minutes,
which means you will finish a 200-page book in 80 minutes (or 1 hour and 20 minutes).
In a year, there are 365 days. If you read for 20 minutes per day, then you will read for a total of
365 * 20 = 7300 minutes (or 121.67 hours) in a year.
Since you can finish a 200-page book in 80 minutes, you can read 7300/80 = 91.25 books in a year.
However, since you cannot read a fraction of a book, the maximum number of 200-page books you can read in a year is 91 books.
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Complete question is :
If you increase your reading speed so that each page takes you 30 seconds less than it did before,
and you begin reading 20 minutes per day,
How many 200 page books can you now read in a year?
The equation y=mx+b
is used to express the equation of a line. Which solution is a correct way to solve this equation for m
in terms of y
?
The correct way to solve this equation for m in terms of y is m = b - y/x
How to determine the valueIt is important to note that subject of formula is the variable that is made to stand alone in an equation.
It is described as the variable that is being worked out in an equation.
The subject of formula in an equation is worked to stand alone on on end of the equality sign.
Example of equations
x = y - 2
The variable 'x' is the subject of formula
From the information given, we have that;
y= mx+b
collect the terms
mx = b - y
Divide by the coefficient
m = b - y/x
The equation for m is m = b - y/x
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The complete question:
The equation y=mx+b is used to express the equation of a line. Which solution is a correct way to solve this equation for m in terms of y?
Joshua's mail truck travels 14 miles every day he works
and is not used at all on days he does not work. At the
end of his 100th day of work the mail truck shows a
mileage of 76,762. Model Joshua's truck mileage as a
function of the number of days he has worked. When
will he reach 100,000 miles?
Solving the equation, Joshua will reach 100,000 miles after approximately 1,760 days of work.
To model Joshua's truck mileage as a function of the number of days he has worked, we can use the following equation:
Mileage (M) = 14 * Number of days worked (D) + Initial Mileage (I)
First, we need to determine the initial mileage on the mail truck. To do this, we can use the information given for his 100th day of work:
76,762 = 14 * 100 + Initial Mileage
76,762 = 1,400 + Initial Mileage
Initial Mileage (I) = 76,762 - 1,400
Initial Mileage (I) = 75,362
Now we can rewrite the equation as:
Mileage (M) = 14 * Number of days worked (D) + 75,362
To find when Joshua will reach 100,000 miles, we can set M equal to 100,000 and solve for D:
100,000 = 14 * D + 75,362
24,638 = 14 * D
D ≈ 24,638 / 14
D ≈ 1,759.857
Since Joshua cannot work a fraction of a day, he will reach 100,000 miles after approximately 1,760 days of work.
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Una persona observa una torre desde una distancia de 100m con un angulo de elevación de 70, con que función trigonométrica obtendrías la altura de la torre? Calcula la altura de la torre
The height of the tower is: 274.7m
How to solveTo find the height of the tower, we will use the tangent trigonometric function.
The tangent function relates the angle of elevation to the ratio of the opposite side (height of the tower) to the adjacent side (distance from the observer to the tower).
In this case, the angle of elevation is 70°, and the distance from the observer to the tower is 100 meters.
The formula we will use is:
tan(θ) = opposite / adjacent
tan(70°) = height / 100m
To calculate the height, we will rearrange the formula:
height = 100m * tan(70°)
Using a calculator, we find that tan(70°) ≈ 2.747.
Therefore, the height of the tower is: 274.7m
height ≈ 100m * 2.747 ≈ 274.7m
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The question in English is:
A person observes a tower from a distance of 100m with an elevation angle of 70, with which trigonometric function would you obtain the height of the tower? Calculate the height of the tower
Find the mean absolute deviation (MAD) of the data in the pictograph below. Baskets
The key says one basketball picture equals two baskets. The key says one basketball picture equals two baskets. A picture graph labeled Baskets each student made. The vertical axis is labeled Baskets made. The horizontal axis is labeled Student. The names from left to right on the horizontal axis are Reynaldo, Marcelle, Allie, and Fernando. There are two basketball pictures above Reynaldo. There are four basketball pictures above Marcelle. There are three basketball pictures above Allie. There are five basketball pictures above Fernando
The mean absolute deviation (MAD) of the data in the pictograph is equals to the one basketball.
We have a data in the pictograph. In mathematics, a pictograph is a pictorial representation of data using images, icons. It is also known as a pictogram. We have a pictograph, in which the vertical axis is labeled Baskets made and the horizontal axis is labeled Student. Here, one basketball picture equals two baskets. Mean absolute deviation (MAD) is a statistical measure of the average absolute distance between each data value and the mean of a data set. It is a parameter or statistic that measures the spread, or variation, in data.
Mean is defined as the sum of data values divided by number of values.
