When X and Y axes are rotated through an angle θ to form X' and Y' axes, then the relation between the axes and the angle is given by,
[tex]\rm{X=X'\cos\theta-Y'\sin\theta}[/tex]
[tex]\rm{Y=X'\sin\theta+Y'\cos\theta}[/tex]
Then x and y are replaced in terms of x' and y' as,
[tex]\rm{x=x'\cos\theta-y'\sin\theta}[/tex]
[tex]\rm{y=x'\sin\theta+y'\cos\theta}[/tex]
Here θ = π/6. Then,
[tex]\rm{x=x'\cos\left(\dfrac{\pi}{6}\right)-y'\sin\left(\dfrac{\pi}{6}\right)=\dfrac{x'\sqrt3-y'}{2}}[/tex]
[tex]\rm{y=x'\sin\left(\dfrac{\pi}{6}\right)+y'\cos\left(\dfrac{\pi}{6}\right)=\dfrac{x'+y'\sqrt3}{2}}[/tex]
We are asked to find the transformed equation for,
[tex]\longrightarrow\rm{P:x^2+2\sqrt3\,xy-y^2=2a^2}[/tex]
We will give substitutions for x and y to find it.
[tex]\small\text{$\longrightarrow\rm{P':\left(\dfrac{x'\sqrt3-y'}{2}\right)^2+2\sqrt3\left(\dfrac{x'\sqrt3-y'}{2}\right)\left(\dfrac{x'+y'\sqrt3}{2}\right)-\left(\dfrac{x'+y'\sqrt3}{2}\right)^2=2a^2}$}[/tex]
Simplify this equation to get,
[tex]\small\text{$\longrightarrow\rm{P':(x')^2-(y')^2=a^2}$}[/tex]
Hence the transformed equation is,
[tex]\longrightarrow\rm{\underline{\underline{x^2-y^2=a^2}}}[/tex]
What is the answer of X² 6x 16?
The answers to the equation x2 – 6x = 16 are x = -2 and x = 8.
What is the answer of x2 6x 16?Two approaches to the equation x2 - 6x = 16 will be examined. First, factoring is used; second, the quadratic formula.
We shall employ the subsequent steps when factoring a solution:
Put all of the terms that are not zero on one side of the equation and zero on the other.
Add the non-zero component of the equation.
Solve the equation by setting each factor to 0.
Getting all non-zero terms on one side of the equation is the first thing want to do.
They are 8x and -2x, respectively. Therefore, (x+8) are the factors of x2 + 6x - 16 = 0. (x-2).
Therefore, the answers to the equation is x2 – 6x = 16 are x = -2 and x = 8.
To learn more about equations refer to:
https://brainly.com/question/2972832
#SPJ4
Write the equation of the line containing the points (-2,4) and (-4, 10)
What is the value of the polynomial 4x² 7x 5 when x 3?
When X=3 a polynomial has a value of 61 and when X=3 a value of 143.
Given that,
Let p(x)=3x²−4x²+7x−5 is our polynomial.
Polynomials are algebraic expressions made up of coefficients and variables. Occasionally, variables are referred to as indeterminates. For polynomial expressions, addition, subtraction, multiplication, and positive integer exponents are all possible; however, division by variable is not. x2+x-12 is an illustration of a single-variable polynomial. Three terms are involved in this example: x2, x, and -12.
Substitute x=3 in above expression,
∴p(3)=3(3) 3−4(3) 2 +7(3)−5
=3(27)−4(9)+21−5
=81−36+21−5
=61
Now,
p(−3)=3(−3)
3 −4(−3) 2 +7(−3)−5
=3(−27)−4(9)−21−5
=−81−36−21−5
=−143
So, value of polynomial when X=3 is 61 and when x=−3 is −143
To learn more about the polynomial click here :
brainly.com/question/11536910
#SPJ4
Does (3, 1) make the equation y = x true?
Answer:
no
Step-by-step explanation:
x=3, y=1
3≠1
therefore x≠y
What is the formula for the sum of the interior angle measures of a polygon
Answer: (n-2) * 180
Step-by-step explanation:
The formula for the sum of the interior angle measures of a polygon with n sides is (n - 2)180 degrees.
eight identical machines produce the same parts at the same rate. the 8 machines complete the required number of parts in 6.5 hours. how many hours does is take 12 machines to produce the same number of parts?
