The area of the region bounded above by y = 3x and below by y = 9x^2 is 270 square units.
To use Green's Theorem to calculate the area of the region bounded above by y = 3x and below by y = 9x^2, we need to first find a vector field whose divergence is 1 over the region.
Let F = (-y/2, x/2). Then, ∂F/∂x = 1/2 and ∂F/∂y = -1/2, so div F = ∂(∂F/∂x)/∂x + ∂(∂F/∂y)/∂y = 1/2 - 1/2 = 0.
By Green's Theorem, we have:
∬R dA = ∮C F · dr
where R is the region bounded by y = 3x, y = 9x^2, and the lines x = 0 and x = 6, and C is the positively oriented boundary of R.
We can parameterize C as r(t) = (t, 3t) for 0 ≤ t ≤ 6 and r(t) = (t, 9t^2) for 6 ≤ t ≤ 0. Then,
∮C F · dr = ∫0^6 F(r(t)) · r'(t) dt + ∫6^0 F(r(t)) · r'(t) dt
= ∫0^6 (-3t/2, t/2) · (1, 3) dt + ∫6^0 (-9t^2/2, t/2) · (1, 18t) dt
= ∫0^6 (-9t/2 + 3t/2) dt + ∫6^0 (-9t^2/2 + 9t^2) dt
= ∫0^6 -3t dt + ∫6^0 9t^2/2 dt
= [-3t^2/2]0^6 + [3t^3/2]6^0
= -54 + 324
= 270.
Therefore, the area of the region bounded above by y = 3x and below by y = 9x^2 is 270 square units.
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Find the coordinates of the vertices of the figure after the given transformation.
Dilation of 0.25
Answer:
Option 3
Step-by-step explanation:
Assuming the dilation is centered at the origin, it is known that for a dilation of scale factor [tex]k[/tex] centered at the origin, [tex](x,y) \to (kx,ky)[/tex].
After dilation of with a scale factor of 0.25 the pre image coordinates
V (1, 1), W (2, -2) T ( 4, 2) and U(3, 3) transformed to
V' (0.25, 0.25)
W' (0.5, -0.5)
T' (1, 0.5)
U' (0.75, 0.75)
How to find the coordinates after dilationDilation is a method of transformation that magnify or shrink the preimage depending on the scale factor
In dilation, transformation rule is as follows
(x, y) for a scale factor of r → (rx, ry)
following similar the rule for r = 0.25 we have
preimage image
V (1, 1) → (0.25 * 1, 0.25 * 1) → V' (0.25, 0.25)
W (2, 2) → (0.25 * 2, 0.25 * -2) → W' (0.5, -0.5)
T (4, 2) → (0.25 * 4, 0.25 * 2) → T' (1, 0.5)
U (3, 3) → (0.25 * 3, 0.25 * 3) → U' (0.75, 0.75)
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Which of the following is not a Pythagorean triplet 3 4 5 6 8 10 8 15 16 10 24 26?
Answer:
8, 15, 16
Step-by-step explanation:
8, 15, 16 is not a Pythagorean triple because it doesn't satisfy the Pythagorean Theorem:
[tex]a^2 + b^2 = c^2[/tex]
When we plug this set of numbers into the theorem:
[tex]8^2 + 15^2 \stackrel{?}{=} 16^2[/tex]
[tex]289 \neq 256[/tex]
we get an inequality, not an equality which would make this set of numbers a Pythagorean triple.
Note that all of the other given triples satisfy the Pythagorean Theorem.
What is the formula of area and perimeter of triangle?
Answer: area: 1/2 length * width, perimeter: add the three sides
Step-by-step explanation:
How do you find the area of a triangle with 3 vectors?
The area of the triangle is equal to: Area = (|A| x |B| x sin(θ))/2
To calculate the area of a triangle with 3 vectors, we will use the vector triple product formula. This formula states that the area of the triangle is equal to the magnitude of the cross product of two of the vectors divided by two.
Let's say we have three vectors A, B, and C. The area of the triangle is then equal to the magnitude of the cross product of vector A and B, divided by two.
The magnitude of the cross product of A and B can be calculated as follows:
Magnitude of (A x B) = |A| x |B| x sin(θ)
Where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.
