Answers
Answer:
equation: (x -1)²/64 -(y -4)²/80 = 1foci: (-11, 4), (13, 4)Step-by-step explanation:
You want the steps to finding the equation of the hyperbola with center (1, 4), vertices (-7, 4) and (9, 4) and that includes the point (-11, -6).
Equation of a hyperbolaThe standard-form equation of a hyperbola with center (h, k) and semi-axes 'a' and 'b' is ...
[tex]\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1[/tex]
The "linear eccentricity" 'c' is the distance from the center to a focus, and satisfies the equation ...
c² = a² +b²
The vertices are (h±a, k) and the foci are (h±c, k).
ApplicationThe center of the hyperbola is given as (1, 4). The distance from the right vertex to the center is ...
a = 9-1 = 8
The equation thus far is ...
(x -1)/8² -(y -4)/b² = 1
The value of 'b' can be determined from the given point:
(-11 -1)²/8² -(-6 -4)²/b² = 1
9/4 -100/b² = 1
5/4 = 100/b²
b² = 100/(5/4) = 80
The linear eccentricity is ...
c² = a² +b²
c² = 64 +80 = 144
c = √144 = 12
The foci are (1±12, 4) = (-11, 4) and (13, 4).
The equation is ...
(x -1)²/64 -(y -4)²/80 = 1
Step summaryThe given information was used to find semi-major axis 'a'. Together with the given center value (h, k), and the given point, the equation was written and solved for b².The value of 'c' was found from a² and b², and used to find the locations of the foci.
Answer:
[tex]\dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{80}=1[/tex]
Center = (1, 4)Vertices = (-7, 4) and (9, 4)Foci = (-11, 4) and (13, 4)Step-by-step explanation:
Standard equation of a horizontal hyperbola (opening left and right):
[tex]\boxed{\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1}[/tex]
where:
Center = (h, k)Vertices = (h±a, k)Co-vertices = (h, k±b)Foci = (h±c, k) where c² = a² + b²Given:
Center = (1, 4)Vertices = (-7, 4) and (9, 4)Point on the hyperbola = (-11, -6)Therefore:
h = 1k = 4Find the value of a using the x-values of the vertices:
[tex]\begin{aligned}\implies h-a &= -7\\1-a &= -7\\a &= 8\end{aligned}[/tex]
[tex]\begin{aligned}\implies h+a &= 9\\1+a &= 9\\a &= 8\end{aligned}[/tex]
Therefore, a = 8.
Substitute the values of h, k and a into the formula:
[tex]\implies \dfrac{(x-1)^2}{8^2}-\dfrac{(y-4)^2}{b^2}=1[/tex]
[tex]\implies \dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{b^2}=1[/tex]
Substitute the given point on the hyperbola (-11, -6) into the equation and solve for b²:
[tex]\implies \dfrac{(-11-1)^2}{64}-\dfrac{(-6-4)^2}{b^2}=1[/tex]
[tex]\implies \dfrac{(-12)^2}{64}-\dfrac{(-10)^2}{b^2}=1[/tex]
[tex]\implies \dfrac{144}{64}-\dfrac{100}{b^2}=1[/tex]
[tex]\implies \dfrac{144}{64}-1=\dfrac{100}{b^2}[/tex]
[tex]\implies \dfrac{144}{64}-\dfrac{64}{64}=\dfrac{100}{b^2}[/tex]
[tex]\implies \dfrac{80}{64}=\dfrac{100}{b^2}[/tex]
[tex]\implies \dfrac{5}{4}=\dfrac{100}{b^2}[/tex]
[tex]\implies b^2=\dfrac{4 \cdot 100}{5}[/tex]
[tex]\implies b^2=\dfrac{400}{5}[/tex]
[tex]\implies b^2=80[/tex]
Therefore, the equation of the hyperbola is:
[tex]\boxed{\dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{80}=1}[/tex]
To find the foci, first find c where c² = a² + b²:
[tex]\implies c=\sqrt{a^2+b^2}[/tex]
[tex]\implies c=\sqrt{64+80}[/tex]
[tex]\implies c=\sqrt{144}[/tex]
[tex]\implies c=12[/tex]
Substitute the found value of c, along with the values of h and k into the foci formula:
[tex]\implies (h+c, k)=(1+12, 4)=(13,4)[/tex]
[tex]\implies (h-c,k)=(1-12,4)=(-11,4)[/tex]
Therefore, the foci are (-11, 4) and (13, 4).
Related Questions
The names or notes of all the white notes on a piano are written in pieces of paper and placed in a cup. There are 7 sets of the notes D E F and G and 8 sets of the notes A B and C. If a person draws one note out of the cup without looking what are the chances of them getting a c
Answers
The probability that the person obtains a note with C written in it is of:
0.1538 = 15.38%.
