The value of the improper integral ∫∫r^2e^(-4r^2) dxdy using polar coordinates is (π/8).
We start by expressing the given integral in polar coordinates as follows:
∫∫r^2e^(-4r^2) dxdy = ∫∫r^2e^(-4r^2) r dr dθ
The limits of integration for r are 0 to infinity and for θ are 0 to 2π. Hence, the integral becomes:
∫0^(2π) ∫0^∞ r^3 e^(-4r^2) dr dθ
We can evaluate the integral using the substitution u = 4r^2, du = 8r dr, and limits of integration from 0 to infinity. This gives:
(1/8) ∫0^(2π) ∫0^∞ e^(-u) du dθ
Solving the inner integral with limits 0 to infinity gives (1/8) ∫0^(2π) 1 dθ = π/4
Therefore, the value of the given integral in polar coordinates is (π/8).
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Andrea is standing in the ground and can see the too of a cliff that is 215 feet tall. If the angle of elevation feom Andrea’s eyw level to the top of the cliff is 42 degrees and her eye level is 5.25 feet above groud, hiw far is Andrea from the base of the cliff? Round your answer to the nearest tenth.
Andrea's distance from the base of the cliff is given as follows:
233 feet.
What are the trigonometric ratios?The three trigonometric ratios are defined as follows:
Sine of angle = length of opposite side divided by the length of the hypotenuse.Cosine of angle = length of adjacent side divided by the length of the hypotenuse.Tangent of angle = length of opposite side divided by the length of the opposite side.For this problem, we have a right triangle in which:
The base angle is of 42º.The adjacent side is the ground distance.The opposite side is the height of 215 - 5.25 = 209.75.Hence the distance is obtained as follows:
tan(42º) = 209.75/d
d = 209.75/tan(42º)
d = 233 feet.
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Kevin ran at a rate of 13 km/h. Convert his speed to meters per minute
Answer:
Step-by-step explanation:
m/s =kph divided by 3.600000
m/s =13 divided by3.600000
m/s =3.611111
Answer: 216.67 meters/ min
Step-by-step explanation: Converting 13 km/h to meters per minute
First, we will convert 13 km to meters
So, we know 1 km = 1000 meters
Then 13 km/ h = x
x = 1000 multiplied 13
x = 13,000 meters/ h
Now, we will convert to hours to minutes
We know 1 h = 60 min
Then 13,000 meters/ 60 min
That = 216.667 meters/ min
After approximating
Answer = 216.7 meters / min
10(b+2)+3=73 distributive property
Answer:
b = 5
Step-by-step explanation:
10(b+2) + 3 = 73
10b + 20 + 3 = 73
10b = 50
∴ b = 5
NEED ANSWER PLS!
WORTH 60 POINTS!
Select the correct answer.
Given that a function, g, has a domain of -20 ≤ x ≤ 5 and a range of -5 ≤ g(x) ≤ 45 and that g(0) = -2 and g(-9) = 6, select the statement that could be true for g.
A.
g(0) = 2
B.
g(-13) = 20
C.
g(7) = -1
D.
g(-4) = -11
Answer:
7
Step-by-step explanation:
7
What is a real image for Class 7?
In optics, an image is defined as the group of focus points of light rays coming from an object. A real image is a collection of focus points actually made by converging rays, while a virtual image is a collection of focus points made by extensions of diverging rays.
The real image is produced by the intersection of light rays. It can be obtained on a screen. Hence, projectors form real images.
The virtual image is formed when the light rays appear to be originating from a point but do not actually meet. It can be seen by human eyes.
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A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 447 gram setting. Based on a 24 bag sample where the mean is 444 grams and the standard deviation is 10, is there sufficient evidence at the 0.1 level that the bags are underfilled or overfilled? Assume the population distribution is approximately normal.
Step 3 of 5: Specify if the test is one-tailed or two-tailed.
The test that we have here is the one tailed test.
How do we know that the test is one tailed?From the direction of change of the parameter, we would be able to tell if the test is a one tailed test or not. That is, if the parameter is increasing or decreasing.
This test is looking to determine if the bags are underfilled (decreased) or if they are overfilled (increased).