Therefore, the area of the triangle is equal to:
Area = (|A| x |B| x sin(θ))/2
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Question 1 (Multiple Choice Worth 1 points)
(03.01 MC)
Simplify √5 (8+3√6).
8√5 +3√30
O 5√10
130
O√40+15√6
So, the answer after simplifying the given square root equation is 8√5 + 3√30.
Define square.A square has four equal sides, and four vertices, and is a closed, two-dimensional object. It has parallel sides on either side. A square is also equivalent to a rectangle with equal length and width.
What are vertices?A vertex is a location in geometry where two or more curves, lines, or edges converge. As a result of this definition, vertices are the intersection of two lines that create an angle as well as the corners of polygons and polyhedra.
Solving,
√5 (8+3√6)
= √5 * 8 + √5 * 3√6
= 8 * √5 + √5 * 3√6
= 8√5 + 3√5*6
= 8√5 + 3√30
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I need help on yo expand and simplify
After simplification, the formula 4 (2x - 1) + 3 (2x + 5) becomes 14x + 11.
How does expanded form work?A number's arithmetic value for each digit can be expressed using the expanding form. Expanded numbers are numerical expressions that consist of the place value of each digit added to the total of the digits. To improve place value comprehension, expanded form is an excellent strategy.
It is given that,
4 (2x - 1) + 3 (2x + 5)
We have to expand and simplify the expression,
= 4 (2x - 1) + 3 (2x + 5)
Multiply the term out of the bracket into the bracket as
= 8x - 4 + 6x + 15
Add comparable terms together so that we can execute mathematical operations like addition and subtraction on the result.
= 14x + 11
As a result, the simplified expression for 4 (2x - 1) + 3 (2x + 5) will be 14x + 11.
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HELP MEEEEEEEEE PLEASEEE
The coordinates of the points after the translation are:
Q' = (5, 4)R' = (6, 1)What are the coordinates of Q' and R'?All the points are translated on the same way, so let's analyze the translation of P to P'.
The coordinates of Pare (5, 8), and P' is at (3, 4)
Then the translation is:
(3, 4) - (5, 8) = (3 - 5, 4 - 8) = (-2, -4)
So we just need to subtract 2 to the x-value and 4 to the y-value.
The coordinates of Q and R are:
Q = (7, 8)
R = (8, 5)
Then the translated points are:
Q' = (7 - 2,8 - 4) = (5, 4)
R' = (8 - 2, 5 - 4) = (6, 1)
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help!
is this correct?
Answer:3/4
Step-by-step explanation:
1/4 multiplied 3/1 =3/4
what are the excluded factors of 2/x+3
Answer:
x=3
Step-by-step explanation:
An equation's omitted factors are the values of x for which the result is unknown. This indicates that if the denominator or x(x-3) equals 0, your solution is undefined.
Set up this equation and solve for the potential possibilities of x to find these numbers: x(x-3)=0
Divide both sides by (x-3) to find one of the values:
x(x-3)/(x-3)=0(x-3)
Because (x-3) cancels on the left side of the equation and everything divided by 0 is zero on the right, you should get: x=0.
Now that you've determined one of the excluded values, work out the other:
To begin, split both sides by x:
x(x-3)=0
x-3=0
Finally, multiply both sides by -3 to get: x=3
Select all the tranformation that apply to thi quadratic function:
y=1/3(x2)24
The transformations that apply are Horizontal stretch or compression by a factor of 3, Vertical shift up by 24 units, Reflection over the x-axis, The leading coefficient of 1/3 is a vertical compression and The exponent of 2 on x is a parabola.
Vertical Stretch/Compression by a factor of 3: The coefficient in front of the x^2 term is 1/3, which stretches or compresses the graph vertically by a factor of 3.
Horizontal Stretch/Compression by a factor of 24: The exponent on the x term is 24, which stretches or compresses the graph horizontally by a factor of 24.
Reflect over the x-axis: The coefficient in front of the x^2 term is positive, so the graph is reflected over the x-axis.
Shift up or down: The constant term is not present, so the graph does not shift up or down.
Shift left or right: The constant term is not present, so the graph does not shift left or right.
