Answers
Due to length restrictions, we kindly invite to check the explanation herein for further details of the hyperbola.
How to analyze an hyperbola
Herein we have an hyperbola whose axis of symmetry is parallel to the y-axis and the major semiaxis length is in the y-direction. By analytical geometry, we know that eccentricities of hyperbolae are greater than 1.
a) The formula for eccentricity is:
e = √(a² + b²) / a (1)
Where:
a - Major semiaxis lengthb - Minor semiaxis lengthIf we know that a = 4 and b = 3, then the eccentricity of the hyperbola is:
e = √(4² + 3²) / 4
e = 5 / 4
b) The coordinates of the two vertices of the hyperbola are:
V(x, y) = (h, k ± a) (2)
Where (h, k) are the coordinates of the center of the hyperbola.
V₁ (x, y) = (0, 4), V₂ (x, y) = (0, - 4)
The coordinates of the foci of the hyperbola are:
F(x, y) = (h, k ± c), where c = √(a² + b²). (3)
c = √(4² + 3²)
c = 5
F₁ (x, y) = (0, 5), F₂ (x, y) = (0, - 5)
The equations of the asymptotes of the hyperbola are:
y = ± (a / b) · x
y = ± (4 / 3) · x (4)
And the equations of the directrices of the hyperbola are:
y = k ± (2 · a - c)
y = 0 ± (8 - 5)
y = ± 3 (5)
The graph is presented in the image attached below.
c) The parametric equations for the hyperbola are the following formulae:
y = ± a · cosh t → y = ± 4 · cosh t (6)
x = b · sinh t → x = 3 · sinh t (7)
d) First, we determine the slopes of the two tangent lines by implicit differentiation:
m = (16 · x) / (9 · y)
m = (16 · 2.3) / [9 · (± 4.807)]
m = ± 0.851
Second, we find the intercept of each tangent line:
(x, y) = (2, 4.807)
b = 4.807 - 0.851 · 2
b = 3.105
y = 0.851 · x + 3.105 (8)
(x, y) = (2, - 4.807)
b = - 4.807 - (- 0.851) · 2
b = - 3.105
y = - 0.851 · x - 3.105 (9)
e) The definite integral of the arc length of the hyperbola is presented below:
[tex]s = \int\limits^{2}_{1} {\sqrt{\left(\frac{dx}{dt} \right)^{2}+\left(\frac{dy}{dt} \right)^{2}}} \, dt[/tex]
If we know that dx / dt = a² · sinh² t and dy / dt = b² · cosh² t, then the definite integral for the arc length is:
[tex]s = \int\limits^2_1 {\sqrt{a^{2}\cdot \sinh ^{2}t +b^{2}\cdot \cosh^{2}t}} \, dt[/tex] (10)
f) We apply the following substitutions on (1): x = r · cos θ, y = r · sin θ. Then, we have the polar form by algebraic handling:
r(θ) = (a · b) / (b² · sin² θ - a² · cos² θ) (11)
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Related Questions
What is foreshortening?
a.
a technique used to create light and shadow
b.
a technique used to create depth in an image
c.
a technique that draws attention to the foreground
d.
a technique of shading with black pencil
Please select the best answer from the choices provided
A
B
C
D
Answers
Answer:
c
Step-by-step explanation:
Foreshortening is a technique used in perspective to create the illusion of an object receding strongly into the distance or background.
Foreshortening is a technique used to create the illusion of depth and three-dimensionality in a two-dimensional image.
Option B is the correct answer.
What is foreshortening?Foreshortening is a technique used in art to create the illusion of depth and three-dimensionality in a two-dimensional image.
It involves distorting the shapes of objects or figures to make them appear closer or farther away from the viewer, often by shortening the lengths of their features or changing the angle from which they are viewed.
Foreshortening is often used in drawing, painting, and other visual arts to create a sense of perspective and realism.
