The mass of the wire that lies along the curve r and has density δ is (7/6)((3/32)π⁶+1). (option c)
Let's start with C1. We're given the curve in parametric form, r(t) = (6 cos t)i + (6 sin t)j, 0 ≤ t ≤(π/2). This curve lies in the xy-plane and describes a semicircle of radius 6 centered at the origin. To find the length of the wire along this curve, we can integrate the magnitude of the tangent vector, which gives us the speed of the particle moving along the curve:
|v(t)| = |r'(t)| = |(-6 sin t)i + (6 cos t)j| = 6
So the length of the wire along C1 is just 6 times the length of the curve:
L1 = 6∫0^(π/2) |r'(t)| dt = 6∫0^(π/2) 6 dt = 18π
To find the mass of the wire along C1, we need to integrate δ along the length of the wire:
M1 =[tex]\int _0^{L1 }[/tex]δ ds
where ds is the differential arc length. In this case, ds = |r'(t)| dt, so we can write:
M1 = [tex]\int _0^{(\pi/2) }[/tex]δ |r'(t)| dt
Substituting the given density, δ = 7t⁵, we get:
M1 = [tex]\int _0^{(\pi/2) }[/tex] 7t⁵ |r'(t)| dt
Plugging in the expression we found for |r'(t)|, we get:
M1 = 7[tex]\int _0^{(\pi/2) }[/tex]6t⁵ dt = 7(6/6) [t⁶/6][tex]_0^{(\pi/2) }[/tex] = (7/6)((1-64)π⁵+1)
So the mass of the wire along C1 is (7/6)((1-64)π⁵+1).
Now let's move on to C2. We're given the curve in vector form, r(t) = 6j + tk, 0 ≤ t ≤ 1. This curve lies along the y-axis and describes a line segment from (0, 6, 0) to (0, 6, 1). To find the length of the wire along this curve, we can again integrate the magnitude of the tangent vector:
|v(t)| = |r'(t)| = |0i + k| = 1
So the length of the wire along C2 is just the length of the curve:
L2 = ∫0¹ |r'(t)| dt = ∫0¹ 1 dt = 1
To find the mass of the wire along C2, we use the same formula as before:
M2 = [tex]\int _0^{L2}[/tex] δ ds = ∫0¹ δ |r'(t)| dt
Substituting the given density, δ = 7t⁵, we get:
M2 = ∫0¹ 7t⁵ |r'(t)| dt
Plugging in the expression we found for |r'(t)|, we get:
M2 = 7∫0¹ t⁵ dt = (7/6) [t⁶]_0¹ = (7/6)(1/6) = (7/36)
So the mass of the wire along C2 is (7/36).
To find the total mass of the wire, we just add the masses along C1 and C2:
M = M1 + M2 = (7/6)((1-64)π⁵+1) + (7/36) = (7/6)((3/32)π⁶+1)
Therefore, the correct answer is (c) (7/6)((3/32)π⁶+1).
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How many square kilometers are in 1,750,000
square meters?
Answer:1.75
Step-by-step explanation:
What is the correlation coefficient for the data shown in
the table?
О о
0 1
04
05
Solving provided question, we can say that the correlation coefficient for data shown will be as [tex]\sqrt{x} =\sqrt{y} = > \sqrt{x^2} = \sqrt{y^2}[/tex]
what is correlation coefficient?In statistics, the Pearson's correlation coefficient, often known as Pearson's r, Pearson's product-moment correlation coefficient, bivariate correlation, or simply correlation coefficient, is a metric for the linear correlation between two data sets. A correlation coefficient is a statistical indicator of how well changes in one variable's value predict changes in another variable's value. The values rise or fall jointly for variables with positive correlation. When evaluating the strength and direction of linear relationships between two sets of variables, correlation coefficients are used. If both variables are regularly distributed, use Pearson's correlation coefficient; if not, use Spearman's.
here,
the correlation coefficient for data shown will be as
[tex]\sqrt{x} =\sqrt{y} \\\sqrt{x^2} = \sqrt{y^2}[/tex]
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What is the rule for the reflection?
ryx(x, y)-(-y, -x)
ryx(x, y)-(-y, -x)
ryx(x, y) (y,x)
ryx(x, y) → (V, x)
While reflecting about the line y=x, option (c) [tex]r_{y=x}(x,y) = (y,x)[/tex] is correct and while reflecting about the line y= -x, option (b) [tex]r_{y=-x}(x,y) = (-y,-x)[/tex] is correct.
