Given a triangle ABC at points A = (-6, 3) B=(-4,7) C = (-2, 3), and a first transformation of up 2 and right 3, and a second transformation of down 1 and left 6, the final point B" is located at (-7, 8), which is option (a).
What is transformation?In mathematics, a transformation refers to a change or mapping of one set of values to another set. In geometry, a transformation involves changing the position, size, or orientation of a geometric object.
To find the location of the final point B", we need to apply the two transformations given to the initial point B (-4, 7).
First transformation: up 2 and right 3
This means we add 2 to the y-coordinate (to move the point up) and add 3 to the x-coordinate (to move the point right).
So, the new location of B after the first transformation is: (-4 + 3, 7 + 2) = (-1, 9)
Second transformation: down 1 and left 6
This means we subtract 1 from the y-coordinate (to move the point down) and subtract 6 from the x-coordinate (to move the point left).
So, the new location of B" after the second transformation is: (-1 - 6, 9 - 1) = (-7, 8)
Therefore, the final point B" is located at (-7, 8), the correct option is a.
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400
300
200
100
2
4
6
8
X
Is the graphed function linear?
Yes, because each input value corresponds to
exactly one output value.
O Yes, because the outputs increase as the inputs
increase.
O No, because the graph is not continuous.
O No, because the curve indicates that the rate of
change is not constant.
It exists not linear the curve indicates that the rate of change exists not constant.
What is a linear Function?
Linear functions exist as those whose graph exists as a straight line. A linear function exists that can be characterized by the equation
y = mx + c, Where m exists the slope and c exists the intercept on the
y-axis. A linear function contains one independent variable and one dependent variable.
Therefore, the correct answer is option c) No, because the curve indicates that the rate of change is not constant.
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A right circular cylinder has a diameter of 12 in and a height of 12 in. if water is flowing in at the rate of 4π in3 per minute, find the rate of change of the height when the height is 4 in?
The rate of change of the height is [tex]\frac{1}{9}[/tex] in/min.
What is a right circular cylinder?A right circular cylinder is a cylinder that has a closed circular surface having two parallel bases on both the ends and whose elements are perpendicular to its base. It is also called a right cylinder.
Volume of the cylinder = [tex]\pi r^{2}h[/tex]
Rate of change of height is [tex]\frac{dh}{dt}[/tex] = ?
At any time t,
Volume, V = [tex]\pi r^{2}h[/tex]
Since r does not change with time, then r is a constant, so,
V = [tex]\pi (6)^{2}h[/tex]
V = [tex]36\pi h[/tex]
Differentiate both sides with respect to time t,
[tex]\frac{dV}{dt} = 36\pi \frac{dh}{dt}[/tex] -------------(i)
Since [tex]\frac{dV}{dt}[/tex]is given as 4[tex]\pi[/tex] cu.in. per min,
[tex]4\pi = 36\pi \frac{dh}{dt}[/tex]
[tex]\frac{dh}{dt} = \frac{4\pi }{36\pi }[/tex]
[tex]\frac{dh}{dt} = \frac{1 }{9 }[/tex] in/min.
Hence, The rate of change of the height is [tex]\frac{1}{9}[/tex] in/min.
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In the power function f(x) = -2x, what is the end behavior of f(x) =3^-x 2 as x goes to [infinity]?
The end behavior of the power function is as x tends to infinity, f(x) tends to zero.
In this question,
The power function is [tex]f(x)=3^{-x}2[/tex]
The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.
The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
The graph below shows the behavior of the function f(x).
The above equation has the degree of 1, which is odd and the leading coefficient has the positive coefficient.
Then the end behavior is
As x -> ∞,
⇒ [tex]\lim_{x \to \infty} f(x)[/tex]
⇒ [tex]\lim_{x \to \infty} 3^{-x}2[/tex]
⇒ [tex]2\lim_{x \to \infty} 3^{-x}[/tex]
⇒ [tex]2\lim_{x \to \infty} 3^{-\infty}[/tex]
⇒ 2(0)
⇒ 0
Thus as x → ∞, f(x) → 0.
Hence we can conclude that the end behavior of the power function is as x tends to infinity, f(x) tends to zero.
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Among Davids classmates, there are twice as many boys as girls. Which of the following numbers may represent the number of all the children in his class?