We have,
Foreshortening is a technique used to create the illusion of depth and three-dimensionality in a two-dimensional image.
It involves distorting the shapes of objects and figures to make them appear closer or farther away from the viewer, often by shortening the lengths of their features or changing the angle from which they are viewed.
Foreshortening can be used in drawing, painting, and other visual arts to create a sense of perspective and realism.
Thus,
Foreshortening is a technique used to create the illusion of depth and three-dimensionality in a two-dimensional image.
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Find two positive numbers whose difference is 9 and whose product is 2950.
Answers
Answer: 50 and 59
I divided 2950 by 25 because I knew it would somehow factor into that since it ends in 50. Then, I got 25 and 118. From there, I just divided 118 by 2 since I knew that it would end in a 9 and I multiplied 25 by 2. Not sure if this is really helpful, but you also try using a GCF calculator and look up all the factors of that number next time!
Find the equation of the line that passes through point (6, 1) with the x-intercept of 2.
Answers
Answer:
○ [tex]y = \frac{1}{4}x - \frac{1}{2}[/tex]
Step-by-step explanation:
To find the equation of the line, let's first consider the points whose coordinates we have been given:
• (6, 1)
• (2, 0).
The point (2, 0) is what is called the x-intercept, which is the point where the line crosses the x-axis. This means that at this point, the y-coordinate of the line is 0.
Next, let's calculate the slope (gradient) of the line using the formula:
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
where:
m = gradient,
[tex](x_2, y_2)[/tex] and [tex](x_1, x_2)[/tex] = points on the line.
Using the formula:
[tex]m = \frac{1 - 0}{6 - 2}[/tex]
⇒ [tex]m =\bf \frac{1}{4}[/tex]
Finally, now that we have two points on the line as well as the line's slope, we can use the following formula to find the equation of the line:
[tex]\boxed{y - y_1 = m(x - x_1)}[/tex]
You can use any of the points on the line as [tex]y_1[/tex] and [tex]x_1[/tex].
Using (2, 0):
[tex]y - 0 = \frac{1}{4}( x - 2)[/tex]
⇒ [tex]y = \frac{1}{4}x - \frac{1}{2}[/tex]
Therefore the equation of the line is [tex]y = \frac{1}{4}x - \frac{1}{2}[/tex].
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Finding the sums
Find the sum of the first 7 terms of the following geometric series.
1/36-1/12+1/4-...
what is the sum?
Answers
The sum of the first 7 terms of the geometric series is 15.180
Sum of geometric seriesThe formula for calculating the sum of geometric series is expressed according to the formula. below;
GM = a(1-r^n)/1-r
where
r is the common ratio
n is the number of terms
a is the first term
Given the following parameters from the sequence
a = 1/36
r = -3
n = 7
Substitute
S = (1/36)(1-(-3)^7)/1+3
S = 1/36(1-2187)/4
S = 15.180
Hence the sum of the first 7 terms of the geometric series is 15.180
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Can segments with lengths of 6, 7, and 9 form a triangle? Why or why not
Answers
Answer:
Yes
Step-by-step explanation:
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side or largest side. So, 7+6 = 13 which is greater than 9
Answer:
Yes, triangle with sides 6, 7 ,9 can be formed.
Step-by-step explanation:
The sum of any two sides of the triangle should be greater than the third side. If so, triangle can be formed.
6 + 7 = 13 > 9
7+ 9 = 16 > 6
6 +9 = 15 > 7
Identify the equation in slope-intercept form for the line containing the points (−4,1) and (2,3).
y=1/3x+7/3
y=1/4x+2
y=1/2x−4
y=1/3x−5/3
Answers
The slope-intercept form for the line is y = 1/3 x -5/3. and the option D is correct option.
According to the statement
we have given that the points (−4,1) and (2,3) and we have to find the slope-intercept form.
And we have to find the equation of line.