What is the rules of reflection in coordinate geometry?The x-coordinate and y-coordinate are reversed when a point is reflected over the line y = x. The x-coordinate and the y-coordinate switch places and become negated if you reflect over the line y = -x (the signs are changed). The line y = x is the point of reflection of the point (x,y) across (y, x).
Firstly, the rule for reflecting a point about the line y=x is:
[tex]r_{y=x}(x,y) = (y,x)[/tex]
While reflecting about the line y=x, we get the reflected points by swapping the coordinates.
So, Option 3 is correct.
Now, the rule for reflecting a point about the line y= -x is:
[tex]r_{y=-x}(x,y) = (-y,-x)[/tex]
While reflecting about the line y= -x, we get the reflected points by multiplying '-1' with the swapped coordinates.
So, Option 2 is correct.
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For my higher grade and i dint get this problem
Answer:
y = 9
Step-by-step explanation:
[tex]\frac{2}{6}[/tex] = [tex]\frac{y}{27}[/tex] ( cross- multiply )
6y = 2 × 27 = 54 ( divide both sides by 6 )
y = 9
Answer:
Step-by-step explanation: From the question 2/6 = y/ 27
Cross multiply
It becomes 2 * 27 = 6 * y
54 = 6y
Make y the subject
y = 54/ 6
y = 9
What is the formula of factor theorem?
A polynomial f(x) has a factor (x-a) if and only if f(a) = 0, according to the factor theorem. It's one of the techniques for factoring a polynomial.
Since, the response above already covered the theorem's formulation.
Here, we shall demonstrate the factor theorem formula, which allows us to factor the polynomial.
When the polynomial f(x) is divided by (x-c), f(c) equals 0.
Remainder Theorem demonstrates that f(x)=(x-c)q(x)+f (c)
where the quotient polynomial is q(x) and the target polynomial is f(x).
Because f(c) = 0, f(x) = (x-c)q(x)+f (c)
(x-c)q(x)+0 f(x) = (x-c)q (x)
Therefore, the polynomial f's factor (x-c) exists (x).
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I’m literally tired of this, math is literally riddles
Answer:
JM is the answer
Step-by-step explanation:
i was going to explain but my brain is in shambles right now
A carousel spins at a rate 2 miles in 7 minutes
What is the speed in miles per minute
Answer:
0.29 miles per minute.
Step-by-step explanation:
Speed, s, is defined by the distance, d, divided by the time, t ⇒ [tex]s = \frac{d}{t}[/tex].The distance the carousel travels is 2 miles.The time it takes for the carousel to travel the total distance is 7 minutes.Therefore, [tex]Speed = \frac{2 miles}{7 minutes}[/tex].Speed is equal to 0.285 or 0.29 miles per minute. A common mistake would be to divide the amount of time by the distance, resulting in an incorrect answer.Please let me know if this helped!!!
Find the measure of angle “x” in degrees. Round to two decimals places as necessary!! HELPP PLS
Answer:
x ≈ 50.06°
Step-by-step explanation:
using the sine ratio in the right triangle
sin x = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{23}{30}[/tex] , then
x = [tex]sin^{-1}[/tex] ( [tex]\frac{23}{30}[/tex] ) ≈ 50.06° ( to 2 decimal places )
Is z =9 a solution to 3z+8=25
Answer:
No, z =9 isn't a solution to 3z+8=25
Step-by-step explanation:
Let's see if it is...