How to calculate a probability?The probability of an event in an experiment is obtained as the number of desired outcomes in the context of the experiment divided by the number of total outcomes in the context of the experiment.
The total number of notes in this problem is composed as follows:
7 notes of each of D, E, F and G.8 notes of each of A, B and C.Hence the total number is of:
Total = 7 x 4 + 8 x 3 = 52 notes.
The desired number of outcomes is the number of notes with C, hence:
Desired = 8 notes.
Thus the probability is calculated as follows:
Probability = Desired/Total = 8/52 = 0.1538 = 15.38%.
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Michelle sells new and used kayaks. She makes a 10.5% commission on every kayak she sells and she also gets paid $495.50 per week.
a) Develop an equation for the way Michelle is paid.
b) Last month, the total value of the kayaks she sold was $19 425. Use the equation you’ve developed to find out how much Michelle was paid for that month.
Answers
Part a
The equation for the way Michelle is paid is
Total weekly payment y = 495.50 + 0.105x
Part b
The total payment in a month = $4021.625
She makes a 10.5% commission on every kayak she sells
Consider the total value of kayaks she sold = x
The rate of commission = 10.5%
The amount of commission = x × (10.5/100)
= 0.105x
The weekly fixed payment = $495.50
Total weekly payment y = 495.50 + 0.105x
Part b
The total value of the kayaks she sold = $19425
Total number of weeks in a month = 4
Then the total payment in a month = (4 × 495.50) + (0.105×19425)
= 1982 + 2039.625
= $4021.625
Hence,
part a
The equation for the way Michelle is paid is
Total weekly payment y = 495.50 + 0.105x
Part b
The total payment in a month = $4021.625
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If the function f(x) is thje absolute value of 3 more than the value of x. Select all the true statements. A. The x-intercept of the function is 3.
Answers
The absolute value function is represented by f(x) = |x+3| and the
x-intercept of the function is -3.
The function is represented as absolute value of 3 more than x.
hence the function 3 more than x is given by
f(x)=x+3
Now for absolute function:
f(x)=|x+3|
The graph of the function will have no negative value for f(x).
The graph will pass through the point (0,3) which is the y-intercept.
The x-intercept is at (-3,0) .
The domain of the function is all real vales of x.
The range of the function is (0,∞)
The parent function is given by y=|x| .
An procedure that establishes the absolute value of a number or variable is known as a modulus function. It produces the variable count's size.
It is also known as an absolute value function. No matter what input was given to this function, a positive result is always the result.
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What is the value of x?
Thank you.
Answers
Answer:
x would be 3 by 2
I am 90% sure but I hope this helps!
Determine which set of side measurements could be used to form a triangle.
8, 14, 20
7, 8, 15
2, 19, 22
1, 4, 8
Answers
The set of side measurements that could be used to form a triangle are 8, 14, 20.
How can the set of side measurements be identified?The concept that will be used is the concept of triangle inequality theorem.
To know if the given 3 side lengths are a triangle, then the use of the triangle inequality theorem comes into play.
This rule tells us that when we sum 2 sides of a triangle the value must be greater than the third side.
Then this rule can be applied as
The first option; 8, 14, 20
a+b>c = 8+14>20
a+c>b = 8+20>14
b+c>a = 20+14>8
Therefore, first option is correct.
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1.3 x 10 to the power of negative 3
Answers
Answer:
The answer to this problem is 0.0013.
Which is the graph of y = 3¥x+2 + 3?
Answers
The graph of the equation y = 3x + 2 + 3 is shown below .
In the question ,
it is given that ,
the equation of the line is y [tex]=[/tex] 3x [tex]+[/tex] 2 [tex]+[/tex] 3 ,
we need to draw the graph of it .
Do draw the graph we need at least two points .
for the first point
put x = 0
we get ,
y = 3(0) + 2 + 3
y = 0 + 2 + 3
y = 2 + 3
y = 5
the first point is (0,5)
for the second point put y = 0,
we get ,
0 = 3x + 2 + 3
3x = - 2 - 3
3x = -5
x = -5/3 ,
So , the second point is (-5/3 , 0)
the graph of the given equation is shown below .
Therefore , The graph of the equation y [tex]=[/tex] 3x + 2 [tex]+[/tex] 3 is shown below .
The given question is incomplete , the complete question is
What is the graph of the equation y = 3x + 2 + 3 ?
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An entry level architect's annual pay is $47,008 based on 52 weeks per year. Due to the economy, his company is having to cut back on the number of weeks that it employs its architects. If the firm cuts the work year to 48 weeks but keeps the same rate of pay, how much should the architect expect his annual pay to decrease? Round your answer to the nearest dollar.
Answers
After the firm work year reduced to 48 weeks at the same rate, the architect annual pay decrease to 43,392 dollars.