The number that could represent the number of students in class is = X + 2x
Calculation and f the total number of studentsLet us take the number of boys to be = X
Then the number of girls is twice that of the boys. That is, y = 2x
Therefore the total number of students in the class z = X + 2x
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An apartment building contains 12 units consisting of one- and two-bedroom apartments that rent for $360 and $450 per month, respectively. When all units are rented, the total monthly rental is $4,950. What is the number of two-bedroom apartments?
Answer:
There are 7 two bed bedroom apartments.
Step-by-step explanation:
Solution Given:
let the one bedroom apartment be x and two bedroom apartment be y.
By the question we can make an equation as
x+y=12
or y=12-x............................[1]
Again:
$360x+$450y=$4,950
Now substituting value of y, we get
$360x+$450(12-x)=$4,950
Opening Bracket
$360x+$5400-$450x=$,4,950
solving equation
-90x=-450
dividing both side by -90,we get
x=5
again,
substituting value of x in equation 1,we get
y=12-5=7 two bed bedroom apartments.
Let, x represents the number of one bedrooms and 12 - x represents the number of two bedrooms.
According to the question,
[tex]360x + 450(12 - x) = 4950[/tex]
[tex]360x + 5400 - 450x = 4950[/tex]
[tex]5400 - 4950 = 450x - 360x[/tex]
[tex]450 = 90x[/tex]
[tex] \frac{450}{90} = x[/tex]
[tex]5 = x[/tex]
Therefore, one bedrooms = x = 5
For number of two bedrooms which is equal to 12 - x
= 12 - 5
= 7
So your required answer is 7
The diameter of the wheel in wades el camino is 15 inches. If the wheel made 5 complete revaluations how far would the wheel have rolled? Round your answer to the nearest hundredth of an inch
Step-by-step explanation:
solution
here
15×5
=75inches
A fourth-degree polynomial with a leading coefficient of 1 has gone through several transformations, including a vertical compression by a scale factor of 1/3 and a reflection across the x-axis. Two of the zero polynomials are -i and 3i.
Find a y-intercept of this polynomial, if it exists
The y-intercept of the resulting polynomial is equal to - 3.
How to derive a fourth-degree polynomial generated by rigid transformations
Herein we assume that the other two zeros of the polynomial are i and - i 3, otherwise the polynomial will have at least a complex number as coefficient of the expression. This is because we need to find values from a Cartesian plan, whose ordered pairs are real numbers.
Initially, we have the following expression by algebra properties:
f(x) = (x + i) · (x - i) · (x + i 3) · (x - i 3)
f(x) = (x² + 1) · (x² + 9)
f(x) = x⁴ + 10 · x² + 9
Then, we proceed to use the two rigid transformations described in the statement. Please notice that rigid transformations are transformations applied on polynomials such that Euclidean distance is conserved:
Vertical compression
f'(x) = (1 / 3) · f(x)
f'(x) = (1 / 3) · (x⁴ + 10 · x² + 9)
f'(x) = (1 / 3) · x⁴ + (10 / 3) · x² + 3
Reflection across the x-axis
g(x) = - f'(x)
g(x) = - (1 / 3) · x⁴ - (10 / 3) · x² - 3
The y-intercept is found for x = 0:
g(0) = - (1 / 3) · 0⁴ - (10 / 3) · 0² - 3
g(0) = - 3
The y-intercept of the resulting polynomial is equal to - 3.
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If you were to take a cross section of a right circular cone parallel to its base, what shape would you get?
The cross section of a right circular cone parallel to its base is a circle.
What is a three dimensional shape?A three dimensional shape is a shape that has length, width and height. Examples of three dimensional shapes are cone, cylinder, prism and pyramid.
A two dimensional shape is a shape that have both length and width. Examples of two dimensional shape are circle, rectangle, square.
A cone is a three-dimensional solid geometric shape having a circular base and a pointed edge at the top
The cross section of a right circular cone parallel to its base is a circle.
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Find the values of a and b.
Answer:
a = 144
b = 67
Step-by-step explanation:
Where a transversal meets parallel lines, consecutive interior angles are supplementary.
Anglesa° +36° = 180° . . . . . . consecutive angles are supplementary
a = 144 . . . . . . . . . . divide by ° and subtract 36
b° +113° = 180° . . . . . . consecutive angles are supplementary
b = 67 . . . . . . . . . . divide by ° and subtract 113
**The equation y = 15x + 100 can be used to model the amount of money, y, that has
been saved by Jared for his bike based on the number of months that have passed. What
does the coordinate (0, 100) mean in the context of the problem?
If the equation is y=15x+100 then the coordinate (0,100) of the equation shows that Jared had $100 when he started his saving.