[tex]\sf 3z+8=25[/tex]
Subtract 8 from both sides:-
[tex]\sf 3z+8-8=25-8[/tex]
[tex]\sf 3z=17[/tex]
Divide both sides by 3:-
[tex]\sf \cfrac{3z}{3}=\cfrac{17}{3}[/tex]
[tex]\sf z=\cfrac{17}{3}\;\; or \;\; z=5.6666....[/tex]
Therefore, no z = 9 isn't a solution to 3z+8=25.
____________________
Hope this's what you're looking for!
Have a great day!
Valentina is being careful with her spending so she can later purchase a house. She has allocated no more than $320 each month for both travel expenses and eating out. On average, every time she eats out it costs $16, and every travel journey costs $5.
Let x be the number of times she eats out and y represent the number of travel journeys each month.
Write an inequality representing the relationship between x and y, with y as the subject.
On solving the provided question we cans say that the inequality is 16x+5y320
What is inequality?An inequality in mathematics is a relationship between two expressions or values that is not equal. Thus, imbalance leads to inequality. An inequality creates the link between two values that are not equal in mathematics. Egality is distinct from inequality. When two values are not equal, most commonly use the not equal sign (). Different inequalities are used to contrast values, no matter how little or large. Many simple inequalities may be resolved by modifying the two sides until the variables are all that remain. But a number of things contribute to inequality: Negative values on both sides are divided or added. Trade off the left and right.
Firm the question, the amount she spends on food every month is 16x and the amount he spends on travel every month is 5y.
Therefore, the total amount she spends every month is 16x+5y.
Since she can't spend more than $320 each month, the inequality is
16x+5y320.
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need help with finding solving for X
Answer:
15
Step-by-step explanation:
Since the sum of all angles in a triangle is 180, we can make an equation
111 + 44 + 2x - 5 = 180
2x - 5 = 25
2x = 30
x = 15
Why do all exponential functions go through 0 1?
All exponential functions go through the point (0,1) because the value of x when it equals to 0, in the general form of exponential function f(x) = [tex]a^x[/tex], results in [tex]a^0[/tex] = 1, which is why all exponential functions have the point (0,1) on their graph.
All exponential functions go through the point (0,1) because of the properties of exponential functions. The general form of an exponential function is f(x) = [tex]a^x[/tex], where a is a positive constant. When x = 0, the function evaluates to [tex]a^0[/tex] = 1, which is why all exponential functions have the point (0,1) on their graph.
Another way to understand this is that the exponential function is defined as the function that describes how a quantity grows or decays at a fixed percentage rate over time. The function f(x) = [tex]a^x[/tex], where a is a constant greater than 0 and less than 1, describes the decay of a quantity over time. For example, if the interest rate is 10%, the value of a = 0.1 and the function f(x) = [tex]0.1^x[/tex]. When x = 0, the function evaluates to [tex]0.1^0[/tex] = 1, which means that the initial value of the quantity is 1.
It's important to note that the point (0,1) is called the y-intercept, and it is a special point in the graph of exponential functions, and it's always present in the graph of exponential functions.
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The complete Question is:
Why do all exponential functions go through point(0,1)?
Find the value of X and Y. Write your radical in simplest form. Exact answers only.
to
28
Answer:
X = 45, Y = 14√2
Step-by-step explanation:
We are given that the diagram is a square, and is cut in half using the length that is 28. Therefore x = 45
Now for y, we know that we have a 45 45 90 triangle with hypotenuse of 28, and the formula for that is the legs have length of x, and hypotenuse of x√2
y = x
=> 28 = x√2
x = 14√2
=> y = 14√2
Solve 3x 1 = 15 for x using the change of base formula log base b of y equals log y over log b. −0.594316 1.405684 1.464973 3.469743
Using the change of base formula log base b of y equals log y over log b x = 4.69743.
To solve 3x + 1 = 15 for x, we can use the change of base formula, which states that log base b of y equals log y over log b. In this case, we can use log base 3 on both sides of the equation to solve for x.