How to find the annual pay decrease?An entry level architect's annual pay is $47,008 based on 52 weeks per year. Due to the economy, his company is having to cut back on the number of weeks that it employs its architects
The company cuts the work year to 48 weeks but keeps the same rate of pay. The amount the architect should expect his annual pay to decrease can be calculated as follows:
Let's find the rate.
Therefore,
52 weeks = 47008
1 week = ?
cross multiply
rate = 47008 / 52
rate = 904 dollars per weeks
He collects the same rate after the company cuts the work year to 48 weeks.
Therefore,
annual pay of the architect after the cut = 48 × 904
annual pay of the architect after the cut = 43,392 dollars
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for any integer p, show that p^3+11p will always be a multiple of 6
Answers
By the principle of mathematical induction, it has been proved that
[tex]p^3 + 11p[/tex] is divisible by 6
What is Principle of mathematical induction?
Let the statement be given by P(n). At first P(n) is proved to be true for 1. Then it is assumed that P(n) is true for all n. If P(n) is true for (n+1) then by Principle of Mathematical induction, P(n) is true for all natural numbers
The given expression is [tex]p^3 + 11p[/tex]
For p = 1
[tex](1)^3 + 11 \times 1\\12[/tex]
which is a multiple of 6
Let it be true for p = m
[tex]m^3 + 11m[/tex] is a multiple of 6
[tex]m^3 + 11m = 6k[/tex], where k is any integer
For p = m + 1
[tex](m+1)^3 + 11(m+1) \\m^3+3m^2+ 3m +1+11m+11\\m^3+11m +3m^2+3m+12\\m^3+11m + 3m(m+1) + 12[/tex]
Now,
[tex]m^3 + 11m[/tex] is divisible by 6
We know that the product of two consecutive integers are divisible by 6
[tex]3m(m+1)[/tex] is divisible by 6
12 is divisible by 6
[tex](m+1)^3 + 11(m+1)[/tex] is divisible by 6
So it is true for p = m + 1
By the principle of Mathematical induction, it is true for all natural numbers and eventually for all integers
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A rectangular garden has an area of 72 square yards. The dimensions of the
garden are not prime numbers. What are the possible dimensions of the
garden
Answers
Answers:
1 yard by 72 yards
4 yards by 18 yards
6 yards by 12 yards
8 yards by 9 yards
==========================================================
Explanation:
Let's list out all the ways to multiply to 72 using two values
1*72
2*36
3*24
4*18
6*12
8*9
Then cross off something like 2*36 since 2 is prime. We also cross off 3*24 since 3 is prime.
We're left with this list
1*72
4*18
6*12
8*9
which represent the possible dimensions of this rectangle.
Keep in mind that 1 is not prime.
Status:
Frank borrowed $56022 from the bank at 4.0% for 69 days to expand his bike shop.
Find the total amount Frank pays back to the bank.
Answers
The total amount Frank pays back to the bank = $210642.72
Assuming the given interest rate- that is the simple interest rate.
so the future value A for the simple interest rate is given by,
A = P(1 + rt)
here, P = $56022
r = 0.04
t = 69
Substitute these values in above formula.
A = 56022 (1 + 0.04 * 69)
A = 56022 * 3.76
A = $210642.72
Therefore, the total amount Frank pays back to the bank = $210642.72
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55555555555555555555555555555555555555555555555555555555555555555555555555x0
Answers
Answer:
Step-by-step explanation:
Ok:
1. Anything times zero always equals zero.
So: 55555555555555555555555555555555555555555555555555555555555555555555555555x0=
Answer:0
Add.
3/5+7/8+3/10
Enter as a mixed number in simplest form.
Answers
Answer:71/40 or 1 31/40 or 1.775
Step-by-step explanation:1 31/40 is the mixed number
A store is having a sale on chocolate chips and walnuts. For 3 pounds of chocolate chips and 2 pounds of walnuts, the total cost is $14. For 5 pounds of
chocolate chips and 6 pounds of walnuts, the total cost is $28. Find the cost for each pound of chocolate chips and each pound of walnuts.
Cost for each pound of chocolate chips: $
Cost for each pound of walnuts: $
Answers
Cost for each pound of chocolate chips is equal to $3.5.
Cost for each pound of walnuts is equal to $1.75
How to determine the cost for each pound of chocolate chips and each pound of walnuts?In order to solve this word problem, we would assign variables to the cost for each pound of chocolate chips and each pound of walnuts respectively, and then translate the word problem into algebraic equation as follows:
Let the variable c represent the amount of cost for each pound of chocolate chips.Let the variable w represent the cost for each pound of walnuts.For 3 pounds of chocolate chips and 2 pounds of walnuts, we have:
3c + 2w = 14 .....equation 1.
