To evaluate the surface integral [tex]\( \iint_S xy \, dS \)[/tex] over the triangular region with vertices[tex]\((1,0,0)\), \((0,8,0)\), and \((0,0,8)\):[/tex]
1. **Equation of the plane**:
8x + y + z = 8.
2. **Parametrize the surface**:
[tex]\[ \mathbf{r}(x, y) = (x, y, 8 - 8x - y). \][/tex]
3. **Normal vector**:
[tex]\[ \mathbf{N} = (8, 8, 1), \quad |\mathbf{N}| = \sqrt{129}. \][/tex]
4. **Surface integral setup**:
[tex]\[ \iint_D xy \sqrt{129} \, dA, \quad \text{with} \quad D: \int_0^1 \int_0^{8(1-x)} xy \, dy \, dx. \][/tex]
5. **Evaluate the integral**:
[tex]\[ \sqrt{129} \int_0^1 32x (1-x)^2 \, dx = \frac{8 \sqrt{129}}{3}. \][/tex]
Thus, the value of the surface integral [tex]\( \iint_S xy \, dS \) is \( \frac{8 \sqrt{129}}{3} \).[/tex]
To evaluate the surface integral [tex]\( \iint_S xy , dS )[/tex], where ( S ) is the triangular region with vertices (1, 0, 0) , (0, 8, 0) , and (0, 0, 8) , we can follow these steps:
1. Determine the equation of the plane containing the triangle:
The vertices of the triangle are (1, 0, 0), (0, 8, 0), and (0, 0, 8). The equation of the plane can be found using these points.
The general form of a plane equation is (Ax + By + Cz = D).
Substituting the points:
[tex]- For \((1, 0, 0)\): \(A(1) + B(0) + C(0) = D \rightarrow A = D\).[/tex]
[tex]- For \((0, 8, 0)\): \(A(0) + B(8) + C(0) = D \rightarrow 8B = D \rightarrow B = \frac{D}{8}\).[/tex]
[tex]- For \((0, 0, 8)\): \(A(0) + B(0) + C(8) = D \rightarrow 8C = D \rightarrow C = \frac{D}{8}\).[/tex]
So, (D = A), [tex]\(B = \frac{A}{8}\), and \(C = \frac{A}{8}\).[/tex]
Using (A = 8) (chosen for simplicity):
- (A = 8),
- (B = 1),
- (C = 1),
- (D = 8).
Thus, the plane equation is (8x + y + z = 8).
2. Parametrize the surface:
We can use (x) and (y) as parameters. From the plane equation, we get (z = 8 - 8x - y).
Let ( (x, y) ) be the parameters. The surface can be parametrized as:
[tex]\[ \mathbf{r}(x, y) = (x, y, 8 - 8x - y). \][/tex]
3. Find the normal vector and its magnitude:
The normal vector [tex]\( \mathbf{N} \)[/tex] can be found from the cross product of the partial derivatives of [tex]\(\mathbf{r}\)[/tex] with respect to (x) and (y).
[tex]\[ \mathbf{r}_x = \frac{\partial \mathbf{r}}{\partial x} = (1, 0, -8), \][/tex]
[tex]\[ \mathbf{r}_y = \frac{\partial \mathbf{r}}{\partial y} = (0, 1, -1). \][/tex]
[tex]\[ \mathbf{N} = \mathbf{r}_x \times \mathbf{r}_y = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & -8 \\ 0 & 1 & -1 \\ \end{vmatrix} = (8, 8, 1). \][/tex]
The magnitude of [tex]\( \mathbf{N} \)[/tex] is:
[tex]\[ |\mathbf{N}| = \sqrt{8^2 + 8^2 + 1^2} = \sqrt{64 + 64 + 1} = \sqrt{129}. \][/tex]
4. Surface integral:
The surface integral is given by:
[tex]\[ \iint_S xy \, dS = \iint_D xy |\mathbf{N}| \, dA, \][/tex]
where (D) is the projection of (S) on the (xy)-plane.
From the vertices, (D) is a triangular region with vertices (1,0), ((0,8), and (0,0).