The test's alternative hypothesis is written in a way that it predicts the direction of the difference between two groups. The test is one-tailed if the alternative hypothesis predicts a direction of change
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Does 2 dilated mean?
2 dilated means that the measurement of an area has increased or expanded in size.
This can be a linear measurement such as the diameter of a circle, or the area of a square or rectangle. The formula for the area of a circle is A=πr2 where r is the radius of the circle. A 2 dilated measurement would mean that the radius has doubled, so the formula would be A=π(2r)2 A=4πr2. This would mean that the area of the circle has increased by a factor of 4. Similarly, if the length and width of a rectangle were 2 dilated, the area would be A=2L2W, which would be four times the original area.
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Who is the father of geometry?
Euclid is the father of geometry
Euclid and René Descartes was often called the father of modern mathematics because he created analytic geometry and the Cartesian coordinate system.
What is modern mathematics?
Different methodologies have been used in contemporary mathematics. It focuses mostly on entities, whereby interconnections follow themes and patterns.
René Descartes was a person who was very famous for his philosophy of science and mathematics. He was considered a person who evaluate and made various new development in the field of math as well as time in separate fields like geometry and algebra with new properties that is developed by him.
He had the main idea of having finite and infinite particles or numbers in geometry. And also used various scientific methods to prove that.
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What is an R value that tells you how well a line of best fit fits the data?
Correlation Coefficient (r) o A model is deemed to be "excellent fit" if r is near to 1 (or -1). o The model is "not a good match" if r is close to 0. o The model has a "perfect fit" if r = 1, with all data points falling on the line. o The two variables do not have a linear connection if r = 0.
What is Correlation Coefficient?The strength of the linear link between two separate variables, x and y, is shown by correlation coefficients.
A positive relationship is shown by a linear correlation coefficient greater than zero.
Indicating a negative association is a value less than zero. The two variables x and y do not have any relationship when their values are 0, to sum up.
The correlation coefficient (r2) is a statistic that expresses how closely two variables' movements are related to one another.
To determine the linear relationship between two variables, the Pearson product-moment correlation, the most popular correlation coefficient, is used.
However, this correlation value might not necessarily be a reliable indicator of reliance in a non-linear connection.
Hence, Correlation Coefficient (r) o A model is deemed to be "excellent fit" if r is near to 1 (or -1).
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Somebody help please I don’t understand!!
The unit rate of the 76th player is [tex]1\frac{1}{3}[/tex] miles per hour.
The unit rate of the 91st player is [tex]1\frac{1}{9}[/tex] miles per hour.
The unit rate of the 44th player is [tex]1\frac{1}{2}[/tex] miles per hour.
What is an improper fraction?
An improper fraction is one in which the denominator is greater than or equal to the numerator. 5/2 and 8/5, for example, are incorrect fractions. Every fraction is made up of two parts: the numerator and the denominator. Proper fractions and improper fractions are the two main types of fractions in mathematics based on the numerator and denominator values.
For 76 number of players:
In 1/4 hour he can run 1/3 mile
In 1 hour he can run (1/3)/(1/4)mile
=4/3 miles
Convert it into a mixed number:
= [tex]1\frac{1}{3}[/tex] miles per hour
For 91 number of players:
In 3/4 hour he can run 5/6 mile
In 1 hour he can run (5/6)/(3/4)mile
=10/9 miles
Convert it into a mixed number:
= [tex]1\frac{1}{9}[/tex] miles per hour
For 44 number of players:
In 1/2 hour he can run 3/4 mile
In 1 hour he can run (3/4)/(1/2)mile
=6/4 miles
=3/2 miles
Convert it into a mixed number:
= [tex]1\frac{1}{2}[/tex] miles per hour
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In the of 76th player's unit rate is 1 1/3 miles per hour , 91st player's unit rate is 1 1/9miles per hour and the 44th player's unit rate is 1 1/2miles per hour.
What is meant by an unit rate?When expressed as a fraction, a unit rate has a denominator of one unit. To write a rate as a unit rate, divide the rate's numerator and denominator by the rate's denominator.
For a total of 76 players:
He can run 1/3 mile in 1/4 hour.
He can run (1/3)/(1/4) mile in 1 hour.