So, For this purpose,
The given points are:
(−4,1) and (2,3)
And the slope m become
[tex]m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]
So, put the values in it
then m= 3-1 / 2+4
m = 1/3
And and b point becomes (2+3) / (−4+1)
Then B = -5/3
Then the general equation of slope intercept form is y = mx +b
Then
y = 1/3 x -5/3.
So, The option D is correct and the slope-intercept form for the line is y = 1/3 x -5/3.
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Mary walks from corner D to corner E to corner F on a city block that measures 264 feet wide and 900 feet long. Kate walks diagonally across the city block from corner D to corner F. How much less distance did Kate walk than Mary to get to corner F? Round your answer to the nearest whole foot. Enter only the number.
Answers
Step-by-step explanation:
we have a rectangle, that is then cut in half diagonally into 2 right-angled triangles.
in our problem it does not matter which side is the larger and which is the shorter one, because the distance calculation for both cases involve one short and one long side. the sequence does not matter.
Mary walks one short and one large side to get to the opposite corner (from D to F) :
264 + 900 = 1,164 ft
Kate walks diagonally. and that diagonal is the Hypotenuse of the right-angled triangle with a short and a long side of the rectangle.
so, Pythagoras applies :
c² = a² + b²
with c being the Hypotenuse (the side opposite of the 90° angle), and a and b are the legs.
in our case that means
diagonal² = 264² + 900² = 69,696 + 810,000 =
= 879,696
diagonal = sqrt(879,696) = 937.9211054... ft ≈ 938 ft
so, Kate walked
1,164 - 938 = 226 ft
less than Mary.
Dmitri wants to cover the top and sides of this box with glass tiles that are 1 cm square. How many tiles will he need? Dmitri will need [Blank] glass tiles. The dimentions are...26 , 15, 8.
Answers
The number of tiles needed to cover the surface area of the box excluding the bottom is: 1,046 tiles.
What is the Surface Area of a Box?The surface area of a box is the area surrounding all its faces. A box has 6 rectangular faces. Therefore, the total surface area of the box equals the sum of all 6 rectangular faces.
What is the Surface Area of a Box?SA = 2(lw + lh + hw), where:
l = lengthw = widthh = height of the boxThe image attached below shows the box Dmitri wants to cover. Since the bottom of the box would be excluded, therefore:
The surface area to be covered = surface area of the box - area of the bottom rectangular face
The surface area to be covered = 2(lw + lh + hw) - (l)(w)
l = 26
w = 15
h = 8
Substitute
The surface area to be covered = 2(l×w + lh + hw) - (l)(w) = 2·(15·26+8·26+8·15) - (26)(15) =
The surface area to be covered = 1436 - 390 = 1,046 cm
Area of one tile = 1 cm square
Number of tiles needed = 1,046/1
Number of tiles needed = 1,046 tiles.
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what the answers
help me'
Answers
Answer:
47°
Step-by-step explanation:
If angles are complementary, that means those angles are added up to 90°.
Therefore,
43 + x = 90
Take 43 to the right side to make x as the subject.
x = 90 - 43
x = 47°
Note that a complementary angle adds up to 90 degrees.
In this problem, we get the equation:
43° + x = 90°
So, we need to find the value of x by doing:
90° – 43° = 47°
x = 47°
43° + 47° = 90°
We get the final result as:
x = 47°
Hope this helps :)
2w+w^3-1/2w^2 for w=2
Answers
The required value is 8
How can we find value?
We will find the value by putting the value of w in the given function to get the required value.
We can find the value of given function foe w=2 as shown below:
Given, function is 2w+w^3-1/2w^2
Let A=2w+w^3-1/2w^2
for w=2
A=2(2)+(2)^3-1/2(2)^2
= 2+8-1/2(4)
=2+8-2
=8
Hence, the required value is 8.
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HOW MANY DIFFERENT COMMITTEES CAN BE FORMED FROM 4 TEACHERS AND 20
STUDENTS IF THE COMMITTEE IS TO CONSIST OF TWO TEACHERS AND THREE
STUDENTS?