5. Set up the integral in (xy) coordinates:
We can describe (D) as:
[tex]\[ \iint_D xy \, \sqrt{129} \, dA. \][/tex]
The bounds for (x) and (y) are:
[tex]\[ \int_0^1 \int_0^{8(1-x)} xy \, dy \, dx. \][/tex]
6. Evaluate the integral:
[tex]\[ \sqrt{129} \int_0^1 \int_0^{8(1-x)} xy \, dy \, dx. \][/tex]
Evaluate the inner integral with respect to (y):
[tex]\[ \int_0^{8(1-x)} xy \, dy = x \left[ \frac{y^2}{2} \right]_0^{8(1-x)} = x \left( \frac{(8(1-x))^2}{2} \right) = x \left( \frac{64(1-x)^2}{2} \right) = 32x (1-x)^2. \][/tex]
So the integral becomes:
[tex]\[ \sqrt{129} \int_0^1 32x (1-x)^2 \, dx. \][/tex]
Let ( u = 1-x ). Then ( du = -dx ), and the limits change from ( x = 0 ) to ( u = 1 ), and ( x = 1 ) to ( u = 0 ):
[tex]\[ \sqrt{129} \int_1^0 32(1-u) u^2 (-du) = \sqrt{129} \int_0^1 32(1-u) u^2 \, du. \][/tex]
Expand and integrate:
[tex]\[ \sqrt{129} \int_0^1 32 (u^2 - u^3) \, du = 32 \sqrt{129} \left[ \frac{u^3}{3} - \frac{u^4}{4} \right]_0^1 = 32 \sqrt{129} \left( \frac{1}{3} - \frac{1}{4} \right). \][/tex]
Simplify:
[tex]\[ 32 \sqrt{129} \left( \frac{4}{12} - \frac{3}{12} \right) = 32 \sqrt{129} \left( \frac{1}{12} \right) = \frac{32 \sqrt{129}}{12} = \frac{8 \sqrt{129}}{3}. \][/tex]
Thus, the value of the surface integral[tex]\( \iint_S xy \, dS \) is \( \frac{8 \sqrt{129}}{3} \).[/tex]
Find the component of a velocity vector of an airplane that is traveling 150 mph at 45° below horizontal.
Answer:
none of the above
Step-by-step explanation:
All of the answer choices agree that the magnitude of the components is 150/√2 ≈ 106.1. The units of these components should be "mph", the same as the units of the magnitude of the given vector.
We only know the angle with respect to horizontal. We don't know whether that angle is measured with respect to the positive x-axis or the negative x-axis. Both of those are horizontal, and there is nothing in the problem statement that restricts the airplane to be traveling in one direction or the other.
Possible answers are ...
<106.1 mph, -106.1 mph> . . . . . . "horizontal" is +x direction
<-106.1 mph, -106.1 mph> . . . . . "horizontal" is -x direction
(The units are not degrees (°).)
Identify the volume and surface area of the hemisphere in terms of π. HELP ASAP!! I do not understand! I will mark brainliest if you are correct!!
Answer:
the answer is the 3 one I'm 90% sure
The volume and the surface area of the hemisphere in terms of π are 3888π in³ and 972π in² respectively.
How to find the volume and surface area of a hemisphere:The volume of a hemisphere can be found using the formula given below:
Volume = (2/3)πr^3
The surface area of a hemisphere can be found using the formula given below:
Surface Area = 3πr^2
We can find the volume and surface area as shown below:The figure is provided. From the figure, we can see that the radius of the hemisphere is 18 inches.
The volume of the hemisphere can be found as shown below:
Volume = (2/3)π*18*18*18 in³
= 3888π in³
The surface area of a hemisphere can be found as shown below:
Surface Area = 3π*18*18 in²
= 972π in²
We have found the volume and the surface area of the hemisphere. The volume and the surface area of the hemisphere are 3888π in³ and 972π in² respectively.
Therefore, we have found that the volume and the surface area of the hemisphere in terms of π are 3888π in³ and 972π in² respectively. The correct answer is option B.
Learn more about hemispheres here: https://brainly.com/question/333717
#SPJ2
Neil has 3 partially full cans of white paint.They contain 1/3 gallon,1/5 gallon,and 1/2 gallon of paint.About how much paint does neil have in all
I don’t know what answer Is I wish I could help
Help. Calculus question.
Answer:
64π/5 cubic units.
Step-by-step explanation:
The line x = 2 and y = x^3 intersect at the point (2 , 2^3) = (2, 8).
The required volume = volume of the cylinder with height 8 and radius 2 - the volume of shape form between the curve and the y axis when revolved about the y-axis.
Using the disk method for volumes of revolution:
Note that x = y^1/3 so:
the second volume = Integral π ((y^1/3)^2) dy between y = 0 and y = 8.