= 4/3 miles
Change it to a mixed number:
= 1 1/3 miles per second.
For a total of 91 players:
He can run 5/6 mile in 3/4 hour.
He can run (5/6)/(3/4) mile in an hour.
= 10/9 miles
Change it to a mixed number:
= 1 1/9 miles per hour.
For a total of 44 players:
He can run 3/4 mile in half an hour.
He can run (3/4)/(1/2) mile in 1 hour.
=6/4 miles
=3/2 miles
Change it to a mixed number:
= 1 1/2 miles per hour.
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my square orchid garden abuts my house so that the house itself forms the northern boundary. the fencing for the southern boundary costs $5 per foot, and the fencing for the east and west sides costs $3 per foot. find the total cost of the fencing as a function of the length (in feet) of a side x.
The total cost of the fencing as a function of the length of the x side is 11x
What are algebraic operations?They are the set of numbers and symbols that are related by the different mathematical operation signs such as addition, subtraction, multiplication, division among others.
To solve this problem we must perform the following algebraic operations with the given information
Information about the problem:
southern = 5$east = 3$west = 3$side = xTotal cost = ?Calculating the total cost:
Total cost = (southern + east + west) * side
Total cost = (5 + 3 + 3)*x
Total cost = 11x
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Calculator What is the volume of this square pyramid? 48 cm³ 96 cm³ 144 cm³ 288 cm² 4 cm 8cm 5 cm 3 cm
This square pyramid has a volume of 48 cm³.
Explain about the square pyramid?The term "square pyramid" refers to a three-dimensional geometric form with a square base and four faces or sides that are triangular and meet at a single point (called the vertex). Its five faces give it the name "pentahedron."
The five faces of a square pyramid, four of which are triangles, are square in shape. An upright pyramid's base is also square-shaped.
Although the square pyramid and square prism both have a square base, the four lateral faces of the former are triangular and the latter are rectangular.
V=13s2h, where s is the length of the square base's sides, is a formula that may be used to calculate the volume of a pyramid whose base is square.
v = 1/3 * b² *h
v = 1/3 * 6² *4
v = 1/3 *36 *4
V = 48cm³
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Can all linear and exponential functions be defined with both recursive and explicit equations? Explain your answer.
Step-by-step explanation:
I never thought about this before, but yes, they can.
linear functions :
explicit :
y = ax + b
recursive :
xn+1 = xn + a
because
a(x + 1) + b = ax + b + a
ax + a + b = ax + a + b
exponential functions :
explicit :
y = a^x
recursive :
xn+1 = xn × a
because
a^(x + 1) = a^x × a = a^(x + 1)
remember,
a^x = a×a×a×a×a×...×a×a×a
x times a multiplied by itself.
Pleaseeee help me thx
When the length of JK is 9.5 feet and the angle L is 61 degrees trigonometric ratio of the tangent, solving yields: KL=9.5ft/1.804 KL=5.27 ft.
what is trigonometry ?The relationship between triangle side lengths and angles is investigated in the branch of mathematics known as trigonometry. The application of geometry in astronomical study gave rise to the area's first appearance during the Hellenistic era, roughly in the third century BC. Exact methods mathematics focuses on certain trigonometric functions and how they might be applied to calculations. In trigonometry, there are six widely used trigonometric functions. Their respective names and acronyms are sine, cosine, tangent, cotangent, secant, and cosecant (csc). Trigonometry is the study of the properties of triangles, especially of right triangles. The features of all geometric shapes are studied in geometry, though.
given
△JKL
JK=9.5ft
∠L=61∘
Find:KL
We may determine the outcome using the tangent's trigonometric ratio.
Tan = JK/KL = 61
When we separate KL, we get KL=9.5ft/tan61.
When the length of JK is 9.5 feet and the angle L is 61 degrees trigonometric ratio of the tangent, solving yields: KL=9.5ft/1.804 KL=5.27 ft.
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The correct question is :-
Given triangle JKL, find the length of KL. The length of JK is 9.5 ft and the angle L is 61 degrees.
Conider the problem, in 3/4 cup of oatmeal, there i no 1/10 of the recommended daily value of iron. What fraction of the daily recommended value of iron i in 1 cup of oatmeal. (1 multiplication and diviion equation)
The fraction which is recommended as value of iron in 1 cup of oatmeal as per the given relation is equal to 2/15.