Answers
Answer:
6840
Step-by-step explanation:
multiply combinations
n!/(r!(n − r)!)
n choose r
n (objects) = |n
r (sample) = r
(4 teachers, CHOOSE 2)*(20 students, CHOOSE 3)
C(4,2) times C(20,3) =
C(n,r)=?
C(n,r) = C(4,2)
4! / (2!(4-2)!)
4! /2! x 2!
= 6
C(n,r)=?
C(n,r) = C(20,3)
20! / (3!(20-3)!)
20! / 3! x 17!
= 1140
6 * 1140 = 6840
alegbracom Edwin McCravy
Find the mean of the following data set: 9, 12, 7, 13, 21, 10, 12, 4
Answers
Answer:
11
Step-by-step explanation:
9 + 12 + 7 + 13 + 21 + 10 + 12 + 4
21 + 20 + 31 + 16
41 + 47
88
88 / 8 = 11
Quadratic equations…….
Answers
Step-by-step explanation:
Greetings !Completing the square is a method that is used for converting a quadratic expression of the form ax² + bx + c to the vertex form a(x - h)² + k. The most common application of completing the square is in solving a quadratic equation.
It's useCompleting the Square is a technique which can be used to find maximum or minimum values of quadratic functions. We can also use this technique to change or simplify the form of algebraic expressions. We can use it for solving quadratic equations.
Example:[tex]x {}^{2} + 6 = - 2 \\ x {}^{2} + 6x + 9 = 7 \\ (x + 3) {}^{2} = 7[/tex]
which of the following functions is graphed below
Answers
The function that is graphed is y = |x -2| + 3. The correct option is D. y = |x -2| + 3
Absolute value graphFrom the question, we are to determine the function for the given graph
From the graph,
When x = 2, y = 3
The equation that satisfy this condition is
y = |x -2| + 3
NOTE: This function satisfies all the points on the graph
Hence, the function that is graphed is y = |x -2| + 3
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PLS ANSWER BOTH PARTS I WILL GIVE BRAINLIEST PLSSS HURRRYYYY
Answers
Answer: slope is 1/3
Step-by-step explanation: make a graph of y=1/3x+9
Ronaldo specializes in creating sewing patterns for dresses, and wants to set up his own website to sell electronic copies. He pays $120 for a domain name and web hosting services for the first year. Ronaldo makes $14.25 for each electronic pattern that he sells. How many patterns must he sell to cover the cost of the website for the first year? Round your answer to a whole number.
Answers
Answer:
8
Step-by-step explanation:
120/14.25=8.4 rounded off to a whole number is 8
The legs of a right triangle measure 3 cm and 8 cm. What is the length of the hypotenuse?
Answers
lim
n->∞ 5n^2/7n³-4
Answers
[tex]\displaystyle \lim_{n\to \infty} ~~ \cfrac{5n^2}{7n^3-4}\qquad \implies \lim_{n\to \infty} ~~ \stackrel{\textit{using L'Hopital's rule}}{\cfrac{\frac{d}{dn}[5n^2]}{\frac{d}{dn}[7n^3-4]}}\implies \lim_{n\to \infty} ~~\cfrac{10n}{21n^2}\implies 0[/tex]
now, how do we get 0?
well, simple, hmmmm let me put a few values in
[tex]\cfrac{10(10)}{21(10)^2}\implies \cfrac{100}{2100}~~,~~ \cfrac{10(10000)}{21(10000)^2}\implies \cfrac{100000}{2100000000}~~,~~ ...[/tex]
in math lingo, on every iteration or change of the variable "n", the numerator is increasing but at a lower rate than the denominator, the denominator is increasing much more on every iteration, namely the denominator is moving faster and thus as "n" increases ever to infinity, the fraction becomes smaller and smaller and smaller, ever getting closer to 0.