= 3/5 y^5/3 * π between 0 and 8
= 96π/5.
The required volume = π 2^2 *8 - 96π/5
= 32π - 96π/5
= 64π/5.
If you know that 4 of an item costs a certain amount, and you want to find out how much one costs, which operation will you use?
A) addition
B) division
Eliminate
C) matriculation
D) multiplication
Answer:
The answer would be division (B)
Step-by-step explanation:
4x equals the total amount
x equals the cost of one item
Therefore you'd divide total cost by 4 and that would find the cost of one item
Answer:
We will use division operation
Step-by-step explanation:
Let x be the cost of 4 items
So, cost of 1 item = [tex]\frac{x}{4}[/tex]
So, we have used division operation
So, to find out how much one costs, we will use division operation
So, Option B is true
Hence we will use division operation
The circumcenter of a triangel is the point equidistant from the vertices of the triangel. True or false
Answer:
True
Step-by-step explanation:
The circumcenter is the center of the circumscribing circle, the circle that intersects each of the vertices. Since the points on a circle are equidistant from its center, the vertices of a triangle are equidistant from the circumcenter.
Final answer:
The circumcenter of a triangle is the point that is equidistant from the vertices of the triangle.
Explanation:
True
The circumcenter of a triangle is the point that is equidistant from the vertices of the triangle. This means that the distance from the circumcenter to each vertex of the triangle is the same.
For example, in an equilateral triangle, the circumcenter is the point where all the perpendicular bisectors of the sides intersect.
if f(x)=x-2 which of the following is the inverse of f(x) brainly
Answer:
The inverse of f(x) is [tex]f ^ {- 1}(x) = x + 2[/tex]
Step-by-step explanation:
To find the inverse of the function [tex]f (x) = x-2[/tex], perform the following steps:
1) do [tex]y = f (x)[/tex]
[tex]y = x-2[/tex]
2) Solve the equation for the variable x.
[tex]y + 2 = x -2 +2[/tex]
[tex]y + 2 = x[/tex]
3) exchange the variable x with the variable y
[tex]y + 2 = x[/tex] ----> [tex]x + 2 = y[/tex]
4) Change the variable y by [tex]f ^{- 1}(x)[/tex]
Finally the inverse function is:
[tex]f ^ {- 1} (x) = x + 2[/tex]
Answer:
f-1(x)=x+2
Step-by-step explanation:
A square and a rhombus have the following in common EXCEPT:
A. Opposite pairs of sides are parallel
B. Opposite pairs of sides are congruent
C. Pairs of interior and exterior angles are supplementary
D. All interior angles are 90 degree
D.All interior angles are 90 degrees
Let's analyze each choice to determine which one is the exception, that is, a property that is not shared between a square and a rhombus:
A. Opposite pairs of sides are parallel:
This is true for both a square and a rhombus. By definition, both a square and a rhombus have opposite sides that are parallel.
B. Opposite pairs of sides are congruent:
This property also holds true for both squares and rhombi. In a square, all four sides are equal in length. In a rhombus, opposite sides are equal in length.
C. Pairs of interior and exterior angles are supplementary:
Again, this is a common feature of both squares and rhombuses. In any parallelogram (which includes both squares and rhombuses), each pair of interior and exterior angles on the same side are supplementary, totaling 180 degrees.
D. All interior angles are 90 degrees:
This is where we can find the exception. In a square, all four interior angles are indeed 90 degrees. However, this is not the case for a rhombus. A rhombus does not necessarily have right angles; its angles can be of any measure as long as opposite angles are equal and the sum of the angles is 360 degrees, which is true for any quadrilateral.
Therefore, the correct answer to the question is:
D. All interior angles are 90 degrees
This is the property that is not common between a square and a rhombus, making it the exception.
Please help me with this
Answer:
A = 146.624 cm
Step-by-step explanation:
Variable a, is one base and variable b is the other base. Variable h is the hight of the trapezoid.
Answer:
146.624cm^2
Step-by-step explanation:
do the split method again :D (10.5cm+21.1cm)(9.28cm/2)
Renee is creating a rectangular garden in her backyard. The length of the garden is 8 feet. The perimeter of the garden must be at least 30 feet and no more than 32 feet. Use a compound inequality to find the range of values for the width w of the garden.