First, we take the log base 3 of both sides of the equation:
log base 3 (15) = log base 3 (3x + 1)
Next, we can use the property of logarithms, which states that log base b (xy) = log base b (x) + log base b (y):
log base 3 (15) = log base 3 (3) + log base 3 (x + 1/3)
We can solve for x by subtracting log base 3 (3) from both sides of the equation:
log base 3 (15) - log base 3 (3) = log base 3 (x + 1/3)
We can then use the change of base formula to solve for x:
log 15/3 = log x + 1/3 over log 3
We can then solve for x by taking the antilog of both sides of the equation:
15/3 = x + 1/3
We can then solve for x by subtracting 1/3 from both sides of the equation:
15/3 - 1/3 = x
Finally, we can solve for x by dividing both sides of the equation by 3:
x = (15/3 - 1/3) / 3
Therefore, x = 4.69743.
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Question 1 (5 points)
The notation "f(x)" (f of x) is another way of representing y-values in a function
True
False
The given statement "The notation "f(x)" (f of x) is another way of representing y-values in a function" is true because it represents the range or output values.
What is a function?In Mathematics, a function is a mathematical expression which can be used for defining and showing the relationship that exist between two or more variables in a data set.
This ultimately implies that, a function typically indicates the relationship between input values (domain or x-values) and output values (range or y-values) of a data set, including the ways in which the elements in a table are uniquely mapped (paired).
In conclusion, we can reasonably infer and logically deduce that the output values (range or y-values) of a function are plotted on the y-coordinate of a graph and they are primarily denoted by either the notation f(x) or y.
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Solve for x.
6x = 18
Enter your answer in the box.
x =
Answer:
x = 3
Step-by-step explanation:
6x = 18
Divided both sides by 6
x = 3
Let's check
6(3) = 18
18 = 18
So, x = 3 is the correct answer.
Answer:
x = 3
Step-by-step explanation:
1) 6x = 18
2) Divide by 6.
6x = 18
6 6
x = 3
given h(x)=-9x+5 find h(8)
whats colder? -0.9 or -1.2
assuming they are the same unit, -1.2 is colder
when a number is negative, the further the number is from zero the less (colder) it is
What is
49 increased by twice a number m
Answer:
The result can be shown in multiple forms.
Exact Form: m = 49/2
Decimal Form: m = 24.5
Mixed Number Form: m = 24 1/2 (twenty-four and a half as a fraction)
Step-by-step explanation:
1. Divide each term in 2 m = 49 by 2:
2m/2 = 49/2
2. Simplify the left side.
m = 49/2
The result can be shown in multiple forms.
Exact Form: m = 49/2
Decimal Form: m = 24.5
Mixed Number Form: m = 24 1/2 (twenty-four and a half as a fraction)
The 49 increased by twice a number m is 49+2m
What are expressions in maths?An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles.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: An expression, also known as an algebraic expression, is a mathematical statement that consists of numbers, variables, and an arithmetic operation between them. An expression is (Number/variable, Math Operator, Number/variable). In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.Given data :
49+2m
Increase means “go up” which is another way to say addition
Twice means “two times” or “doubled”
A number would be a variable since it’s unknown
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Adam bought granola bars at the store. The
expression 6p + 5n gives the number of bars
in p boxes of plain granola bars and n boxes of
granola bars with nuts. What are the terms of
the expression?
How do you find the coordinates of an object in an image?
Finding the coordinates of an object in an image involves using image processing techniques such as object detection and image segmentation.
Object detection involves identifying the presence of an object in an image, while image segmentation involves isolating the object in the image.
Once the object is identified and isolated, its coordinates can be determined by using the position of the object's bounding box or contour.
Bounding box is a rectangular box that surrounds the object, its coordinates can be determined by the position of its top-left corner and bottom-right corner.
Contour is the boundary of the object, its coordinates can be determined by the points that are part of the boundary.
Computer Vision libraries such as OpenCV, Tensorflow or PyTorch can be used to perform object detection and image segmentation. They also provide pre-trained models which can be used for object detection and image segmentation without the need to train a model from scratch.