It becomes
[tex]$$\begin{aligned}5: 8: & 6.5: x \\5 \times x & =8 \times 6.5 \\x & =\frac{8 \times 6.5}{5} \\x & =\frac{52}{5} \\x & =10.4\end{aligned}$$[/tex]
Therefore,
5 machines will take $10.4$ hour to produce the same number of parts.
What is equal rate?Equivalent interest rates are interest rates that produce the same future value after one year. For example, 10% compounded quarterly and 10.125% compounded semiannually are equivalent nominal interest rates. If you calculate the future value of $100 invested at either rate for one year, you will obtain $110.38.
learn more about same rate to visit this link
https://brainly.com/question/2491293
#SPJ4
These are two separate questions
Answer:
Hi,
The first one is the second one, the one on the top right. This is because you add two branches every step.
I'm not so sure about the second question. sorry ! hope it helps tho
Suppose you invested $1200 for four years. You earned $312 in simple
interest. What is the interest rate?
Answer: 6.5%
Step-by-step explanation: 312/4 = 78
78/1200 = 0.065
0.065 x 100 = 6.5%
Write an expression to represent: The sum of three and the quotient of a number t and two.
Answer:
3+(t/2) or 3+t/2
Step-by-step explanation:
sum is addition
quotient is division
so, a quotient of T and 2 is t/2 (t divided by 2), and a sum of that and 3 is 3+(t/2)
Adding the parenthesis is optional :)
Which equation represents the same relationship
Answer:
The linear function which is passing through the points (-20, -268) and (-1, -40) is y + 268 = 12(x + 20). Then the correct option is C.
What is the linear system?
A Linear system is a system in which the degree of the variable in the equation is one. It may contain one, two, or more than two variables.
The table represents some points on the graph of a linear function.
Let the linear relationship is given below.
it possible to create an equation for a circle given exactly two points. describe exactly how one would do this. provide an example if necessary.
It is impossible to create an equation for a circle given exactly two points.
Let's consider the two basic equations of circle, i.e, the standard equation and general equation.
Now, Standard equation of circle with center (h, k) and radius r is represented as
(x−h)² +(y−k)² =r²
Note here we need values of three unknowns, h, k, r to get equation of a certain circle.
The other general equation of circle is of the form
x² + y² + 2gx + 2fy + c = 0
Again, values of three unknowns (g, f and c) are necessary for obtaining equation of a certain circle. So in both cases, values of three unknowns are required to have an equation of a distinct circle. In order to calculate, the values of these three unknowns, we must first require three conditions that would be produced by three equations. Solving those equations, allows us to construct the equation of circle. So if only two points are given, then these gives two equations with three unknowns. It would be impossible to get three distinct values from two equations. So, there is no possible distinct circle for two points.
To learn more about Equation of circle , refer:
https://brainly.com/question/1506955
#SPJ4
are these good or okayish grades??
Answer:
these are okayish grades but great effort.
Plot the graph of y = x³ - 2x for -2 ≤ x ≤2 on the axes below.
y =
-2
3
-1
0
1
2
The value of y are when x = -2, -1, 0, 1, 2 are -4, 1, 0, -1, 4. The graph of the function attached.
What is coordinate plane?
A Cartesian coordinate system in a plane is a system of coordinates that uniquely identifies each point by a pair of numerical coordinates. These numerical coordinates are the signed distances from two fixed perpendicular oriented lines to the point, measured in the same unit of length.
Given function is y = x³ - 2x
Putting x = -2 in the equation y = x³ - 2x:
y = (-2)³ - 2×(-2)
y = -4
Putting x = -1 in the equation y = x³ - 2x:
y = (-1)³ - 2×(-1)
y = 1
Putting x = 0 in the equation y = x³ - 2x:
y = (0)³ - 2×(0)
y = 0
Putting x = 1 in the equation y = x³ - 2x:
y = (1)³ - 2×(1)
y = -1
Putting x = 2 in the equation y = x³ - 2x:
y = (2)³ - 2×(2)
y = 4
The points on the equations are (-2,-4), (-1, 1), (0,0), (1,-1), (-2,4).