Rotation: The graph does not rotate as it is a parabola in the standard form y=ax^2 + bx + c, it does not have any trigonometric function in it.
Therefore, The transformations that apply are Horizontal stretch or compression by a factor of 3, Vertical shift up by 24 units, Reflection over the x-axis, The leading coefficient of 1/3 is a vertical compression and The exponent of 2 on x is a parabola.
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On a standardized test with a normal distribution, the mean
is 20 and the standard deviation is 2.6. In which interval
would the greatest number of scores occur?
Answer: The interval in which the greatest number of scores would occur is between (20 - 2.6) and (20 + 2.6), which is approximately 17.4 and 22.6 respectively. This interval is also known as the "bell curve" or the "normal distribution curve" and it is centered at the mean (20) and covers approximately 68% of all possible scores. Most of the data points would fall within one standard deviation of the mean, hence the greatest number of scores would occur in this interval.
Step-by-step explanation:
Part A
Melissa is planning a rectangular vegetable garden with a square patch for tomatoes. She wants the length of the garden to exceed three times the length of the tomato patch by 2 feet. She also wants the garden’s width to exceed the width of the tomato patch by 5 feet.
Let x represent the length, in feet, of the square tomato patch.
Write expressions to represent the length and width of Melissa’s vegetable garden in terms of x.
Enter the correct answer in the box.
y = ____
The length of the vegetable garden as a function of its length will be given as: y = 3x + 2
What is the expressions to represent the length and width of Melissa’s vegetable garden in terms of x?The expression of the garden width in terms of the width of the tomato patch is given as:
y = x + 5
Let the length of the garden be y
Let the length of the tomato patch be x
If Melissa wants the length of the garden to exceed three times the length of the tomato patch by 2 feet, the length of the vegetable garden as a function of its length will be given as:
y = 3x + 2
Therefore, If she also wants the garden's width to exceed the width of the tomato patch by 5 feet, the expression of the garden width in terms of the width of the tomato patch is given as:
y = x + 5
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Jordan plotted the graph below to show the relationship between the temperature of his city and the number of cups of hot chocolate he sold daily.
Part A: In your own words, describe the relationship between the temperature the city and the number cups hot chocolate sold.
Part B: Describe how you can make the line of best fit. Write the approximate slope and y-intercept the line best fit. Show your work, including the points that you use to calculate the slope and y-intercept
Every 10 degrees higher in temperature, the slope would be around -3 so draw a line from (40,20) to (80,8) .
what is slope ?How steep a line is defined by its slope. Gradient overflow is a term used to describe a gradient-based mathematical equation (the change in y divided by the change in x). The slope is calculated by dividing the horizontal change (run) between two points by the vertical change (rise) between the same two points. The slope-intercept form of an equation is used to express the equation of a straight line, which is written as y = mx + b. The line's y-intercept is situated where the slope is m, b is b, and (0, b). For instance, the slope and y-intercept of the equation y = 3x - 7 (0, 7).
givenI won't use (40,2) because it is an extreme case.
If I were to draw a line from those points, I would roughly estimate that it would have the same number of dots above and below the line, indicating that it is a line of best fit.
Every 10 degrees higher in temperature, the slope would be around -3 so draw a line from (40,20) to (80,8) .
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How do you find the pattern rule of a sequence?
An example of a pattern rule is "add 3" which means that to find the next term in a sequence, you add 3 to the previous term.
To write a rule for a sequence, you need to identify the pattern. This involves looking at the differences between the numbers and seeing if there is a pattern of increasing or decreasing numbers. Once you have identified the pattern, you can then write a rule that expresses that pattern.
Examples of pattern sequences include the Fibonacci sequence (1,1,2,3,5,8,13,21,34, etc.), the even numbers (2,4,6,8,10, etc.), and the multiples of 3 (3,6,9,12,15, etc.).
To describe the pattern of a sequence, you need to look at the differences between the numbers and the pattern of increases or decreases. For example, in the Fibonacci sequence, the pattern is that each number is the sum of the two numbers before it so 1+1=2, 1+2=3, 2+3=5, etc.
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Original function: F= 9/5C+32
Inverse function: C=F−32·5/9
Suppose we want to find the temperature in Celsius when it is 10 degrees Fahrenheit and when it is 90 degrees Fahrenheit. Which equation would you use? Why?