In conclusion, finding the coordinates of an object in an image requires using image processing techniques such as object detection and image segmentation to identify and isolate the object, and then determining its coordinates by using the position of the object's bounding box or contour.
This process can be aided by using Computer Vision libraries such as OpenCV, Tensorflow or PyTorch.
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Can the line segments 1.5 cm 2 cm 2.5 cm be the sides of a right angled triangle?
It is possible to construct a triangle whose sides are 1.5 cm 2 cm and 2.5 cm because the sum of two sides are greater than three sides.
In the given question, we have to check that it is possible to construct a triangle whose sides are 1.5 cm 2 cm and 2.5 cm.
To check whether the given sides can make a triangle or not, we have to check that the sum of two sides always greater than the third side.
To check this we firstly add the 1.5 and 2
1.5 + 2 > 2.5
3.5 > 2.5
Now we add 2.5 and 2
2.5 + 2 > 1.5
4.5 > 1.5
Now we add 1.5 and 2.5
1.5 + 2.5 > 2
4 > 2
It is possible to construct a triangle whose sides are 1.5 cm 2 cm and 2.5 cm because the sum of two sides are greater than third sides.
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PLEASE HELP
if f(x)=8x-2/2x what is f (I) equeal to?
A.3
b. 5
c.6
d.8
The value of f(1) is equal to 7
What are functions?Function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given here f(x)=8x-2/2x putting x=1 we get
f(1)= 8-2/2
= 7
Hence f(1) is equal to 7
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(Two-Step Linear Inequalities MC) Which graph represents the solution to the inequality 2(b + 2) > 24? number line with open point at 10 with arrow pointing left number line with closed point at 10 with arrow pointing left number line with closed point at 10 with arrow pointing right number line with open point at 10 with arrow pointing right
The graph that represents the solution to the inequality is a number line with closed point at 10 with arrow pointing left
What is a graph?You should be aware that inequality is is a mathematical statement showing that two opposite things are not equal. The graph is a diagram that illustrates the statement.
The given inequality is 2(b + 2) > 24
Opening the brackets to get
2b+4≥24
2b≥24-4
This implies that 2b≥20
Making b the subject
b≥20/2
b≥10
Therefore, the graph is a number line with closed point at 10 with arrow pointing left
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What is the solution to 9|x + 8| > 45?
x < –3 or x > 13
–3 < x < 13
x < –13 or x > –3
–13 < x < –3
Answer:
x<-13 or x>-3
Step-by-step explanation:
solve 9 abs(x + 8)>45 = x<-13 or x>-3
Morgan buys 4 1/3 crates of apples for 39$. How much does 1 carton of apples cost?
Answer:
$9
Step-by-step explanation:
because we would take the total which is $39 so we divide 39/ 4 1/3 = to 9 so for one crate it is going to be $9 a crate
hope this helps
what is 64 divided by 48000
and I need work please
How do you find the ratio of area of two circles?
The ratio of the area of two circles can be found by dividing the area of one circle by the area of the other.
The formula for the area of a circle is A = πr^2, where r is the radius of the circle. To calculate the ratio of the areas of two circles, take the area of one circle and divide it by the area of the other:
Ratio of area of two circles = A1/A2 = (πr1^2)/(πr2^2)
For example, if the radius of one circle is 3 and the radius of the other is 5, then the area of the first circle is 28.27 and the area of the second circle is 78.54. The ratio of the areas of these two circles is then 28.27/78.54, which is 0.36.
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What is the solution Y=x-5
The solution of the expression for x is y+ 5.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the expression is y = x - 5. The solution of the expression for x will be,
y = x - 5
x = y + 5
Therefore, the solution of the expression for the value of x will be x = y + 5.
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Help me Evaluate the expression n4 for n = 0.3
Answer:
0.0081
Step-by-step explanation:
n^4
n = 0.3
We just need to put 0.3 in for n and solve
(0.3)^4 = 0.3 x 0.3 x 0.3 x 0.3 = 0.0081