Plot the points and joint them by curve.
To learn more about graph, click on below link:
https://brainly.com/question/25270636
#SPJ1
How to know what integers and whoke numbers are
Answer:
integers have signs like -&+ while whole numbers are ALWAYS positive
Justine just ran her first race. Her distance from the finish line decreased as she ran.
This situation can be modeled as a linear relationship.
What does the y-intercept of the line tell you about the situation?
A. justine finished the race in 40 minutes
B. when justine started the race she was 10 kilometers from the finish line
C. after running for 20 minutes justine was 6 kilometers from the finish line
D. justine ran 4 kilometers every 20 minutes
Choices B and C are both correct statements.
Choices A and D are both false.
The y-intercept of the line tells us about the situation when Justine started the race she was 10 kilometers from the finish line.
What are y-intercept and x-intercept?The points where a line crosses an axis are known as the x-intercept and the y-intercept, respectively.
Given, a line whose y-intercept is at y = 10 which is represented by the distance remaining in kilometers. Since x shows the time spent running that means at x = 0 she started running.
Therefore, The y-intercept of the line reveals that Justine was 10 kilometers from the finish line when she started the race.
Learn more about y-intercept and x-intercept here:
brainly.com/question/14180189
#SPJ2
5* Le terrain de M. Campo mesure 1250 m².
C'est 450 m² de plus que celui de M. Vergne.
Quelle est la superficie du terrain de
M. Vergne ?
Answer:
800 m^2
Step-by-step explanation:
= 1250 - 450
= 800 m^2 est la superficie de la terre de M. Vergne
j'espère que ma réponse t'aidera.
Benjamin threw a rock straight up from a cliff that was 40 ft above the water. If the height of the rock h, in feet, after t
seconds is given by the equation h = - 16t* + 12t + 40, how long will it take for the rock CO hit the water?
Answer:
Step-by-step explanation:
To find the time it takes for the rock to hit the water, we need to find the value of t when h is equal to zero. This is because the rock will hit the water when it reaches a height of zero feet above the water.
Substituting zero for h in the equation h = -16t^2 + 12t + 40, we get:
0 = -16t^2 + 12t + 40
This equation is a quadratic in t, so we can use the quadratic formula to solve for t:
t = (-12 +/- sqrt(12^2 - 4 * (-16) * 40)) / (-32)
Simplifying, we get:
t = (-12 +/- sqrt(144 + 640)) / (-32)
t = (-12 +/- sqrt(784)) / (-32)
t = (-12 +/- 28) / (-32)
t = -40/-32 or t = 16/-32
t = 1/8 or t = -1/2
Since the time cannot be negative (because the rock was thrown upwards), the only solution that makes sense is t = 1/8 seconds.
Therefore, it will take 1/8 seconds for the rock to hit the water.
Which of the following is the value of the discriminant for 2x² 5x 7 0?
The value of the discriminant for 2x² + 5x + 7 = 0 is -31.
The discriminant of a quadratic equation is the part of the equation used to determine the number and type of solutions. It is calculated by subtracting 4 times the coefficient of the squared term, a, from the coefficient of the linear term, b, squared. This is then divided by four times the coefficient of the squared term, a.
In this case, the equation is 2x² + 5x + 7 = 0.
The discriminant for this equation is calculated as follows:
Discriminant = b² - 4ac
Discriminant = (5)² - 4(2)(7)
Discriminant = 25 - 56
Discriminant = -31
Therefore, the value of the discriminant for 2x² + 5x + 7 = 0 is -31.
Learn more about discriminant here:
https://brainly.com/question/15884086
#SPJ4
How many odd integers are there from 1 to 100?
There are 50 odd integers from 1 to 100.
What are odd integers?Odd integers are the numbers that cannot be divided by 2 evenly. It cannot be divided into two separate integers evenly. If we divide an odd integer by 2, then it will leave a remainder. In other words, numbers in the form of [tex]2k + 1[/tex], where [tex]k[/tex] ∈ [tex]Z[/tex] (i.e. integers) are called odd numbers.