plssssssssssssss help
Answers
Answer:
b. 30 m²
Step-by-step explanation:
find the area of the rectangle and triangle first
area of rectangle = length x height
aR = 10 x 6
aR = 60
area of triangle = base x height x 1/2
aT = 10 x 3 x 1/2
aT = 30
now, to find the area of the shaded area subtract the triangle from the rectangle
area = rectangle - triangle
a = 60 - 30
a = 30
Area Of Triangle = BasexHeight÷2
Area of Rectangle = LengthxHeight
each triangle adds up to 7.5 because 3x5÷2
But since theres two triangles it adds up to 15
The rectangle is 3x10 which is 30
Add both digits up and u get 45.
ZA =
Round your answer to the nearest hundredth.
Answers
Answer:
Rounded to the nearest tenth would be 75.5 degrees
Step-by-step explanation:
We are driving to Las Vegas. The sign says that it is one hundred forty-five miles to Las Vegas.
How many kilometers is it to Las Vegas?
Answers
Answer:
233.35488 km (Rounded to 233 km)
Step-by-step explanation:
If you convert miles to kilometers then you would use this conversion factor:
1 mile = 1.609344 km (or 1.61 km)
so in this case,
145 mile = 1.609344 km x 145
145 mile = 233.35488 km
I could also round it to 233 km.
Since your on a trip... i hope u enjoy ur journey to Las Vegas!
What is the length of the line segment PQ on the coordinate grid?
A fourquadrant coordinate grid from negative 50 to positive 50 in increments of 10 is drawn. Point P is plotted on the ordered pair negative 20, 10. Point Q is plotted on the ordered pair 10, 10. The points are connected by a line.
20 units
30 units
40 units
50 units
Answers
Using the formula for the distance between two points, the length of the line segment PQ is of 30 units.
What is the distance between two points?Suppose that we have two points, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The distance between them is given by:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
The coordinates of the endpoints for this problem are given by:
P(-20,10).Q(10,10).Hence the distance is given by:
D = sqrt[(-20 -10)² + (10 - 10)] = 30 units.
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Answer: 30 units
Step-by-step explanation:
In the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be StartFraction pi Over 4 EndFraction times the volume of the pyramid that it fits inside. A cone is inside of a pyramid with a square base. The cone has a height of h and a radius r. The pyramid has a base edge length of 2 r. Which statement best describes where the StartFraction pi Over 4 EndFraction comes from in the formula derivation? It is the ratio of the area of the square to the area of the circle from a cross section. It is the ratio of the area of the circle to the area of the square from a cross section. It is the difference of the area of the square and the area of the circle from a cross section. It is the sum of the area of the square and the area of the circle from a cross section.
Answers
In the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be D. the the sum of the area of the square and the area of the circle from a cross section.
How to illustrate the information?It should be noted that a volume simply means the amount of three dimensional space that's enclosed by a closed surface.
It is typically meaured on cubic units.
In this case, in the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be the the sum of the area of the square and the area of the circle from a cross section.
In conclusion, the correct option is D.
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Answer:
B. It is the ratio of the area of the circle to the area of the square from a cross section.
Step-by-step explanation:
Solve the following proportion for v.
11/v = 5/13 Round your answer to the nearest tenth.
Answers
Drag the tiles to the correct boxes to complete the reasoning of the proof. Not all tiles will be used.
May this help other people in the near future!!!!!
Answers
The missing reasons for each of the given statements are respectively;
1) Substitution
2) Definition of a Straight angle because a straight angle is 180°.
3) Congruent Angles have equal measures
How to prove angles in a triangle?
We are given that;
m∠1 = 90°
m∠1 + m∠2 = m∠5
∠3 ≅ ∠4
Statement 1; 90° + m∠2 + m∠4 = 180°
Reason 1; Substitution
Statement 2; m∠1 + m∠2 + m∠3 = 180°
Reason 2; Definition of a Straight angle because a straight angle is 180°.