Given an equation y=15x+100 that shows the amount of money saved by Jared for his bike based on the number of months that have passed.
We are required to find what the coordinate (0,100) mean in the context of the equation.
Equation is like relationship between two or more variables that are expressed in equal to form. Equations of two variables look like ax+by=c.It may be linear equation, quadratic equation, cubic equation or many more dependng on the power of variable present in that equation.
When we put x=0 in the equation we get the following:
y=15*0+100
y=100
It means that he had $100 in the beginning when he just started doing his savings.
Hence if the equation is y=15x+100 then the coordinate (0,100) of the equation shows that Jared had $100 when he started his saving.
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please help i dont know to do it please help
Answer:
5/10
Step-by-step explanation:
0.5 is 5/10 as a decimal. Any number that can be written as a fraction or a ratio is a rational number, so, since 0.5 can be written as 5/10, it’s a rational number.
0.7 as a fraction or mixed number
5. During the season, the Rangers played 25 games. They won 14 of the games.
What fraction and percent of the games did they lose? Show your work in the
space below. Remember to check your solution.
to find the fraction of losses, we need to divide the total amount of losses by the total amount of games played.
if we know that the Rangers won 14 games but they played 25, we can write an equation to find the amount of games lost.
14 + n = 25, where n equals the amount of losses.
now we isolate by moving the 14 to the other side. to do this, we must subtract 14 from both sides, leaving us with n = 11. now we know they lost 11 games.
to find the fraction of lost games, all we need to do is divide 11 by 25 without completing the division. that leaves us with 11/25.
the Rangers lost 11/25 games during the season
hope this helped!!
If two 12-sided dice are rolled what is the probability that both numbers will be even?
Answer:
probability result of sample space would result in 12/144
Step-by-step explanation:
as result there is 12 numbers on each dice, blth have exact same numbers, resulting in 144 combinations ti be thrown as sample space event outcome wluld be;
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12
2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 2/9 2/10 2/11 2/12
3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 3/9 3/10 3/11 3/12
4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 3/9 4/10 4/11 4/12
5/1 5/2 5/3 5/4 5/5 5/6 5/7 5/8 5/9 5/10 5/11 5/12
6/1 6/2 6/3 6/4 6/5 6/6 6/7 6/8 6/9 6/10 6/11 6/12
7/1 7/2 7/3 7/4 7/5 7/6 7/7 7/8 7/9 7/10 7/11 7/12
8/1 8/2 8/3 8/4 8/5 8/6 8/7 8/8 8/9 8/10 8/11 8/12
9/1 9/2 9/3 9/4 9/5 9/6 9/7 9/8 9/9 9/10 9/11 9/12
10/1 10/2 10/3 10/4 10/5 10/6 10/7 10/8 10/9 10/10 10/11 10/12
11/1 11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9 11/10 11/11 11/12
12/1 12/2 12/3 12/4 12/5 12/6 12/7 12/8 12/9 12/10 12/11 12/12
giving 12 matching numbers in a 144 possible rolled outcome
mathematic formula would be P(A) = n(A)/n(S)
As P(A)= probabolity of event A
n(A)= number of favouravle outcome
n(S)= total number of outcome in sample space
giving
12=2/144=0,0138888889
Which number best represents the slope of the graphed line?
A graph is a line that extends from second quadrant to fourth quadrant through left parenthesis 0, 2 right parenthesis, and left parenthesis 0.5, 0 right parenthesis.
The number that best represents the slope of the graphed line that is given is: -4.
What is the Slope of a Line?The slope of a line is the ratio of the vertical distance to the horizontal distance across the line. It is calculated using the formula:
Slope (m) = rise/run = change in y / change in x = (y2 - y1)/(x2 - x1)
Given the two points on a coordinate plane as shown in the mage below, let:
(x1, y1) represent (0, 2)
(x2, y2) represent (0.5, 0)
Plug in the values
Slope (m) = change in y / change in x = (0 - 2) / (0.5 - 0)
Slope (m) = -2/0.5
Slope (m) = -4
Therefore, we can state that, the number that best represents the slope of the graphed line that is given is: -4.
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The actual answer is C.1/2
chust mee am a doktor
Anja collected data about the number of dogs 9th-grade students own. She created this histogram to represent the data and determined that it is skewed right.
A graph shows number of dogs labeled 0 to 2 on the horizontal axis and frequency on the vertical axis. 0 dogs at frequency 20. 1 dog at frequency 15. 2 dogs at frequency 12.