Answer:
The appropriate compound inequality is then 14 ft ≤ W ≤ 16 ft
Step-by-step explanation:
30 ft perimeter: P = 30 ft = 2L + 2W = 2(8 ft) + 2W
Solving for W, we get: 30 ft - 16 ft = 14 ft. The minimum width, W, is 14 ft.
32 ft perimeter:
P = 32 ft = 2L + 2W = 2(8 ft) + 2W
Solving for W, we get: 32 ft - 16 ft = 16 ft. The minimum width, W, is 16 ft.
The appropriate compound inequality is then 14 ft ≤ W ≤ 16 ft
Answer:
Range of width = [7,8]
Step-by-step explanation:
Let w be the width of garden,
The length of garden, l = 8 feet
We have perimeter = 2 x ( length + width)
Perimeter = 2 x ( 8 + w)
The perimeter of the garden must be at least 30 feet and no more than 32 feet.
That is
30 ≤ 2 x ( 8 + w) ≤ 32
15 ≤ 8 + w ≤ 16
7 ≤ w ≤ 8
So range of width = [7,8]
I have no idea how to do this. I can’t cooperate with the imaginary number, please help me
Answer:
Step-by-step explanation:
This is a third degree polynomial because we are given three roots to multiply together to get it. Even though we only see "2 + i" the conjugate rule tells us that 2 - i MUST also be a root. Thus, the 3 roots are x = -4, x = 2 + i, x = 2 - i.
Setting those up as factors looks like this (keep in mind that the standard form for the imaginary unit in factor form is ALWAYS "x -"):
If x = -4, then the factor is (x + 4)
If x = 2 + i, then the factor is (x - (2 + i)) which simplifies to (x - 2 - i)
If x = 2 - i, then the factor is (x - (2 - i)) which simplifies to (x - 2 + i)
Now we can FOIL all three of those together, starting with the 2 imaginary factors first (it's just easier that way!):
(x - 2 - i)(x - 2 + i) = [tex]x^2-2x+ix-2x+4-2i-ix+2i-i^2[/tex]
Combining like terms and canceling out the things that cancel out leaves us with:
[tex]x^2-4x+4-i^2[/tex]
Remembr that [tex]i^2=-1[/tex], so we can rewrite that as
[tex]x^2-4x+4-(-1)[/tex] and
[tex]x^2-4x+4+1=x^2-4x+5[/tex]
That's the product of the 2 imaginary factors. Now we need to FOIL in the real factor:
[tex](x+4)(x^2-4x+5)[/tex]
That product is
[tex]x^3-4x^2+5x+4x^2-16x+20[/tex]
which simplifies down to
[tex]x^3-11x+20[/tex]
And there you go!
If the mean of a normal distribution is 18, what is the median of the distribution?
A) 22
B) 26
C) 14
D) 18
D. Normal distributions are symmetric, so the mean is the same as the median.
Answer:
18
Step-by-step explanation:
If 3(x+2)=5(x-8) what is the value of x+2?
A. 23
B.25
C.40
D.46
Answer:
B. 25
Step-by-step explanation:
In order to find out the value of x + 2, you need to know what x is. Let's solve for it then sub it back in to evaluate the expression.
3(x+2)=5(x-8) distributes out to give you
3x + 6 = 5x - 40. Get like terms together on opposite sides of the equals sign:
46 = 2x and divide to get x = 23. Now that we know that x = 23, that means that x + 2 is the same as 23 + 2 which is 25.
PLEASE HELP ASAP 36 PTS + BRAINLIEST TO RIGHT/BEST ANSWER
Answer:
(b) 3x -5
Step-by-step explanation:
You can slog through the entire long division process, or you can realize that the product of the quotient constant and the divisor constant must match the dividend constant. If we let q represent the quotient constant, you must have ...
4q = -20
q = -20/4 = -5
The only answer choice with a constant of -5 is choice B, 3x -5.
For questions 1-2, compute the modulus and argument of each complex number. Plot and label (with A-G corresponding to 1-2) each complex number in the complex plane given.
1. -2i
2. -5
Answer:
1. modulus: 2; argument: 3π/2 (radians)
2. modulus: 5; argument: π (radians)
Step-by-step explanation:
The modulus is the magnitude of the number. When the number is aligned with one of the axes, it is simply the absolute value of the non-zero component. The argument is the arctangent of the imaginary part divided by the real part, with respect given to signs. Again, when the number is aligned with one of the axes, that angle will be some multiple of π/2 radians, where arg(i) = π/2; arg(-1) = π; arg(-i) = 3π/2; arg(1) = 0.