The inverse function is used to find the temperature in Celsius,
the temperature in celsius is -12.22°C and 32.22°C.
What is an expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows:
Expression: (Math Operator, Number/Variable, Math Operator).
Given expression for temperature calculation
Original function: F = 9/5C + 32
Inverse function: C = (F − 32)5/9
where C is temp. in Celsius and F is temp. in Fahrenheit,
so when the temperature is given in Fahrenheit we have to use the inverse function formula to calculate the temperature in degree celsius,
because it will be easy to calculate.
temperature = 10°F
C = (F − 32)5/9
C = (10 - 32)5/9
C = -22 x 5/9
C = -12.22°C
when temperature = 90°F
C = (90 - 32)5/9
C = 32.22°C
Hence the temperature is -12.22°C and 32.22°C.
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PLEASE!!!! I NEED HELP WITH MY HOMEWORK!!!!!!
(GEOMETRY)!!!
Answer:N= 10 fo sho m -14 maybe
Step-by-step explanation:
this triangle is an isosceles triangle because of the 90 45 45 angles. This means that the two side lengths are equal and 10 is equal to n. to calculate m is maybe square root of 2 times 10 because this image.
Do the lengths 5 12 and 13 form a right triangle?
The lengths 5, 13 and 12 can form a right angled triangle as they follow Pythagoras theorem.
The lengths can form a right triangle if they follow Pythagoras theorem. The longest side will be hypotenuse which is 13. Remaining will be base and height.
Pythagoras theorem -
Perpendicular² = Base² + Height²
Keep the values in formula
13² = 12² + 5²
Taking square on each side of the equation
169 = 144 + 25
Performing addition on Right Hand Side of the equation
169 = 169
Thus, the righ triangle can be formed based on the mentioned dimensions.
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Which graph could be the graph of f(x)
The graph which could be the graph of the polynomial f( x) with roots-8,-5, 1, and 2 is Graph 4.
What is Polynomial function?
A polynomial function is a function that involves onlynon-negative integer powers or only positive integer expounders of a variable in an equation like the quadratic equation, boxy equation, etc. For illustration, 2x 5 is a polynomial that has exponent equal to 1.
Given that;
Root of polynomial = 4.
This means that the polynomial f( x) has 4 roots.
Now, when f( x) = 0, we're told that the 4 roots are;
x = -8
x = -5
x = 1
x = 2
Now, the roots of a polynomial on a graph are the points where the graph wind crosses the x- axis which are called the x-intercepts. Looking at all the given graphs, the only bone that crosses the x-axis at the points x = -8, x = -5, x = 1 and x = 2 is the last graph which is Graph 4. In conclusion, the graph of the polynomial f( x) with roots-8,-5, 1, and 2 is Graph 4.
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. Which ordered pairs in the form (x, y) are solutions to the equation 7x – 5y = 28?
Choose all answers that are correct.
A. (−6, −14)
B. (−1, −7)
C. (4, 10)
D. (7, 9)
Answer:
A and B
Step-by-step explanation:
to determine which pairs are solutions , substitute the x and y coordinates into the left side of the equation and if equal to the right side then solution.
A (- 6, - 14 )
7(- 6) - 5(- 14) = - 42 + 70 = 28 ← true , so solution
B (- 1, - 7 )
7(- 1) - 5(- 7) = - 7 + 35 = 28 ← true , so solution
C (4, 10 )
7(4) - 5(10) = 28 - 50 = - 22 ≠ 28 , not a solution
D (7, 9 )
7(7) - 5(9) = 49 - 45 = 4 ≠ 28 , not a solution
Triangle HIJ has ∠I = 17° and ∠H = 84°. Find m∠ J.
CALC HELP SEE PHOTO!!!
=================================================
Explanation:
Matrix C has 3 rows and 2 columns. The size or dimensions can be written in shorthand as 3x2.
Matrix D has dimensions 2x1, aka 2 rows and 1 column.
-------------------
Write "3x2" and "2x1" side by side in that exact order.
