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
A cannonball is shot straight upward with a velocity of 80 ft/sec. Its height after t seconds is given by f (t) = 80t - 16 t to the second power. Round your answers to 1 decimal place if necessary.
How high will the cannon ball go?
How many seconds will it take to reach this maximum height?
Answers
Check the picture below, so the ball's path is pretty much like so, and it reaches its hightest at its vertex, so
[tex]~~~~~~\textit{initial velocity in feet} \\\\ h(t) = -16t^2+v_ot+h_o \quad \begin{cases} v_o=\textit{initial velocity}&80\\ \qquad \textit{of the object}\\ h_o=\textit{initial height}&0\\ \qquad \textit{of the object}\\ h=\textit{object's height}&h\\ \qquad \textit{at "t" seconds} \end{cases} \\\\\\ f(t)=80t-16t^2\implies f(t)=-16t^2+80t+0 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{-16}x^2\stackrel{\stackrel{b}{\downarrow }}{+80}x\stackrel{\stackrel{c}{\downarrow }}{+0} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)[/tex]
[tex]\left(-\cfrac{ 80}{2(-16)}~~~~ ,~~~~ 0-\cfrac{ (80)^2}{4(-16)}\right) \implies \left( - \cfrac{ 80 }{ -32 }~~,~~0 - \cfrac{ 6400 }{ -64 } \right) \\\\\\ \left( \cfrac{5}{2}~~,~~100 \right)\implies \underset{~\hfill feet ~~ }{\stackrel{seconds\qquad }{\left( 2\frac{1}{2}~~,~~100 \right)}}[/tex]
what is the answer for this question? i really need it
Answers
Answer:
a = 3
b = -9
c = -3
d = 22
e = 49 - 9
f = a^2 -3
Step-by-step explanation:
Please no more questions :(
Determine whether or not the lines are parallel based on the
given angle measures. Explain your answer in the case.
Answers
Answer:
answer
Step-by-step explanation:
To see whether or not two lines are parallel, we must compare their slopes. Two lines are parallel if and only if their slopes are equal. The line 2x – 3y = 4 is in standard form. In general, a line in the form Ax + By = C has a slope of –A/B; therefore, the slope of line q must be –2/–3 = 2/3.
Convert ( 4 , 300 ∘ ) to rectangular form.
Answers
The rectangular representation of the polar point of (4 , 300) is (2,- 2√3)
According to the statement
we have given a coordinates of the rectangle and we have to find the polar coordinates.
So, For this purpose, we know that the
We Use the conversion formulas to convert from polar coordinates to rectangular coordinates which are
x = rcosθ
y = rsinθ
Substitute the given values in it then
x=(4)cos(300)
y=(4)sin(300)
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
x=(4)cos(60) -(1)
y= - (4)sin(60) -(2)
And then
x=(4)cos(60)
x=(4)(1/2)
x = 2 -(3)
and
y= - (4)sin(60)
y= - (4)(√3/2)
y= - 2√3 -(4)
Replace (3) with (1) and (4) with (2)
then it becomes
x = 2 and y= - 2√3
The rectangular representation of the polar point of (4 , 300) is (2,- 2√3)
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E4.2 Simplify the following using suitable properties (a) 109x 50
Answers
109×50
100×50+9×50=5000+450
5450
Answer:
5450
Step-by-step explanation:
You can just use a caculater to get the answer.
The composite figure is made up of a parallelogram and a rectangle. Find the
area.
A. 88 sq. units
B. 64 sq. units
C. 24 sq. units
D. 120 sq. units
Answers
Answer:
A
Step-by-step explanation:
The parallelogram is the length times the height. The height is 8 and the length is 12 - 2 - 2 or 8.
8x8 = 64
The small rectangle is 12 x 2 or 24
64 + 24 = 88
The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 50%. What is the probability that it will rain on exactly one of the three days they are there? Round your answer to the nearest thousandth.