1. the number is aligned with the negative imaginary axis. Its magnitude is |-2| = 2, and its argument is 3π/2.
__
2. the number is aligned with the negative real axis. Its magnitude is |-5| = 5, and its argument is π.
Mikayla is a waitress who makes a guaranteed $50 per day in addition to tips of 20% of all her weekly customer receipts, t. She works 6 days per week. Which of the following functions best represents the amount of money that Mikayla makes in one week?f(t) = 50 + 20tf(t) = 300 + 20tf(t) = 50 + 0.2tf(t) = 300 + 0.2t
$50 per day x 6 days = 300
20% = 0.2, times t = 0.2t
The equation would be: f(t) = 300 + 0.2t
Answer:
Option D:[tex]f(t)=300+0.2 t[/tex]
Step-by-step explanation:
We are given that Mikayla is a waitress
She makes a guaranteed per day =$50
Her tips per day =20 % of weekly tips t
We have to find a function that represents the amount of money makes by Mikayla in one week
[tex]20% of t=\frac{20}{100}t=0.2t[/tex]
She works per week=6 days
She earns in 6 days=[tex]50\times 6[/tex]=$300
Total earning of Mikayla in one week =[tex]300+0.2t[/tex]
[tex]f(t)=300+0.2 t[/tex]
Hence, option D is true.
“V =“ please help me find the answer *pic attached :)
Answer:
The volume of the prism = 27√3/2 inches³ (≅ 23.38 inches³)
Step-by-step explanation:
* Lets revise the volume of the triangular prism
- The prism has 5 faces
- Two triangular bases
- Three rectangular side faces
- The rule of its volume = Area of its base × its height
* Now lets solve the problem
- The bases of the prism are equilateral triangles of side 3 inches
- The side faces of the prism are rectangles of dimensions 3 and 6 inches
∵ Its volume = Area of the its base × its height
- The bases are equilateral triangles
∵ The area of the equilateral triangle = √3/4 s² , where s is the
length of its side
∵ The length of the side = 3 inches
∴ The area of the base = √3/4 (3)² = 9√3/4 inches²
∵ Its height = 6 inches
∴ Its volume = 9√3/4 × 6 = 27√3/2 inches³
* The volume of the prism = 27√3/2 inches³ (≅ 23.38 inches³)
Could you Help me there was 395 Lenin ice cups at thesnack shop. People bought 177 lemon ice cups . How many lemon ice cups are still all the snack shop.
Answer:
There are 288 cups left
Step-by-step explanation:
Take the amount that were there (395) and subtract the amount that were sold (177). that is the amount that are left
395-177 = 218
There are 288 cups left
after selling 177 cups, there are 218 lemon ice cups still at the snack shop.
To calculate how many lemon ice cups are still at the snack shop after some were sold, we need to perform a simple subtraction operation. The initial quantity of lemon ice cups was 395. After selling 177 cups, we subtract 177 from 395 to find out how many are left.
Here's the calculation:
Start with the initial number of lemon ice cups: 395.Subtract the number of lemon ice cups sold: 395 - 177.Calculate the remaining number of lemon ice cups: 218.So, after selling 177 cups, there are 218 lemon ice cups still at the snack shop.
Which point does NOT lie on the graph of y = 2x 3?
(1, 8)
(-1, -2)
(0, 0)
(2, 16)
[tex]\bf (\stackrel{x}{1},\stackrel{y}{8})\qquad y=2x^3\implies 8=2(1)^3\implies 8\ne 2~~\bigotimes \\\\\\ (\stackrel{x}{-1},\stackrel{y}{-2})\qquad y=2x^3\implies -2=2(-1)^3\implies -2=-2~~\checkmark \\\\\\ (\stackrel{x}{0},\stackrel{y}{0})\qquad y=2x^3\implies 0=2(0)^3\implies 0=0~~\checkmark \\\\\\ (\stackrel{x}{2},\stackrel{y}{16})\qquad y=2x^3\implies 16=2(2)^3\implies 16=16~~\checkmark[/tex]
Which value for x makes the following equation TRUE?
-2x - 4 = - 6
Answer:
1
Step-by-step explanation:
-2(1) = -2
then -2 - 4 = -6
-6 = -6
Answer:
1
Step-by-step explanation:
Move all terms that don't contain x to the right side & solve.
Express this number in standard form.
3
.
6
4
3
×
1
0
−
1
=
?