As given in the question,
Daily recommended value of iron in the given cup of oatmeal
3 / 4 cup of oatmeal gives 1/10 daily value of iron.
Fraction recommended as daily value of iron is given by :
⇒ 3 /4 cup of oatmeal = 1 /10 value of iron
⇒ 1 cup of oatmeal = ( 1 × 4 ) / ( 3 × 10 ) value of iron
⇒ 1 cup of oatmeal = ( 4 ) / ( 30 ) value of iron
⇒ 1 cup of oatmeal = ( 2 ) / ( 15 ) value of iron
Therefore, the fraction recommended as daily value of iron is equal to
( 2/15 ).
The above question is incomplete, the complete question is :
In 3/4 cup of oatmeal, there is 1/10 of the recommended daily value of iron. What fraction of the daily recommended value of iron is in 1 cup of oatmeal? ( multiplication and division equation)
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given the function f(x)=-2x+3 complete the table
Answer:
Outputs are-
f(-2) = 7
f(-1) = 5
f(0) = 3
f(3) = -3
Step-by-step explanation:
Given, f(x) = -2x+3
To find, 1) f(-2)
2) f(-1)
3) f(0)
4) f(3)
Solution,
1) Substituting x for -2,
f(-2) = -2 x -2 + 3 = 7
2) Substituting x for -1,
f(-1) = -2 x -1 + 3 = 5
3) Substituting x for 0,
f(0) = -2 x 0 + 3 = 3
4) Substituting x for 3,
f(3) = -2 x 3 + 3 = -3
a designer made two tunnels. Tunnel A is represented by x^2+y^2+28x+5=0 and tunnel b is resented by x^2-36x+16y+68=0, where al measurements are in feet. The designer wants to verify whether a 8 feet wide and 13.5 feet high can pass through the tunnel.
Determine the maximum height of each tunnel, is the truck able to pass through either tunnel without damage, if so which tunnel(s) and why
a) Tunnel A is a circle with center (-14, 0) and radius 12, meaning that both the width and the height are of 12.
b) Tunnel B is a parabola that has a width of 32 feet and a height of 16 feet.
c) The maximum height of Tunnel A is of 12 feet and of Tunnel B is of 16 feet, hence, since the height of the truck is of 13.5 feet, it can pass through Tunnel B without damage, as it's height is greater than 13.5 feet.
What is the equation of a circle?The equation of a circle of center and radius r is given by:
(x-x₀) +(y-y₀) = r²
For Tunnel A, the equation is given by:
x² + y² + 28x + 52 = 0
Completing the squares:
(x + 14)² + y² = -52 + 14²
(x + 14)² + y² = 144.
Tunnel A is a circle with center (-14, 0) and radius 12, meaning that both the width and the height are of 12.
A quadratic equation is modeled by:
y = ax^2 + bx + c
For Tunnel B, the equation is given as follows:
x² - 36x + 16y + 68 = 0.
16y = -x² + 36x - 68.
Simplifying by 16:
y = -0.0625x² + 2.25x - 4.25.
Hence the coefficients are a = -0.0625, b = 2.25, c = -4.25, and the maximum height is:
y₀=(2.25)²-49-0.0625)(-1.25)/4(-0.0625) = 16
The width is the difference between the roots, which, using a calculator, are: 2 and 34, hence the width is of 32 feet.
This implies that the tunnel B is a parabola that has a width of 32 feet and a height of 16 feet.
For item c, we have that:
The maximum height of Tunnel A is of 12 feet and of Tunnel B is of 16 feet, hence, since the height of the truck is of 13.5 feet, it can pass through Tunnel B without damage, as it's height is greater than 13.5 feet.
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Correct question:
An architect has designed two tunnels. Tunnel A is modeled by x2 + y2 + 28x + 52 = 0, and tunnel B is modeled by x2 – 36x + 16y + 68 = 0, where all measurements are in feet. The architect wants to verify whether a truck that is 8 feet wide and 13.5 feet high can pass through the tunnels.