Answers
The probability that it will rain on exactly one of the three days they are there is 0.375.
Given that Hiking Club plans to go camping in a State park where the probability of rain on any given day is 50%.
The binomial distribution is used when there are exactly two outcomes of a trial that are mutually exclusive.
Use binomial probability:
P = ⁿCʳ pʳ (1−p)ⁿ⁻ʳ
where n is the number of trials,
r is the number of successes,
and p is the probability of success.
probability of rain on any given day =p =0.50
total number of ways,n=3
let X is number of days with rain out of 3 days
So, [tex]X\sim Bin(3,0.50)[/tex]
P(x=X)=ⁿCₓpˣ(1-p)ⁿ⁻ˣ
P(x=1)=³C₁(0.50)¹(1-0.50)³⁻¹
P(x=1)=3×0.50×(0.5)²
P(x=1)=1.5×0.25
P(x=1)=0.375
Hence, the probability that it will rain on exactly one of the three days they are there where the probability of rain on any given day is 50% is 0.375.
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A freighter has to go around an oil spill in the Pacific Ocean. The captain sails due east for 30 miles. Then he turns the ship and heads due north for 14 miles. What is the distance and direction of the ship from its original point of course correction? Round each number to the nearest whole number.
Answers
The distance and direction of the ship from its original point of course correction is 33 miles and 23° respectively.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Pythagoras theorem is used to show the relationship between the sides of a right angled triangle.
Let x represent the distance travelled, hence:
x² = 14² + 30²
x = 33 miles
tanФ = 14/33
Ф = 23°
The distance and direction of the ship from its original point of course correction is 33 miles and 23° respectively.
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Which expression represents the
number of reams of paper the company
produced during the second year?
Answers
The expression that represents the number of reams of paper the company produced during the second year is 4.704 × 10¹⁰. The correct option is D. 4.704 × 10¹⁰
Writing an ExpressionFrom the question, we are to determine the expression that represents the number of reams of paper the company produced during the second year
From the given information,
During the first year of operation,
The company produced 8.4 × 10⁹ reams of paper
And
During the second year,
the company produced 5.6 times the number of reams of paper that it produced during the first year.
Thus,
The number of reams of paper the company produced during the second year = 5.6 × 8.4 × 10⁹ reams of paper
The number of reams of paper the company produced during the second year = 47.04 × 10⁹ reams of paper
= 4.704 × 10¹ × 10⁹ reams of paper
= 4.704 × 10¹⁰ reams of paper
Hence, the expression that represents the number of reams of paper the company produced during the second year is 4.704 × 10¹⁰. The correct option is D. 4.704 × 10¹⁰
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I really need help I work a 12 hr shift in less then 3 hrs and if I don't answer this correct I don't graduate and have to redo over 100 questions so please if you answer please make it the right answer please someone show mercy and help me
Answers
Answer:
72 Degrees
Step-by-step explanation:
10 / 50 = 0.2
A circle has 360 degrees.
0.2 X 360 = 72
A lawn-mowing company is trying to grow its business. It had 30 clients when they started its business and wants to increase by 6 new clients each week. Use an
arithmetic sequence to write a function to represent this real-world situation and determine the range of the function for the first four weeks of data.
Of(x)=6x +30,0 ≤ y ≤4
Of(x)=6x+24; 0 < y < 4
Of(x)=6x +30, 30 ≤ y ≤ 48
Of(x)=6x+24; 30 ≤ y ≤ 48
Answers
Answer:
Step-by-step explanation:
Option C.
Step-by-step explanation:
Initial number of clients = 18
Number of Increase per week = 4
We need to use an arithmetic sequence to write a function.
So, the required arithmetic sequence for four weeks is
18, 22, 26, 30
Here, first term is 18 and common difference is 4. The range is [18,30].
The explicit formula of n arithmetic sequence is
where, a is first term and d is common difference.