Sum of data values = 4 + 4×2 + 3×2 + 5×2
= 28
So, mean = 28/4 = 7
Now, | 4 - 7| + |8 - 7| + |6 -7 | + | 10 - 7|
= 3 + 1 + 1 + 3 = 8
So,, mean absolute deviation (MAD) of the data = 8/7 = 1.1 ~ 1. Hence, required value is 1.
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Complete question :
Find the mean absolute deviation (MAD) of the data in the pictograph below. Baskets
The key says one basketball picture equals two baskets. The key says one basketball picture equals two baskets. A picture graph labeled Baskets each student made. The vertical axis is labeled Baskets made. The horizontal axis is labeled Student. The names from left to right on the horizontal axis are Reynaldo, Marcelle, Allie, and Fernando. There are two basketball pictures above Reynaldo. There are four basketball pictures above Marcelle. There are three basketball pictures above Allie. There are five basketball pictures above Fernando
in KLMO, OM-25. What are the coordinates of M and K?
L(2b+4c4d)
ㅁ
M
The value of coordinate M and K are,
M = (25, 0)
K = (2b + 4c - 25, 2d)
Given that;
In KLMO,
OM = 25
L = (2b + 4c , 4d)
Hence, We can formulate;
The value of coordinate of M is,
M = (25, 0)
Since, M lies on x - axis.
Let the coordinate of K is,
K = (x, y)
Hence, Midpoint of LO is same as midpoint of KM.
Midpoint of LO is,
(0 + 2b + 4c / 2, 4d/2)
(b + 2c, 2d)
Midpoint of KM is
(x + 25/2, y + 0/2)
(x + 25/2 , y/2)
By comparing,
x + 25/2 = b + 2c
x + 25 = 2b + 4c
x = 2b + 4c - 25
y/2 = 2d
y = 2d
Thus, the coordinate of K is,
K = (2b + 4c - 25, 2d)
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For the cost function C(x) = 6000 + 242 + 0.005.03 find: A) The production level that will minimize the average cost. B) The minimal average cost.
To find the production level that will minimize the average cost, we need to differentiate the cost function with respect to x and set it equal to zero. So:
C'(x) = 0.005x^2 + 242x + 6000
0 = 0.005x^2 + 242x + 6000
Using the quadratic formula, we get:
x = (-242 ± sqrt(242^2 - 4(0.005)(6000))) / (2(0.005))
x = (-242 ± sqrt(146416)) / 0.01
x = (-242 ± 382) / 0.01
x = -14,000 or 27,000
Since the production level cannot be negative, we can discard the negative solution and conclude that the production level that will minimize the average cost is 27,000 units.
To find the minimal average cost, we need to plug the production level back into the cost function and divide by the production level. So:
C(27,000) = 6000 + 242(27,000) + 0.005(27,000)^2
C(27,000) = 6,594,000
Average cost = C(27,000) / 27,000
Average cost = 6,594,000 / 27,000
Average cost ≈ 244.22
Therefore, the minimal average cost is approximately $244.22.
To answer your question, first, let's correct the cost function, which should be in the form of C(x) = Fixed cost + Variable cost. Assuming it is C(x) = 6000 + 242x + 0.005x^2.
A) To find the production level that will minimize the average cost, we need to first determine the average cost function, which is AC(x) = C(x)/x. So, AC(x) = (6000 + 242x + 0.005x^2)/x.
Now, find the first derivative of AC(x) concerning x, and set it equal to zero to find the minimum point:
d(AC(x))/dx = 0
The first derivative of AC(x) is:
d(AC(x))/dx = (242 + 0.010x - 6000/x^2)
Setting this to zero and solving for x will give us the production level that minimizes the average cost:
242 + 0.010x - 6000/x^2 = 0
Now, you can solve for x using numerical methods, such as Newton-Raphson or others. After solving for x, you will get the production level that minimizes the average cost.
B) To find the minimal average cost, plug the production level x you found in part A into the average cost function, AC(x):
Minimal Average Cost = AC(production level)
This will give you the minimal average cost for the given cost function.
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The sum of five consecutive odd integers is 235. What is the greatest of
these integers?
Answer:
x + x + 2 + x + 4 + x + 6 + x + 8 = 235
5x + 20 = 235
5x = 215, so x = 43
The integers are 43, 45, 47, 49, and 51.
The greatest of these integers is 51.
Christian is rewriting an expression of the form y = ax2 bx c in the form y = a(x – h)2 k. which of the following must be true? h and k cannot both equal zero k and c have the same value the value of a remains the same h is equal to one half –b
The value of 'a' remains the same, 'h' is equal to -b/(2a), and 'h' and 'k' cannot both equal zero.
When rewriting a quadratic expression of the form y = ax^2 + bx + c into the vertex form y = a(x - h)^2 + k, the following must be true:
1. The value of 'a' remains the same in both expressions, as it represents the parabola's vertical stretch or compression.