3.643×10
−1
=?3, point, 643, times, 10, start superscript, minus, 1, end superscript, equals, question mark
Answer:
0.3643
Step-by-step explanation:
3.643×10^-1 = 3.643×(1/10) = 3.643×0.1 = 0.3643
A circle with a diameter of 10 inches is spanned by a central angle of 180 degrees. What is the length of the subtended arc?
since the diameter of the circle is 10, then its radius must be half that or 5.
[tex]\bf \textit{arc's length}\\\\ s=\cfrac{\pi \theta r}{180}~~ \begin{cases} r=&radius\\ \theta =&angle~in\\ °rees\\ \cline{1-2} r=&5\\ \theta=&180 \end{cases}\implies s=\cfrac{\pi (180)(5)}{180}\implies s=5\pi \implies s\approx 15.71[/tex]
Answer:
The correct answer is 15.7 inches
Step-by-step explanation:
Points to remember
Circumference of a circle = 2πr
Where r is the radius of circle
To find the value of arc length
It is given that, diameter = 10 inches
Radius = diameter/2 = 10/2 = 5 inches
Circumference = 2πr
= 2 * 3.14 * 5
= 31.4
central angle of arc =180
Arc length = (180/360) * circumference
= (1/2) * 31.4
= 15.7 inches
Therefore the correct answer is 15.7 inches
For parametric equations x= a cos t and y= b sin t, describe how the values of a and b determine which conic section will be traced.
Step-by-step explanation:
x = a cos t, y = b sin t
cos t = x / a, sin t = y / b
cos² t + sin² t = 1
(x / a)² + (y / b)² = 1
If a = b, the conic section is a circle.
If a and b are different, the conic section is an ellipse.
-The terminal side of the angle contains the following point. Find the values of the six trigonometric functions for the following points. #16
-Given the following, state the six trigonometric ratios. #17 #18
Answer: see attachments
Step-by-step explanation:
Use Pythagorean Theorem to find the missing side (x² + y² = h²)
Use the following formulas to find the trig functions:
[tex]sin\theta=\dfrac{y}{h}\qquad \qquad csc\theta=\dfrac{h}{y}\\\\\\cos\theta=\dfrac{x}{h}\qquad \qquad sec\theta=\dfrac{h}{x}\\\\\\tan\theta=\dfrac{y}{x}\qquad \qquad cot\theta=\dfrac{x}{y}[/tex]
Which polar coordinates represent the same point as the rectangular coordinate (2,-1?
[tex]\bf (\stackrel{a}{2}~,~\stackrel{b}{-1})\qquad \begin{cases} r=\sqrt{a^2+b^2}\\\\ \theta =tan^{-1}\left( \frac{b}{a} \right) \end{cases} \\\\[-0.35em] ~\dotfill\\\\ r=\sqrt{2^2+(-1)^2}\implies r=\sqrt{5} \\\\\\ \theta =tan^{-1}\left( \cfrac{-1}{2} \right)\implies \theta \approx -26.57^o\implies \theta \approx 333.43^o \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill (\sqrt{5}~~,~~333.43^o)~\hfill[/tex]
Answer:
[tex](r,\theta); (\sqrt{5} , tan^{-1}(\frac{x}{y}))\\(r,\theta); (-\sqrt{5} , -tan^{-1}(\frac{x}{y}))[/tex]
Step-by-step explanation:
Here we are given our rectangular coordinates as (2,-1) . We have to convert this into polar coordinates. The formula for conversion into polar form is
[tex]r=\sqrt{x^2+y^2}[/tex]
[tex]\theta=tan^{-1}(\frac{x}{y})[/tex]
Substituting the values of x and y in the above formulas we get
[tex]r=\sqrt{2^2+(-1)^2}\\r=\sqrt{4+1}\\r=\sqrt{5}\\r=-\sqrt{5}\\[/tex]
[tex]\theta=tan^{-1}(\frac{-1}{2})[/tex]
Hence our polar coordinates are
[tex]r=(\sqrt{5},tan^{-1}(\frac{-1}{2}) )\\r=(-\sqrt{5},tan^{-1}(\frac{-1}{2}) )\\[/tex]
Ashley is thinking of two numbers. The first number is four more than twice the second number. The sum of the two numbers is 16. Which system of equations can be used to determine the first number, x, and the second number, y?