Substitute a=18 and d=4 in the above formula.
The function notation is
The function is f(x)=4x+14 and the range is 18 ≤ y ≤ 30.
Therefore, the correct option is C.
A clothing store can make 4 dress shirts out of 14 yards of cotton cloth. At that rate, how many dress shirts can be made out of 63 yards of cotton cloth?
Answers
14x = 4*63
14x = 252
x = 18 dress shirts
Check
4/14 = 0.286
18/63 = 0.286
The volume of this regular pentagonal pyramid is 82.5 cubic meters. What is the height of the pyramid?
Answers
the height of the pentagonal pyramid is 5. 70 meters
Volume of a regular pentagonal pyramidThe formula for determining the volume of a regular pentagonal pyramid is given as;
V=5/12tan(54°)ha^2
Where
a is the base edgeh is the heightWe have the volume to be;
volume = 82. 5 cubic centers
height = h
a = 5m
Substitute the values
[tex]82. 5 = \frac{5}{12}[/tex] × [tex]tan 54[/tex] × [tex]h[/tex] ×[tex]5^2[/tex]
[tex]82. 5 = 0. 42[/tex] × [tex]1. 3764[/tex] × [tex]25[/tex] × [tex]h[/tex]
Make 'h' subject of formula
[tex]h = \frac{82. 5}{14. 45}[/tex]
h = 5. 70 meters
The height of the pentagonal pyramid is 5. 70 meters
Thus, the height of the pentagonal pyramid is 5. 70 meters
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What is the value of y?
OA. 30°
OB. 120°
OC. 60°
60
Answers
Answer:
60
Step-by-step explanation:
60 + y + y = 180
2y = 120
y = 60
What is ZT?
a. 7
b. 14
c. 28
d. 29
Answers
Answer:
B
Step-by-step explanation:
• the opposite sides of a parallelogram are congruent , then
ZY = WX , that is
5x - 6 = 4x + 1 ( subtract 4x from both sides )
x - 6 = 1 ( add 6 to both sides )
x = 7
• the diagonals bisect each other , then
ZT = TX = 2x = 2 × 7 = 14
Proofs 50 points for whole page ( serious answers only or report and 1 star )
Answers
1) Given - This is given in the question.
2) Reflexive Property of Equality - Anything is equal to itself
3) Substitution Property of Equality - Due to the given, substituting 118° for m∠1 and 62° for m∠2 will not change the equality of the equation
4) Simplify - No need to explain as they have already given this
5) Definition of Supplementary Angles - Two angles are supplementary only when the sum of their measures is 180°.
6) Converse of Same-side Interior Angles Theorem - If there is a transversal intersecting two lines and the same-side interior angles formed are supplementary, then the lines are parallel.
Question 2:1) Given - It is given in the question
2) Vertical Angles Theorem - Vertical angles have the same measure, making their angles congruent
3) Transitive Property of Congruence - Since it's given that [tex]\angle1\cong\angle3[/tex] and in step 2 we showed that [tex]\angle3\cong\angle4[/tex], we can say that [tex]\angle1\cong\angle4[/tex]
4) Transitive Property of Congruence - Since we proved that [tex]\angle1\cong\angle4[/tex] and it's given that [tex]\angle4\cong\angle5[/tex], we can say that [tex]\angle1\cong\angle5[/tex]
5) Converse of Alternate Interior Angles Theorem - If a transversal intersects two lines and the measures of alternate interior angles formed are equal, then the lines are parallel.
need heeeelp please
Answers
Step-by-step explanation:
y = -f(x)
so, everything is mirrored around the x-axis.
what was below the x-axis, is now above the x-axis, what was above is now below. and what was on the x-axis, is still on the x-axis.
pick the corner points of f(x), their x-values stay the same, their y-values flip the sign. the points on the x-axis (y = 0) stay the same of course.
and then connect the points, so that the curve is upside-down to the original.