Which statement is true about Anja’s claim?
Anja is correct that the histogram is skewed to the right
According to the statement
we have given that the anja collected the data and created the histogram and after analyzing we have to tell that her claim is correct or not.
So, she collected the data:
A graph shows number of dogs labeled 0 to 2 on the horizontal axis and frequency on the vertical axis. 0 dogs at frequency 20. 1 dog at frequency 15. 2 dogs at frequency 12.
we know that the
Histogram is is an approximate representation of the distribution of numerical data.
and after making the histogram we see that
The histogram is skewed to the right because there is less data on the right side.
So, Anja is correct that the histogram is skewed to the right.
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Answer:
D: She is correct; the histogram is skewed to the right because there is less data on the right side.
7x-x
simplify please
Based on the given task content; the simplification of 7x - x to its simplest form is 6x.
SimplificationSimplify 7x - x
There are two terms in the expressionThe terms are;7x and -x
The terms both have the same variable xThe term -x have an imaginary coefficient of 1So,
7x - x
= 6x
Therefore, 6x is the result of the simplification of 7x - x to its simplest form.
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What order will result in a final output of - 31 when the initial input is 0?? Someone please please helpoo
The order that will result in a final output of -31 when the first input is 0 is as follows; Third → Second → Fourth → First
How to find the order?First gives;
y = -2·x + 34 = 18
Second gives;
y = -x/3 - 10 = -38/3
Third gives;
= -24
Fourth gives;
y = (x - 2)² = 36
We have after several iterations;
Third gives,
= -24
Next we input into the second to get,
y = -(-24)/3 - 10 = 24/3 - 10 = 8 - 10 = -2
Next we input into the fourth to get,
y = ((-2) - 2)² = (-4)² = 16
Next we input into the first to get,
y = -2×16 + 34 = -32 + 34 = 2
Which gives the order as follows;
Third → Second → Fourth → First
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One fifth of the sum of 9 and 6 is multiplied by the quotient of 18 divided by the sum of 2 and 4, find the result.
please tell me.
Answer:
9
Step-by-step explanation:
We can tackle the individual parts and combine
1/5 of the sum of 9 and 6 means 1/5 * (9 + 6)
The quotient of 18 divided by the sum of 2 and 4 is 18 / (2 + 4)
Combing everything gives us (1/5 * (9 +6)) * (18 / (2 + 4)
(1/5 * 15) * 18/6
3 * 3 = 9
A 400 L tank is filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min. a. Write a differential equation (initial value problem) for the mass of the copper sulfate. b. Solve the differential equation
(a) Let [tex]C(t)[/tex] denote the amount (in grams) of copper (II) sulfate (CuSO₄) in the tank at time [tex]t[/tex] minutes. The tank contains only pure water at the start, so we have initial value [tex]\boxed{C(0)=0}[/tex].
CuSO₄ flows into the tank at a rate
[tex]\left(20\dfrac{\rm g}{\rm L}\right) \left(4\dfrac{\rm L}{\rm min}\right) = 80 \dfrac{\rm g}{\rm min}[/tex]
and flows out at a rate
[tex]\left(\dfrac{C(t)\,\rm g}{400\,\mathrm L + \left(4\frac{\rm L}{\rm min} - 4\frac{\rm L}{\rm min}\right) t}\right) \left(4\dfrac{\rm L}{\rm min}\right) = \dfrac{C(t)}{100} \dfrac{\rm g}{\rm min}[/tex]
and hence the net rate of change in the amount of CuSO₄ in the tank is governed by the differential equation
[tex]\boxed{\dfrac{dC}{dt} = 80 - \dfrac C{100}}[/tex]
(b) This ODE is linear with constant coefficients and separable, so we have a few choices in how we can solve it. I'll use the typical integrating factor method for solving linear ODEs.