2. 'h' is equal to -b/(2a), which is derived from completing the square to transform the standard form into the vertex form.
3. 'k' and 'c' do not necessarily have the same value. 'k' is the value of the quadratic function when 'x' equals 'h', which can be found by substituting 'h' back into the original equation and solving for 'y'.
4. 'h' and 'k' cannot both equal zero, unless the vertex of the parabola is at the origin (0,0).
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HELP ME PLEASE I WILL GIVR BRAINLIEST TO THE FASTED CORRECT ANSWER PLEASE HELP ME FAST AND TY
Which equations are true for x = –2 and x = 2? Select two options x2 – 4 = 0 x2 = –4 3x2 + 12 = 0 4x2 = 16 2(x – 2)2 = 0
Answer:
Here we try the method of trial and error to find out if the equations have a common answer as zero
Step-by-step explanation:
Now,
(a) x²-4=0
(b) x²=-4
(c) 3x²+12=0
(d) 4x²=16
(e) 2(x-2)2=0
Here if we check the first equation i.e x²-4=0
Equating - 2²-4=4-4
=0
2²-4= 4-4
=0
So option (a) is true
Now x²=-4
Substituting, - 2²=4 & 2²=4
So here we get 4≠-4
Therefore, (b) is not true
Now 3x²+12=0
3(-2)²+12= 3(4)+12
=12+12= 24
3(2)²+12= 12+12
= 24
4x²=16
Substituting, 4(-2)²= 4(4)= 16
4(2)²= 4(4)= 16
So option (d) is also true
Now, 2(x-2)2=0
Substituting, 2(-2-2)2= 2(-4)2
4(-4)= - 16
2(2-2)2= 2(0)2
=4(0)=0
Here when we put x=-2, we get - 16
when we put x=2, we get 0
So the following equation is true only for x=2 and not x=-2
I hope this helps ;)
Can someone help with this math equation from study island…????
The solution of the exponents is shown below.
What is the solution of the exponents?Exponents are mathematical shorthand for multiplying a number by itself a certain number
We have that;
5^n = 1
5^n = 5^0
n = 0
2) 2^-7/2^n = 2^2
2^-7 - n = 2^2
-7 - n = 2
-n = 2 + 7
n = -9
3) 6^5 * 6^n = 6^1
6^ 5 + n = 6^1
5 + n = 1
n = 1 - 5
n = -4
4) (8^n)^7 = 8^21
8^7n = 8^21
7n = 21
n = 3
5) 4^n = (1/4)
4^n = 4^-1
n = -1
In each of the cases, we have applied the laws of the exponents as we know them.
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The equation of the forms are matched as;
2⁻⁷/2ⁿ = 2², n = -9
6⁵ * 6ⁿ = 6. n = -4
(8ⁿ)⁷ = 8²¹, n = 3
4ⁿ = 1/4 , n = -1
What are index forms?Index forms are simply described as mathematical forms that are used to represent numbers of variables that are too large or small.
To multiply index forms, you need to add the exponents of the same bases.
To divide index forms, you need to subtract the exponents of the same bases.
From the information given, we have that;
2⁻⁷/2ⁿ = 2²
cross multiply the values
2⁻⁷ = 2²⁺ⁿ
Then,
-7 = 2 + n
n = -9
6⁵ * 6ⁿ = 6
take the exponents
5 + n= 1
n =- 4
(8ⁿ)⁷ = 8²¹
We have;
7n = 21
Make 'n' the subject
n = 3
4ⁿ = 1/4
4ⁿ = 4⁻¹
n =-1
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Copy and complete the equation of line B below. y = — 84 NWPца - 0 7- 6+ 5- 4- 3- 2- 1/ -11 -2- -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 ܢܐ ܚ ܩ ܘ ܘ ܢ -3 -4- -5 -6 x +_ -7 -8- Line B 19
The equation of the line passing through the given points is y = 3x-1.
Given that is a line passing through two points (0, 2) and (-1, -1) we need to find the equation of the line using them,
We know that the equation of a line passing through points (x₁, y₁) and (x₂, y₂) is =
y-y₁ = y₂-y₁ / x₂-x₁ (x-x₁)
Here (x₁, y₁) and (x₂, y₂) are (0, 2) and (-1, -1),
Therefore, the required equation is =
y+1 = -1-2/-1 (x-0)
y+1 = 3x
y = 3x-1
Hence, the equation of the line passing through the given points is y = 3x-1.
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find the next two terms in sequence?
Answer:
83 and 99 The sequence is going in +16 so the answer is 83 and 99 sorry if I am a bad explainer I’m new to this app :(
Step-by-step explanation:
Joey needs to buy chocolate milk they only sell it in pint containers how many pint containers of chocolate milk should he buy
To determin how many pint containers of chocolate milk Joey should buy, we need to know how much chocolate milk he wants in total.