Part A: Write the equation for Tunnel A in standard form and determine the conic section. Show your work. (4 points)
Part B: Write the equation for Tunnel B in standard form and determine the conic section. Show your work. (4 points)
Part C: Determine the maximum height of each tunnel. Is the truck able to pass through either tunnel without damage? If so, which tunnel(s) and why? Show your work. (7 points)
Use the quadratic formula to find the solutions to the quadratic equation
below.
x-5x-4 = 0
O A. x =
O
-5+√41
2
B. x = 5±√41
O c. x =
C.
-5+√/29
D. x = 5√2
On solving the provided question we cans say that quadratic equation - x^2-5x-4 = 0 and x = 1 , 3
What is quadratic equation?A quadratic equation is x ax2+bx+c=0, which is a quadratic polynomial in a single variable. a 0. The Fundamental Theorem of Algebra ensures that it has at least one solution since this polynomial is of second order. Solutions may be simple or complicated. An equation that is quadratic is a quadratic equation. This indicates that it has at least one word that has to be squared. The formula "ax2 + bx + c = 0" is one of the often used solutions for quadratic equations. where are numerical coefficients or constants a, b, and c. where the variable "X" is unidentified.
here, we have
quadratic equation -
x^2-5x-4 = 0
x(x-1) -3(x-3) =
(x-1)(x-3) = 0
x = 1 , 3
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Evaluate.
33 + (23)2 I will probally give u brainlest
A.27 4/9
B.9 2/3
C.9 4/9
D.27 2/3
Answer:
69
Step-by-step explanation:
33 + (23)2 =
= 33 + 46
= 69
This is the correct answer to the problem you posted. If you meant something else, you should have written it differently.
Use / for fractions and ^ for exponents.
Prove that, cosecA/(tanA + cot A) =cosA
The proof is given below
We can start by using the definition of the trigonometric functions cosec(A), tan(A), and cot(A):
cosec(A) = 1/sin(A)
tan(A) = sin(A)/cos(A)
cot(A) = 1/tan(A) = cos(A)/sin(A)
Now, we can substitute these definitions into the equation:
cosec(A)/(tan(A) + cot(A)) = 1/sin(A) / (sin(A)/cos(A) + cos(A)/sin(A))
Simplifying the right-hand side gives us:
1/sin(A) / (sin(A)(1/cos(A) + 1/sin(A))) =
1/sin(A) / (sin(A)/sin(A)cos(A) + sin(A)) =
1/sin(A) / (1 + sin(A))
Finally,
multiplying both sides by sin(A) (which is not 0 because A is not a multiple of pi/2)
cosec(A)/(tan(A) + cot(A)) * sin(A) = 1/sin(A) / (1 + sin(A)) * sin(A) = 1 / (1 + sin(A)) * sin^2(A) = cos(A)
Therefore, cosec(A)/(tan(A) + cot(A)) = cos(A)
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What is the relationship between <4 and <15? *
The Least Common Factor of 4 and 15 is 1. The factors of 4 are 1, 2 and 4, and the factors of 15 are 1, 3, 5 and 15. Hence, 1 is the Least Common Factor of 4 and 15.
What's meant by Factor?A factor is a number that fully divides another number.
To put it another way, if adding two whole figures results in a product, also the figures we're adding are factors of the product because the product is separable by them.
Factor simply translates to" doer" in Latin. therefore, a" factor" in English is a" actor,"" element," or" element" in a situation or number.
Success can depend on charm, and having a terrible constitution can affect from inadequate exercise.
Variable factors are those whose employment varies in proportion to affair; further are employed when product rises and smaller when product falls.
Labor, energy, and raw accoutrements employed directly in product are exemplifications of typical variable rudiments.
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epare for
D
Question 3
If 2*x+2*10%-55, x=? (please keep 1 decimal)
The value of x in the given equation is 17.5
What are the equations?A mathematical statement that has a "equal to" symbol between two expressions with equal values is called an equation. as in 3x + 5 Equals 15, for instance. Equations come in a variety of forms, including linear, quadratic, cubic, and others. In this article, let's learn more about math equations.
What are the 3 types of equations?There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form.