Notice the inner '2's match up. This is important. We need the inner numbers to match so that matrix multiplication is possible. The outer values '3' and '1' will help form the dimensions of the final answer matrix. The final answer matrix will have dimensions 3x1
Luckily we don't need to compute the value of CD. We just need to use the justification mentioned in the previous paragraphs to conclude that matrix CD is possible.
-------------------
Something like BC is not possible because B is 3x2 and C is 3x2. We don't have a match of inner values this time. We can then rule out the answer choice "BC". The answer choices "DC" and "CB" are ruled out for similar reasoning.
Evaluate the expression: 5 + 3 × (5 - 2) 2
32
72
36
16
28. From the four numbers below, Maria chose one number. The number is even and none of its digits are the same. The hundreds digit is triple the ones digit. The tens digit is greater than the thousands digit. Which number did Maria choose? A. 3672 B. 4983 C. 2682 D. 8462
Answer:
Maria chose C. 2682
Step-by-step explanation:
how much does the job pay in a 8hour day if they have 5 employees working $25 a hr
Answer:
1000$
Step-by-step explanation:
1000$
Marsha and Kristen are scuba diving. Marsha is at an elevation of -24 feet, and she is descending at a rate of 12 feet per minute. Kristen is at an elevation of -8 feet and she is descending at a rate of 16 feet each minute. The variable t represents the time in minutes.After how many minutes will Marsha and Kristen be at the same elevation? At what elevation will they be at that time?
Marsha and Kristen will be at the same elevation of -72 feet after 4 minutes.
How to find their elevation?
To find out after how many minutes Marsha and Kristen will be at the same elevation, we can set their elevation equations equal to each other:
-24 - 12t = -8 - 16t
Then, we can solve for t by adding 8 to both sides and dividing by 4:
t = 4
So, Marsha and Kristen will be at the same elevation after 4 minutes.
To find out what elevation they will be at that time, we can substitute 4 into either one of the elevation equations and solve for the elevation:
-24 - 12(4) = -24 - 48 = -72
So, Marsha and Kristen will be at the same elevation of -72 feet after 4 minutes.
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Calculate the area of the quadrilateral.
What is the degree of the polynomial 2x2?
In this case, the highest power on any variable is 2. As a result, the degree of polynomial is 2.
What is degree of the polynomial?The degree of a polynomial in mathematics is the highest of the degrees of the polynomial's monomials with non-zero coefficients. A term's degree is the sum of the exponents of the variables in it, and it is thus a non-negative integer. The degree of a polynomial is the greatest or maximum power of a variable in a polynomial equation. A degree represents the polynomial with the greatest exponential power. Finding the degree of a polynomial. 2 x 2 + 2 x - 364 = 0. The maximum degree of the polynomial in the preceding equation.
Here,
Given : 2x^2
Degree of polynomial = Highest power of a variable.
Here, 2 is the highest power on any variable. So, degree of polynomial is 2.
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PLEASE HELP FIND X
1/2 (2x - 4) + 2 = 4(x + 2) - 11
Find the coordiantes of the vertices for the figure after the given transformation
Dilation of 0.25.
Answer:
[tex]I'(-0.5,0.25), J'(0.5,0.5), K'(0.5, -0.25)[/tex]
Step-by-step explanation:
Assuming the dilation is centered at the origin, we can use the rule that:
For a dilation of scale factor [tex]k[/tex] centered at the origin, [tex](x,y) \to (kx,ky)[/tex]For the given equation, find the values of a, b, and c, determine the direction in which the parabola opens, and determine the y-intercept. Decide which table best illustrates these values for the equation: f (x) = negative 9 x squared + 7 x.
Examining the function -9x² + 7x the parabolic curve is opening downwards and the y intercept is 7/9 and 0
Where to look for the y interceptsThe sign of the x2 coefficient determines whether a parabolic curve opens upward or downward in this scenario.
In this function, the coefficient of x2 is a and is -9.
Rising downwards results from a negative coefficient, while opening upwards from a positive coefficient
The intercept of y
-9x² + 7x = 0
x(-9x + 7) = 0
x = 0
-9x + 7 = 0
-9x = -7
x = 7/9
Therfore, Examining the function -9x² + 7x , 0 and 7/9 is the y intercept.
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