[tex]\dfrac{dC}{dt} + \dfrac C{100} = 80[/tex]
The integrating factor is
[tex]\mu = \exp\left(\displaystyle \int \frac{dt}{100}\right) = e^{t/100}[/tex]
Distributing [tex]\mu[/tex] on both sides gives
[tex]e^{t/100} \dfrac{dC}{dt} + \dfrac1{100} e^{t/100} C = 80 e^{t/100}[/tex]
and the left side is now the derivative of a product,
[tex]\dfrac d{dt} \left[e^{t/100} C\right] = 80 e^{t/100}[/tex]
Integrate both sides. By the fundamental theorem of calculus,
[tex]e^{t/100} C = e^{t/100}C\bigg|_{t=0} + \displaystyle \int_0^t 80 e^{u/100}\, du[/tex]
The first term on the right vanishes since [tex]C(0)=0[/tex]. Then
[tex]e^{t/100} C = 8000 \left(e^{t/100} - 1\right)[/tex]
[tex]\implies \boxed{C(t) = 8000 - 8000 e^{-t/100}}[/tex]
Algebra
x^3- 2x^2 + x
Answer:
x(x-1)^2
Step-by-step explanation:
x(x^2-2x+1)
x(x-1)(x-1)
(x-1)(x-1)=(x-1)^2
Box a has a volume of 32 cubic meters. box b is similar to box a. to create box b, box a's dimensions were tripled. what is the volume of box b?
The volume of box B is (A) 864 cubic meters.
What is volume?The amount of three-dimensional space enclosed by a closed surface is expressed as a scalar quantity. For example, the space occupied or contained by a substance or 3D object.Volume is frequently mathematically quantified using the SI-derived unit, the cubic meter.To find the volume of box B:
We have been assigned a volume of 32 cubic meters. The cubic root of 32 yields the box's dimensions, which are 3.1748 x 3.1748 x 3.1748 meters. Meters when the dimensions are tripled: 9.5244 x 9.5244 x 9.5244= 864 cubic meters.
Therefore, the volume of box B is (A) 864 cubic meters.
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The complete question is given below:
Box A has a volume of 32 cubic meters. Box B is similar to box A. To create box B, box A's dimensions were tripled. What is the volume of box B?
a. 864 m3.
b. 288 m3.
c. 96 m3.
d. 32 m3
Ujarak has 100mL t of a strong toothpaste with a 1.1% concentration of sodium fluoride. They have a weaker toothpaste with a 0.3% concentration of sodium fluoride.
What volume, in milliliters, of the weaker toothpaste would Ujarak need to add to the strong toothpaste to create a blend with a 0.8% concentration of sodium fluoride?
The volume of the 0.3% concentration of sodium fluoride required to mix with 100mL of a strong toothpaste with a 1.1% concentration of sodium fluoride to create a blend with a 0.8% concentration of sodium fluoride is 60 mL.
What are mixtures?Mixtures refers to substances which are composed of two or more substances physically combined together.
The 1.1% concentration of sodium fluoride will form a mixture with the 0.3% concentration sodium fluoride.
Let the volume of the 0.3 % concentration toothpaste required be x.
Final volume of toothpaste = (100 + x) mL
Amount of mixture = 0.8 * (100 + x)
100 ml * 1.1 + x * 0.3 = 0.8 * (100 + x)
solving for x
110 + 0.3x = 80 + 0.8x
110 - 80 = 0.8x - 0.3x
30 = 0.5x
x = 60
Therefore, 60 mL of the 0.3% concentration of sodium fluoride is required.
In conclusion, the volume of the weaker solution of the sodium fluoride solution required is obtained from the method of mixtures.
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two buildings are built 10m apart. *the angle of depression from the top of the shorter building (y) to the foot of the taller building (p) is 53. *the angle of elevation from the top of the shorter building to the top of the taller building (x) is 28. *calculate the height (xp) of the taller building. page 262 trigonometry topic 11 of text book
The height of the taller building is 18.6 meters
How to find the side of a right triangle?A right triangle has one of its angles as 90 degrees. The side of a right angle triangle can be found using trigonometric ratios.
Therefore, the height of the taller building is the sum of the height of the smaller building and upper part of the taller building.
tan 53 = opposite / adjacent
tan 53 = x / 10
cross multiply
x = 10 tan 53
x = 10 × 1.32704482162
x = 13.2704482162
x = 13.3 m
tan 28 = y / 10
y = 10 tan 28
y = 5.31709431661
y = 5.3
Therefore, the height of the taller building is 13.3 + 5.3 = 18.6 meters.
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Drag the factors to the correct locations on the image.
Each factor can be used more than once, but not all factors will be used. What is the factored form of this expression? x3 - 6x2 - 9x + 54
Tiles go here
( )( )( )
Tile options:
x - 6
x - 9
x + 9
x - 3
x + 3
x + 6
Answer: x-6, x+3, x-3
Step-by-step explanation;
Split the equation into two.
x2(x-6)-9(x-6)
(x-6) is seen as one of the tiles. Now work on x2-9
x2-9 is a difference of squares meaning the last two are (x+3) and (x-3).