x + y = 16
x = y^2 x 4
i hope this helps :)
good luck
Please please answer this correctly
Answer:
800,000,000
Step-by-step explanation:
Subtract the known numbers from the sum to find the missing number:
800,903,402 - 900,000 -2 -3,000 -400 = 800,000,000
URGENT! Use Gauss's approach to find the following sums 1+3+5+7+...997
Formula please and solve
Answer:
The sum of the first 499 terms is 249001
Step-by-step explanation:
* Lets revise the arithmetic sequence
- There is a constant difference between each two consecutive
numbers
- Ex:
# 2 , 5 , 8 , 11 , ……………………….
# 5 , 10 , 15 , 20 , …………………………
# 12 , 10 , 8 , 6 , ……………………………
* General term (nth term) of an Arithmetic sequence:
- U1 = a , U2 = a + d , U3 = a + 2d , U4 = a + 3d , U5 = a + 4d
- Un = a + (n – 1)d, where a is the first term , d is the difference
between each two consecutive terms n is the position of the
number
- The sum of first n terms of an Arithmetic sequence is calculate from
Sn = n/2[a + l], where a is the first term and l is the last term
* Now lets solve the problem
∵ The terms of the sequence are 1 , 3 , 5 , 7 , ......... , 997
∵ The first term is 1 and the second term is 3
∴ The common difference d = 3 - 1 = 2
∵ The first term is 1
∵ The last term is 997
∵ The common difference is 2
- Lets find how many terms in the sequence
∵ an = a + (n - 1) d
∴ 997 = 1 + (n - 1) 2 ⇒ subtract 1 from both sides
∴ 996 = (n - 1) 2 ⇒ divide both sides by 2
∴ 498 = n - 1 ⇒ add 1 for both sides
∴ n = 499
∴ The sequence has 499 terms
- Lets find the sum of the first 499 terms
∵ Sn = n/2[a + l]
∵ n = 499 , a = 1 , l = 997
∴ S499 = 499/2[1 + 997] = 499/2 × 998 = 249001
* The sum of the first 499 terms is 249001
Final answer:
The sum of the series 1+3+5+7+...+997 is calculated using Gauss's approach for arithmetic series, resulting in a sum of 249500.
Explanation:
To find the sum of the series 1+3+5+7+...+997 using Gauss's approach, we first recognize that this is an arithmetic series where each term increases by a constant difference, in this case, 2. The first term, a, is 1 and the last term, l, is 997. We can find the number of terms, n, in the series using the formula: n = (l - a) / common difference + 1, which in this case is (997 - 1) / 2 + 1 = 499. Then, the sum of an arithmetic series can be found using the formula: Sum = n/2 * (a + l), so the sum of this series is 499/2 * (1 + 997) = 249500.
Evaluate 7 − (−1).
6
−6
8
−8
Answer:
8
Step-by-step explanation:
The minus sign outside parentheses changes the sign of what's inside parentheses when parentheses are eliminated:
7 -(-1) = 7 +1 = 8
_____
Alternate way to think about it
Subtraction is the same as addition of the opposite. The opposite of -1 is +1, so subtracting -1 is the same as adding +1.
Time sensitive question. Find the sum of the first 26 terms of an arithmetic series whose first term is 7 and 26th term is 93.
ANSWER
[tex]S_{26}=1300[/tex]
EXPLANATION
The sum of an arithmetic sequence whose first term and last terms are known is calculated using
[tex]S_{n}= \frac{n}{2} (a + l)[/tex]
From the given information, the first term of the series is
[tex]a = 7[/tex]
and the last term of the series is
[tex]l = 93[/tex]
The sum of the first 26 terms is
[tex]S_{26}= \frac{26}{2} (7 + 93)[/tex]
[tex]S_{26}= 13 (100)[/tex]
[tex]S_{26}=1300[/tex]
The sum of the first 26 terms of the given arithmetic series is 1300, obtained using the sum formula S = n/2 * (a1 + an) for arithmetic series.
Explanation:To find the sum of the first 26 terms of an arithmetic series whose first term (a1) is 7 and the 26th term (a26) is 93, you can use the formula for the sum of an arithmetic series: S = n/2*(a1 + an). Here, n is the number of terms, a1 is the first term, and an is the nth term. In this case, we have n = 26, a1 = 7, and a26 = 93.
The sum, S, of the series is calculated as follows:
S = 26/2 * (7 + 93) = 13 * (100) = 1300.
Therefore, the sum of the first 26 terms of the arithmetic series is 1300.