Calculation:First odd number is 1 from 1 to 100.
Last odd number is 99 from 1 to 100.
Total odd numbers = 99+1 ÷ 2 = 50
There are 50 odd integers from 1 to 100.
To know more about odd integers , check out:
https://brainly.com/question/23709877
#SPJ4
work out the angle of abc
Answer: The angle is 90 degrees.
Step-by-step explanation:
A triangle that has a diameter as its hypotenuse inside a inscribed circle has a right angle opposite of its hypotenuse. This is true for any right triangle inscribed in a circle, because the equation to get measure B is:
(angle ABC) = 1/2 * (angle AOC)
Angle AOC is 180 degrees. Putting 180 into the euqation, we get 90 degrees for angle ABC.
3.75 ÷ 1.50
3
.
75
÷
1
.
50
is equivalent to 375 ÷ 150
375
÷
150
.
The dividend and divisor were both multiplied by .
The quotient of both expressions is?
Both were multiplied by 100 and the quotient of both expressions is 2.5.
What are arithmetic operations?
Arithmetic operations is a branch of mathematics that studies numbers and the operations on numbers that are useful in all other branches of mathematics. It consists primarily of operations like addition, subtraction, multiplication, and division.
We have,
3.75 ÷ 1.50 is equivalent to 375 ÷ 150.
Moving the decimal point 2 places to the right is equivalent to multiplication by 100.
so, both were multiplied by 100.
The quotient is:
3.75 ÷ 1.50 = 2.5
375 ÷ 150 = 2.5
Hence, both were multiplied by 100 and the quotient of both expressions is 2.5.
To learn more about arithmetic operations visit,
brainly.com/question/4721701
#SPJ1
FILL IN THE BLANK. The domain of f is the _____of f −1, and the _____ of f −1 is the range of f.
As per the concept of inverse function, the domain of f is the range of f⁻¹, and the domain of f⁻¹ is the range of f.
The term inverse function in math is referred as the function y=f(x) can be obtained by switching x and y and solving for y and then the value y will be replaced with f⁻¹ as this denotes the inverse of the function f.
Here we need to find the domain of f is the _____of f⁻¹, and the _____ of f⁻¹ is the range of f.
As we all know that the domain refers the set of all inputs of the function and range refers the set of all outputs of the function.
Based on these definition we have identified that the domain of function is always be the range of inverse function f.
Similarly, the range of inverse function f is always the domain of the function f.
To know more about function here.
https://brainly.com/question/28193995
#SPJ4
4(m - 2) = -2(3m + 3)
Answer:
m=1/5
Step-by-step explanation:
SImplify
4m-8=-6m-6
then 10m=2
so m=1/5
4(m - 2) = -2(3m + 3)
4m - 8 = -6m -6
4m + 6m = -6 + 8
10m = 2
m = 1/5 or 0.2
CHECKING:
4(0.2 - 2) = -2[3(0.2) + 3]
4(-1.8) = -2(0.6 + 3)
- 7.2 = -2(3.6)
- 7.2 = - 7.2
Therefore, the value of m is 1/5 or 0.2. We also proved that this is correct.
Step-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-How many times can you claim SSS sickness benefit?
In a given year, a member may get illness benefits for a maximum of 120 days. The total number of permitted compensable days for the next year cannot be increased by any unused fraction being carried forward. For the same condition, the sickness benefit cannot be awarded for more than 240 days.
A daily cash payment is made for each day a member is unable to work as a result of illness or accident.
Qualification Requirements
If a member meets the following criteria, they may be eligible to receive this benefit: Is unable to work due to illness or injury and is confined to a hospital or their home for a minimum of four (4) days;
Has duly notified the SSS directly of the fact of sickness or injury;
Has paid at least three (3) months' worth of payments within the 12-month period immediately preceding the semester of illness or injury.
For more questions on Sickness benefits
https://brainly.com/question/14288129
#SPJ4
Please help I will give brainliest
Answer:
B. 2.2 kilometers
Step-by-step explanation:
⭐What is the change in the cyclist's elevation?