In the given equation:-
2x+20-55=0
x=35/2
x=17.5
Hence the value of x is 17.5
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A 14-foot ladder is set up 4 feet from the base of a building. How far up the building does the ladder reach? Round your answer to the nearest tenth of a foot.
The distance up the building that the ladder would reach, given the ladder height and the base of the building, is 13.4 ft.
How to find the height ?To find out how far up the building the ladder reaches, we can use the Pythagorean theorem. The theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side (the hypotenuse).
Assuming the height that the ladder reaches is x, the formula would be:
x ² + 4 ² = 14 ²
x ² = 14 ² - 4 ²
x ² = 196 - 16
x = √ 180
x = 13. 4 ft
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Use the information about four different waves to answer the question.
Which wave contains the lowest energy?
-wave 2
-wave 1
-wave 3
-wave 4
^
(screenshot is for this one)
X-ray technicians wear protective gear to keep them safe from the X-rays. Why are X-rays dangerous?
They have short wavelengths that can penetrate materials, and their low energy can damage matter.
They have short wavelengths that can penetrate materials, and their high energy can damage matter.
They have long wavelengths that can penetrate materials, and their low energy can damage matter.
They have long wavelengths that can penetrate materials, and their high energy can damage matter.
1) X-Rays are dangerous because: "They have short wavelengths that can penetrate materials, and their high energy can damage matter." (Option B)
2) From the table given, the wave that has the lowest energy is Wave 4.
What is the relationship between Frequency and wavelength?The relationship between frequency and wavelength is inverse. The wavelength of the wave with the highest frequency is the shortest. One-half the wavelength equals twice the frequency. As a result, the wavelength ratio is inverse to the frequency ratio.
To summarize, waves transport energy. The quantity of energy carried by them is proportional to their frequency and amplitude. The greater the frequency, the greater the energy, and the greater the amplitude, the greater the energy.
Also, the lower the frequency, the lower the energy, and the lower the amplitude, the lower the energy. Thus, the wave with the lowest energy is Wave 4.
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Identify or mark the missing side or angle that would make triangle ABC congruent to triangle PDF by SAS.
Answer: Angle C for AAS, Side AC for SAS
Step-by-step explanation:
Your instructions were confusing since in the text it asks for SAS and in the image it asks for AAS so I just gave both.
At the craft show, Mai purchases rings and bracelets. Rings cost `\$1` each and bracelets cost `\$2` each. Mai spends a total of `\$20` on these items. Let `r` represent the number of rings Mai buys. Let `b` represent the number of bracelets Mai buys.
(a) Write an equation relating the two variables.
(b) Rewrite your equation from part a so that it gives `b` as the dependent
(c) Rewrite your equation from part a so that it gives `r` as the dependent variable in terms of `b` as the independent variable. variable in terms of `r` as the independent variable.
Pleaseee hurry. In a really important test.
The equation relating the two variables is r + 2b = 20.
The equation with r as the dependent variable is r = 20 - 2b.
The equation with b as the dependent variable is b = (20 - r)/2.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2xx + 3 = 9 is an equation.
We have,
Number of rings = r
Number of bracelets = b
Cost of one ring = $1
Cost of one bracelets = $2
Total cost = $20
Now,
r + 2b = 20
The equation with r as the dependent variable.
r = 20 - 2b
The equation with b as the dependent variable.
2b = 20 - r
b = (20 - r)/2
Thus,
The equation with r as the dependent variable is r = 20 - 2b.
The equation with b as the dependent variable is b = (20 - r)/2.
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19/2-1/3-9/6=?????????
Answer:
[tex] \frac{23}{3} ....in \: fraction \: or \\ 7.666666...in \: decimal[/tex]
Step-by-step explanation:
Greetings!!!
look at the image attached
Hope it helps!!!
1. What is the standard form of the equation of a circle and what do each of the variables represent?
2. What are the standard forms of the equation of a parabola and what do each of the variables represent?
3. Explain the relationship between the vertex, focus and directrix.
4. What are the standard forms of the equation of an ellipse and what do each of the variables represent?
5. Explain the relationship between the foci, vertices, and endpoints of the minor axis of an ellipse.
6. What are the standard forms of the equation of a hyperbola and what do each of the variables represent?
7. Explain the relationship between the foci, vertices, and conjugate axis of a hyperbola.
Answer:
1) The standard form of a circle's equation is (x - h)² + (y - k)² = r². (h,k) is the center and r is the radius.