Let's assume Joey wants to buy "x" pints of chocolate milk. Then, the total amount of chocolate milk he will get in quarts can be calculated as:
Total amount of chocolate milk = x pints/2 pints per quart = x/2 quarts
If Joey has a specific total amount of chocolate milk he wants to buy, we can solve for "x". For example, if Joey wants to buy 3 quarts of chocolate milk, we can set up the following equation:
x/2 = 3
Multiplying both sides by 2, we get:
x = 6
So, Joey needs to buy 6 pint containers of chocolate milk to get 3 quarts in total.
If Joey has a different desired amount of chocolate milk, we can adjust the equation accordingly to find the number of pint containers he needs to buy
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The guidance department has reported that of the senior class 2. 3% are members of key club 8. 6% are enrolled in AP physics and 1. 9% are in both
The percentage is 9.0% of the senior class are either members of the Key Club, enrolled in AP Physics, or both.
We need to find the percentage of seniors who are either members of the Key Club, enrolled in AP Physics, or both. We can use the formula:
Total percentage = Key Club percentage + AP Physics percentage - Both percentage
Step 1: Identify the given percentages
Key Club percentage = 2.3%
AP Physics percentage = 8.6%
Both percentage = 1.9%
Step 2: Apply the formula
Total percentage = 2.3% + 8.6% - 1.9%
Step 3: Calculate the result
Total percentage = 9.0%
So, 9.0% of the senior class are either members of the Key Club, enrolled in AP Physics, or both.
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Penny needs 12 ounces of a snack mix that is made up of chocolate and almonds. Chocolate cost $3. 50 per ounce and almonds cost $4. 50 per ounce. Penny has $50 to spend and plans to sell it all. X the amount of chocolate and Y is the amount of almonds. Determine which equations you are used to form a system of equations for the scenario
The two equations which can be used to form a system of equations for the scenario are X + Y = 12 and 3.50X + 4.50Y = 50
To solve this problem, we need to form a system of equations. Let X be the amount of chocolate and Y be the amount of almonds. The first equation we can form is based on the total amount of snack mix that Penny needs, which is 12 ounces:
X + Y = 12
The second equation we can form is based on the cost of the ingredients. We know that chocolate costs $3.50 per ounce and almonds cost $4.50 per ounce. If X is the amount of chocolate and Y is the amount of almonds, then the total cost of the snack mix will be:
3.50X + 4.50Y = 50
This equation represents the fact that Penny has $50 to spend on the snack mix. Now we have a system of two equations that we can use to solve for X and Y. We can use substitution or elimination to solve the system and find the values of X and Y that satisfy both equations.
Once we have those values, we can check that they add up to 12 and that the total cost is $50. This system of equations allows us to calculate the amount of chocolate and almonds Penny needs to make the snack mix within her budget.
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Why do you think that credit cards tend to be the entry point for establishing credit for so many consumers?
I believe credit cards tend to be the entry point for establishing credit for so many consumers because they provide an easy and accessible way for individuals to begin building their credit history. Credit card companies report to credit bureaus on a regular basis, which helps establish a credit score and credit history.
Additionally, credit cards offer a convenient way for individuals to make purchases and build their credit at the same time. However, it is important for individuals to use their credit cards responsibly and make timely payments in order to maintain good credit standing.
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At her job, Avery earns $120 per week plus a one-time $300 bonus. Janelle teaches art lessons and earns $24 per week plus a $60 art supply fee for each student she teaches. a. System of equations:
The system of equations to describe the earnings by Avery and Janelle would be:
Avery's earnings: y = 120x + 300
Janelle's earnings: y = 24x + 60s
How to find the system of equations ?The problem provides two scenarios with different methods for earning money. Avery earns a fixed amount of $120 each week, in addition to a one-time bonus of $300. To represent this situation as an equation, we can use the formula:
y = 120x + 300
where y is Avery's total earnings, x is the number of weeks she works, and 300 is the one-time bonus she receives.
For Janelle, her earnings consist of a fixed weekly rate of $24 plus a variable amount based on the number of students she teaches.
We can represent Janelle's earnings as an equation using the formula:
y = 24x + 60s
where y is Janelle's total earnings, x is the number of weeks she works, and s is the number of students she teaches.
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Question
Write the product using exponents.
(−13)⋅(−13)⋅(−13)
Answer:
(-13)^3
Step-by-step explanation:
Exponents can be used for repeated multiplication.
In this case, the number "negative 13" is repeated several times, all connected with multiplication.
There are a total of three "negative 13"s being multiplied together ("negative 13" appears three times on the page).
To rewrite using exponents, we would write one of the following:
(-13)^3
[tex](-13)^3[/tex]