How do you find the maximum or minimum value of a quadratic polynomial?
Marissa is selling paintings for $15 each and bracelets for $7 each. Her goal is to sell at least $800 in products, and she must sell at least 60 bracelets. Which of the following combinations will satisfy these constraints?
The first inequality is represented by the purple area seen in the picture and the second inequality by the black area seen in the picture, the solutions set is the region where the two areas overlap each other.
How to find posible solutions from a system of inequalities
In this question we have two inequalities representing constraints that Marissa should satisfy, provided that she wants to make a profit in selling paintings and bracelets.
The first inequality represents the minimum profit required from the sales of paintings (x) and bracelets (y) and the second one represents the minimum number required of sold bracelets. The two inequalities are described below:
15 · x + 7 · y ≥ 800 (1)
y ≥ 60 (2)
Since there are more than one solution, we decided to define a solutions set with the help of graphing tool. The first inequality is represented by the purple area seen in the picture and the second inequality by the black area seen in the picture, the solutions set is the region where the two areas overlap each other.
Remark
The statement is incomplete and such issue cannot be resolved. We decided to modify the statement as follows:
Marissa is selling paintings for $ 15 each and bracelets for $ 7 each. Her goal is to sell at least $ 800 in products, and she must sell at least 60 bracelets. Please show the set of all possible solution that will satisfy these constraints?
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Answer:
26 paintings and 61 bracelets
Step-by-step explanation:
if you do 26x15=390 and 61x7=427
390+427=817
so the answer is 26 paintings and 61 bracelets
also, I took the test so I know its right
can someone pls solve this
The value of Z in given equation is -55.
According to the statement
We have given that a linear equation which is
2 = 1/5z +13
And from this equation we have to find the value of the z.
So, For this purpose
we know that the
A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept.
So, The equation is
2 = 1/5z +13
1/5z +13-2 = 0
1/5z +11 = 0
z = -11*5
z = -55
here z + 55 =0
So, The value of Z in given equation is -55.
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Use the method of this example to calculate f · dr,c wheref(x, y) = 2xyi (y2 − x2)j(x2 y2)2 and c is any positively oriented simple closed curve that encloses the origin. f · dr
The area around the given curve according to the green theorem is [tex]F. dr = 0[/tex].
According to the statement
we have to find the area enclosed by the simple closed curve that encloses the origin.
So, We know that the
The given equation is
[tex]f(x,y) = \frac{2xyi + (y^{2} - x^{2} ) j}{(x^{2} + y^{2} )^{2} }[/tex]
and
If function is in form of,
[tex]F = Pi + Qj[/tex]
and C is any positively oriented simple closed curve that encloses the origin.
Then,by use of Green's theorem
Do the partial differentiation of the given function
Then
[tex]\frac{dQ}{dx} = \frac{2x^{3} - 6xy^{2}}{(x^{2} + y^{2} )^{3}}[/tex]
and
[tex]\frac{dP}{dy} = \frac{2x^{3} - 6xy^{2}}{(x^{2} + y^{2} )^{3}}[/tex]
On substitution in Green's theorem,
We get the value
[tex]F. dr = 0[/tex]
From this it is clear that the area around the given curve is zero.
So, The area around the given curve according to the green theorem is [tex]F. dr = 0[/tex].
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A bank loans a family $90,000 at 4.5% annual interest rate to purchase a house. The family agrees to pay the loan off by making monthly payments over a 15 year period. How much should the monthly payment be in order to pay off the debt in 15 years? Show all your calculations
Using it's formula, it is found that the monthly payment should be of $688.49 in order to pay off the debt in 15 years.
What is the monthly payment formula?It is given by:
[tex]A = P\frac{\frac{r}{12}\left(1 + \frac{r}{12}\right)^n}{\left(1 + \frac{r}{12}\right)^n - 1}[/tex]
In which:
P is the initial amount.r is the interest rate.n is the number of payments.For this problem, the parameters are given as follows:
A = 90000, r = 0.045, n = 15 x 12 = 180.
Hence:
r/12 = 0.045/12 = 0.00375.
Then we solve for A to find the monthly payment, as follows.
[tex]A = P\frac{\frac{r}{12}\left(1 + \frac{r}{12}\right)^n}{\left(1 + \frac{r}{12}\right)^n - 1}[/tex]
[tex]A = 90000(0.00375)\frac{(1.00375)^{180}}{(1.00375)^{180} - 1}[/tex]
A = $688.49
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