2) The vertex is the midpoint of the directrix and the focus. It involves the parabola changing directions, and at any point the directrix and the focus will be equidistant from the parabola. The focus is a single point inside the curve of the parabola, and the directrix is a line outside the parabola.
3) The standard form of an equation of an ellipse is:
(x − h)2a 2 +( y − k) 2 b 2 = 1 ( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1. (h,k) is the center, a is the distance from the center to the horizontal vertices and b is the distance from the center to the vertical vertices.
4) The foci are (−c,0) and (c,0). The ellipse is the set of all points (x,y) which are the sum of the distances from (x,y) to the foci is constant. (a,0) is a vertex of the ellipse. Finally, the endpoints of the minor axis is a co-vertex of the ellipse.
5) A hyperbola has two standard equations. These equations of a hyperbola are based on its transverse axis and conjugate axis. The standard equation of the hyperbola is [(x² / a²) – (y² / b²)] = 1, where the X-axis is the transverse axis and the Y-axis is the conjugate axis.
6) The foci are located on the line that where the transverse axis is. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect.
Let me know if I didn't answer all of your questions, I hope this helps! :)
Step-by-step explanation:
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We can use some rules for exponents to simplify the expression, we will get:
7⁵/7¹ = 7⁴
How to simplify the expression?Here we need to use two properties about exponents, these are:
The exponent of 1 bassically does nothing:
x¹ =x
The rule for divisions is:
xᵃ/xᵇ = x⁽ᵃ ⁻ ᵇ ⁾
Now we have the expression:
7⁵/7
We can rewrite that as:
7⁵/7¹
Now we can use the second property to get:
7⁵/7¹ = 7⁵ ⁻ ¹ = 7⁴
That is the simplified expression.
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The line with equation y = mx is a tangent to the circle C₁ with equation
(x+4)² + (y-7)² = 13.
(a) Show that m satisfies the equation
3m² + 56m +36=0.
The tangents from the origin O to C₁ touch C₁ at the points A and B.
(b) Find the coordinates of the points A and B.
Another circle C₂ has equation x² + y² = 13. The tangents from the point (4, -7) to C₂ touch it at
the points P and Q.
(c) Find the coordinates of either the point P or the point Q.
On solving the provided question we can say that value equation is y = mx => -7 = m * 4 => m = -7/4 and x² + y² = 13 => x² + (-7)² = 13 => x
What is equation?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence.
Squaring both sides and simplifying we get:
3m² + 56m + 36 = 0
The coordinates of the points A and B are (-12,103) and (-3,-89) respectively.
y = mx => -7 = m * 4 => m = -7/4
x² + y² = 13 => x² + (-7)² = 13 => x
So the equation of the tangent line is y = m(x - 0) + 0, where m = -(x+4)/(y-7)
Now we can substitute the coordinates of the center of the circle into the equation of the tangent line to get y = -(x+4)/(y-7) * x
Squaring both sides and simplifying we get:
3m² + 56m + 36 = 0
b) To find the coordinates of the points A and B, we can use the equation of the tangent line and the equation of the circle.
We know that the equation of the tangent line is y = -(x+4)/(y-7) * x and the equation of the circle is (x+4)² + (y-7)² = 13.
We can substitute y = -(x+4)/(y-7) * x into the equation of the circle and solve for x.
This equation simplifies to 3x² + 56x + 36 = 0, which factors to (x+12)(x+3) = 0
y = -(-12+4)/(y-7) * -12 = 8/(y-7) * 12 = 96/(y-7)
y = 96 + 7 = 103 and y = -96 + 7 = -89
The coordinates of the points A and B are (-12,103) and (-3,-89) respectively.
We know that the equation of the tangent line is y = mx and the equation of the circle is x² + y² = 13.
We can substitute the point (4, -7) for x and y in the equation of the tangent line and the equation of the circle and solve for m and x,y respectively.
y = mx => -7 = m * 4 => m = -7/4
x² + y² = 13 => x² + (-7)² = 13 => x
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