Statement 3; m∠3 = m∠4
Reason 3; Congruent Angles have equal measures
Thus, the missing reasons for each of the given statements are respectively;
1) Substitution
2) Definition of a Straight angle because a straight angle is 180°.
3) Congruent Angles have equal measures
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Put the fractions in order from smallest to largest.
2/6 11/12 2/3
Answers
Answer: 2/6 2/3 11/12
Step-by-step explanation: if you do the common denominator, you will find the numbers 2/6= 4/12 11/12=11/12 2/3=8/12. So your answer should be 2/6 11/12 2/3
Under 20 Between 20 and 40 Over 40
10
20
20
7
19
11
Action
10
Comedy 3
Drama
16
The manager is going to randomly select one person to win a free movie pass.
What is the probability that the person selected is 40 years old or younger? Round the answer to the nearest hundredth. Enter the answ
the box.
Answers
The probability that the person selected is 40 years old or younger is 0.61.
What is the probability?Probability determines the odds that a random event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
The more likely the event is to happen, the closer the probability value would be to 1. The less likely it is for the event not to happen, the closer the probability value would be to zero.
The probability that the person selected is 40 years old or younger = number of people 40 years or younger / total number of people at the movie theatre = (6 + 17 + 18 + 10 + 4 + 6) / (6 + 17 + 18 + 10 + 4 + 6 + 16 + 3 + 20)
61 / 100 = 0.61
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Find the cube root of Z =1+i the power of 5
Answers
Answer:
Cube Root Calculator
Use this calculator to find the cube root of positive or negative numbers. Given a number x, the cube root of x is a number a such that a3 = x. If x positive a will be positive, if x is negative a will be negative. Cube roots is a specialized form of our common radicals calculator.
It will help you alot.
Step-by-step explanation:
. A school band sold 432 tickets for their upcoming spring concert. This represents about 85.5% of the number of tickets sold last year. How many tickets were sold last year? (Round to the nearest whole number)
Answers
original value multiplied by .855 will equate
to 432:
ticketstastyear * 0.855 = 132
Dividing each side by 0.855 results in the
amount of tickets sold last year:
ticketslastyear ~ 505.263
We round to the nearest whole number of
505.
The answer is 505
A manufacturer of widgets wants to test a new widget-producing machine to determine if it can make an average of 25 widgets per second before deciding to invest in the machine. The standard to reject the new machine is if it makes an average of less than 25 widgets per second. Here are data from a small random sample:
25.6, 26.2, 22.5, 20.5, 26.4, 27.4, 23.6, 26.9, 25.7, 24.9
Assuming the population follows a normal distribution, is there evidence that the new machine should be rejected at the 0.01 significance level? State the hypotheses, list and check the conditions, calculate the test statistic, find the p-value, and make a conclusion in a complete sentence related to the scenario.
Answers
Using the t-distribution, it is found that since the p-value is greater than 0.01, there is no evidence that the new machine should be rejected at the 0.01 significance level.
What are the hypothesis tested?At the null hypothesis, it is tested if the average is not less than 25, that is:
[tex]H_0: \mu \geq 25[/tex]
At the alternative hypothesis, it is tested if the average is less than 25, that is:
[tex]H_1: \mu < 25[/tex]
What is the test statistic?The test statistic is given by:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
The parameters are:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.s is the standard deviation of the sample.n is the sample size.Considering the situation described, the values of the parameters are given as follows:
[tex]\overline{x} = 24.97, \mu = 25, s = 2.16, n = 10[/tex]
Hence the test statistic is:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{24.97 - 25}{\frac{2.16}{\sqrt{10}}}[/tex]
t = -0.04
What is the p-value and the conclusion?Using a t-distribution calculator, for a left-tailed test, as we are testing if the mean is less than a value, with 10 - 1 = 9 df and t = -0.04, the p-value is of 0.4844.
Since the p-value is greater than 0.01, there is no evidence that the new machine should be rejected at the 0.01 significance level.
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