Application
I
3. Ray has a stride of about 1.4 m. He runs 9 km daily to train for a marathon. Approximately
how many strides does he need to run 9 km?
Answers
He needs to run with approximately 6429 for a distance of 9 km
How to determine the number of strides?The given parameters are:
Length of stride = 1.4 m
Distance for marathon = 9 km
The number of strides needed is then calculated as:
Number of stride = Distance for marathon/Length of stride
Substitute the known values in the above equation
Number of stride = 9km/1.4m
Convert km to m
Number of stride = 9000m/1.4m
Evaluate the quotient
Number of stride = 6428.57143
Approximate the estimate
Number of stride = 6429
Hence, he needs to run with approximately 6429 for a distance of 9 km
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For the function p defined by p(m) = 2m^-4m+3, find p(1/3).
Answers
p(1/3) = 2(1/3)^2 - 4(1/3) + 3
= 2(1/9) - 4/3 + 3
= 1/3 - 4/3 + 9/3
= -3/3 + 9/3
= 6/3
= 2
I need help asap. everythings in the image
Answers
The function in terms of the time (t) is t=d/r and the value of the time (t) for the given values is 5.
Linear FunctionAn equation can be represented by a linear function. The standard form for the linear equation is: ax+b , for example, y=2x+7. Where:
a= the slope
b=constant term that represents the y-intercept.
For the given example: a=2 and b=7.
In the exercise is asked the equation for the time (t) in the function of the distance (d) and the rate (r).
Thus, if d=rt, we will have t= d/r
Replacing the variables for the given values for finding t. The question gives:
distance (d) =40
rate (r) = 8
Thus, you will have t= 40/8. Then, t=5
Therefore, the answer is the fourth option.
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Riverbed company purchased a delivery truck for $33,000 on January 1, 2022 the truck has an expected salvage value of 2760 and he suspected to be driven 108,000 miles over estimated useful life of 10 years actual miles driven was 16,000 802,022 and 12,000 602,023 how much is the depreciation expense per mile
Answers
Answer:
0.28
Step-by-step explanation:
Depreciation Expense per mile:
= (Cost of delivery truck - Salvage value) ÷ Expected miles driver
=(33000-2760)÷108000=0.28 per mile
see attachment for question
Answers
The percentages for each outcome are given as follows:
a) 34.4%.
b) 29.4%.
What is a percentage?The percentage of an amount a over a total amount b is given by a multiplied by 100% and divided by b, that is:
P = a/b x 100%
For this problem, we already have the percentages, hence we just combine them.
For item a, he can be late in two ways, one that happens 24.4% of the time(waking up late), and the other 10%(stuck in traffic), hence the percentage is of 34.4%, as 24.4% + 10% = 34.4%.
For item b, the percentage stuck in traffic is of 5%, hence the percentage that he will be late is 24.4% + 5% = 29.4%.
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a storage room measures 20 feet by 15 feet. the floor is covered by tiles that cost $7.50 per square foot. what will be the cost of the entire floor with tiles?
Answers
Answer: $2250
Step-by-step explanation:
Given information
Dimension of the room = 20 feet × 15 feet
Cost of tiles = $7.50 / ft²
Derived formula from the given information
Total cost = Floor area × Cost of tiles
Substitute values into the formula
Total cost = Floor area × Cost of tiles
Total cost = (20 × 15) × (7.5)
Simplify by multiplication
Total cost = 300 × (7.5)
[tex]\Large\boxed{Total~cost~=~2250~Dollars}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Joey had eight pencils in the morning. He gave some pencils away to friends who needed them. To determine the number of pencils he has left, ___ the number of pencils he gave away. count difference subtract add
Answers
Hope it help you
Please mark me brianliest
PLEASE HELP ASAP!!! PLEASE ALSO GIVE EXPLANATION WITH EQUATION AND WORDS!!
Answers
Using proportions, it is found that the two cities are 27.5 miles apart.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.