The change in the cyclist's elevation is the value of the height (x) - the value of their starting point (0)Thus, we need to solve for the value of the height (x) of the hill.
Notice that the hill is in the shape of a right triangle. In order to find an unknown side length for right triangles, we need to use a trigonometric function.
⭐What is a trigonometric function?
There are 3 trigonometric functions:sin(θ) = opposite/hypotenusecos(θ) = adjacent/hypotenusetan(θ) = opposite/adjacentθ = an angle measureFirst, let's see what kind of side lengths we are given (opposite, hypotenuse, or adjacent).
We are given a side length of 10km, which is the hypotenuse of the triangle, and we are given a side length of x, which is the opposite of θ (13°).
Therefore, we will use the trigonometric function sin(θ) to solve for x.
Substitute the side lengths and angle measures we are given:
[tex]sin(13) = \frac{x}{10}[/tex]
Set the equation equal to x:
[tex]10sin(13) = x[/tex]
Solve for x. I recommend using a scientific calculator (e.g. Desmos Scientific Calculator):
[tex]2.2 = x[/tex]
∴ The cyclist's change in elevation is 2.2 kilometers.
the volume of the cone in 270 pi
Answer:
[tex] \displaystyle{\frac{2810}{3}} \pi \: \: \text{cm} ^{3} [/tex]
Step-by-step explanation:
The solid shape is composed of a cone and a hemisphere.
Let's work out the radius of the cone since the hemisphere and the cone share the same radius.
Let the radius be r cm.
Volume of cone= [tex] \displaystyle{\frac{1}{3} \pi {r}^{2} h}[/tex]
Substituting the volume and height of the cone:
[tex]270\pi = \displaystyle{\frac{1}{3} (\pi)( {r}^{2} )(10)}[/tex]
Simplify:
[tex] \frac{10}{3} {r}^{2} = 270[/tex]
[tex]81 = {r}^{2} [/tex]
Square root both sides:
[tex]r = \sqrt{81} [/tex]
r= 9
The volume of a hemisphere is half the volume of a sphere.
Volume of hemisphere
[tex] = \displaystyle{ \frac{1}{2}( \frac{4}{3} )(\pi )( {r}^{3}) }[/tex]
[tex] = \displaystyle{ \frac{2}{3}\pi {r}^{3} }[/tex]
Substituting the value of r:
Volume of hemisphere
[tex] = \displaystyle{ \frac{2}{3} (\pi)(10^{3} )}[/tex]
[tex] = \displaystyle{ \frac{2}{3}(1000)(\pi) }[/tex]
[tex] = \displaystyle{\frac{2000}{3} \pi}[/tex]
Volume of the shape
= volume of cone + volume of hemisphere
[tex] = \displaystyle{(270 + \frac{2000}{3} })\pi[/tex]
[tex] = \displaystyle{ \frac{2810}{3} \pi} \: \: \text{{cm}}^{3} [/tex]
To learn more about volume of composite shapes, check out: https://brainly.com/question/23576345
HELP ME PLEASE 10POINTS EACH WAIT FOR NEXT QUESTION
Answer:
2/1
Step-by-step explanation:
Answer:
the answer is 2!
Step-by-step explanation:
you count the rise over run
count from where it meets a whole number to where it meets another whole number and then you have your answer!
for this you count from (3,3) to (2,1) which is 2/1 making it 2
Are all equilateral triangles isosceles as well?
How does performing a rigid motion transformation help you prove congruence?
Only if two figures can be precisely mapped to one another using transformations are two figures said to be congruent.
What is congruence?
A "matching" figure is one that can be placed perfectly on another. The bread has the same size and form when it is stacked. Match refers to objects that are precisely the same size and form. If one of two geometric figures can be consistently overlaid on the other, they are said to be congruent or to be in a congruent relationship.
Only if two figures can be precisely mapped to one another using transformations are two figures said to be congruent. Due to the stiff transformations' preservation of the distance and angle scales, all related sides and angles are congruent.
To know more about congruence visit:
https://brainly.com/question/7888063
#SPJ4
Is zero a real number in math?
Answer:
Zero is a real number in math
Step-by-step explanation:
Zero is considered to be both a real and an imaginary number.