From the given scale, we have that each inch of distance represents 10 miles. Hence the distance in miles for a distance of 2 and 3/4 miles = 2.75 miles is given by:
D = 10 x 2.75 = 27.5 miles.
Hence, the two cities are 27.5 miles apart.
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Newton's Law of Cooling states that the rate of change of the temperature of an object, T, is proportional to the difference of T and the temperature of the region, TR or dT over dt equals k times the quantity T minus T sub R end quantity. An object with core temperature of 1200°F is removed from a fire and placed in a region with a constant temperature of 80°F. After 1 hour, its core temperature is 830°F. What is the object's core temperature 4 hours after it is taken off the fire? (1 point)
Answers
Answer:
305 °F
Step-by-step explanation:
The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.
Cooling ModelThe solution to the differential equation described by Newton's law of cooling is the exponential equation ...
y = ab^t +c
where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.
For this problem, the ambient temperature is c=80, and the differences of interest are ...
a = 1200 -80 = 1120
b = (830 -80)/1120 = 75/112
Using these values in the model gives ...
y = 1120(75/112)^t +80 . . . . . . where y(t) is the core temperature at time t
Note that units of time are hours.
ApplicationWe want y when t=4.
y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212
The core temperature after 4 hours is about 305 °F.
__
Additional comment
The differential equation will have a solution of the form ...
[tex]T-T_R=(T_0-T_R)e^{kt}[/tex]
where k = ln(75/112) ≈ -0.40101
In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.
a linear equation is an equation whose graph is a straight line
Answers
Answer:
True
Step-by-step explanation:
Plotting the point of any linear equation on a graph gives a straight line
Question 10 of 10
What is the slope of the graph shown below?
OA. 5
OB. 2
C. -2
OD. -5
Answers
The slope of the graph is 1 / 5.
How to find the slope of a line?A slope of a line is the change in y coordinate with respect to the change in x coordinate.
Slope describes the steepness of a line. The slope of any line remains constant along the line.
Therefore,
slope = y₂ - y₁ / x₂ - x₁
Hence, let's take two coordinates in the graph,
(2, 2)(7, 3)
x₁ = 2
x₂ = 7
y₁ = 2
y₂ = 3
Therefore,
slope = 3 - 2 / 7 - 2
slope = 1 / 5
Therefore, the slope of the graph is 1 / 5.
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Solve b + 3∕8 = 4∕9 for b.
A. 32/27
B. 59/72
C. 5/72
D. 1/6
Answers
Answer:
The value of b equals to
[tex] \frac{5}{72} [/tex]
Step-by-step explanation:
Hello!steps given below
given expression
[tex]b + \frac{3}{8} = \frac{4}{9} \\ [/tex]
subtract
[tex] \frac{3}{8} [/tex]
from both sides
[tex]b + \frac{3}{8} - \frac{3}{8} = \frac{4}{9} - \frac{3}{8} \\ b = \frac{4}{9} - \frac{3}{8} [/tex]
To, continue find the L.C.M of 9 and 8 thus, it's 72
So, plug 72 and solve the expression as follows
[tex]let \: \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} \\ b = \frac{32 - 27}{72} = \frac{5}{72} [/tex]
Hope it helps!
Copy the problem, mark the givens in the diagram and write a statement/reason proof
Answers
Answer:
Step-by-step explanation:
<E = <T Given
M is the midpoint of TE Given
TM = ME Definition of a midpoint
<TMI = <RME Property of Vertically Opposite Angles
ΔTMI = ΔRME ASA = ASA
MI = ME Corresponding parts of Congruent triangles are Congruent
answer: reasoning:
<E = <T given
M is the midpoint of TE given
TM = ME definition of a midpoint
<TMI = <RME property of Vertically Opposite Angles
ΔTMI = ΔRME ASA = ASA
MI = ME corresponding parts of congruent triangles are Congruent
hope this helps!