For 5 pounds of chocolate chips and 6 pounds of walnuts, we have:
5c + 6w = 28 .....equation 2.
Solving the system of equations by using the elimination method, we have:
3 × (3c + 2w = 14) = 9c + 6w = 42
5c + 6w = 28_
4c = 14
c = 14/4
c = $3.5
For the cost for each pound of walnuts, we have:
3c + 2w = 14
3(3.5) + 2w = 14
10.5 + 2w = 14
2w = 14 - 10.5
2w = 3.5
w = 3.5/2
w = $1.75.
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Suppose you are designing a cardboard box that must have a volume of 18 cubic feet. The cost of the cardboard is $6 per square foot. How much will the material for each box cost?
Answers
Answer:
I think its 108 each Box is going to cost 108
y is directly proportional to x3 When x = 10, y = 250
(a) Find a formula for y in terms of x.
Answers
The formula for y in terms of x is a cubic equation:
y = 0.25*x³
How to find the relation between x and y?We know that y is directly proportional to x³, then we can write the equation:
y = k*x³
Where k is the constant of proportionality.
In this case, we know that when x = 10, the value of y is 250, then we can replace these values in the above equation to find the value of x:
250 = k*10³
250 = k*1,000
Dividing both sides by 1,000 we get:
250/1,000 = k
0.25 = k
Then the equation is:
y = 0.25*10³
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log 4(x+8) + log 4(x+5) = 1
rewrite the equation
Answers
According to the product rule of logarithms, the given equation is written as, log₄(x² + 13x + 40) = 1
Product rule of logarithm
Basically, the product rule states that the sum of two logarithms is equal to the product of the logarithms.
And it is the first law and it can be represented as;
logₐ(x) + logₐ(y) = logₐ(xy)
Given,
Here we have the logarithmic equation log 4(x+8) + log 4(x+5) = 1
And we have to rewrite the equation.
Here we have the logarithmic equation,
=> log₄(x+8) + log₄(x+5) = 1
By using the product property of logarithms,
We have to rewrite the equation as,
=> log₄(x+8)(x+5) = 1
Now, we have to expand (x+8)(x+3) using the FOIL Method.
Then we get,
=> log₄(x² + 5x + 8x + 40) = 1
=> log₄(x² + 13x + 40) = 1
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An investment portfolio is shown below.
Investment Amount Invested ROR
Savings Account $3,200 2.1%
Municipal Bond $4,900 4.5%
Preferred Stock $940 10.5%
Common Stock A $1,675 −3.5%
Using technology, calculate the weighted dollar amount of the savings account.
−$58.63
$58.63
−$67.20
$67.20
Answers
The weighted dollar amount of the savings account as per the an investment portfolio is; $67.20.
What is the percentage?A percentage is a minimum number or ratio that is measured by a fraction of 100.
Given that An investment portfolio is;
Investment Amount Invested ROR
Savings Account $3,200 2.1%
Municipal Bond $4,900 4.5%
Preferred Stock $940 10.5%
Common Stock A $1,675 -3.5%
The weighted dollar amount of the savings account is;
= 2.1% of $3200
= ( 2.1 / 100) × $3200
= ($6720) /100
= $ 67.20
Therefore, the weighted dollar amount of the savings account as per the an investment portfolio is; $67.20.
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John worked 18 hours last week. He earned $216. What was his hourly rate?
Answers
Divide $216 by 18 to get $12
can someone
help with this
Answers
Step-by-step explanation:
let angles of a triangle a=86
b= 180-3z , c= 180-4z
a+b+c = 180 {angle sum property}
86 +180+180-3z-4z=180
446 -7z = 180
-7z = -266
[tex]z = \frac{ - 266}{ - 7} = 38[/tex]
so other angle are
3z = 3 x 38 = 114
4z = 4 x 38 = 152
The equation y – 4 = 6(x + 7) represents a linear function. What is the y-intercept of the graph of the equation?
Answers
Step-by-step explanation:
In general equation, y = mx+c, c represents y-intercept.
So, y-4 = 6x+42
y = 6x+46
So, y intercept will be 46
(b) Tickets for a raffle are placed in a box. The box contains 25 blue tickets and 20 red
tickets. Tickets are drawn at random from the box, one at a time and are not replaced.
What is the probability that:
(i) the first ticket drawn is red and the second ticket drawn is blue?
Answers
Probability of drawn ticket at random with the condition of first ticket drawn is red and for second ticket drawing blue is equal to 0.253.
As given in the question,
Number of blue tickets in a box is equal to 25
Number of red tickets in a box is equal to 20
Total number of tickets in a box = 25 + 20
= 45
Drawing a ticket one at a time and not replaced
Probability of drawing a ticket first is red and second drawn is blue
= ( 20 / 45 ) × ( 25 / 44 )
= ( 500 / 1980 )
= 0.2525..
= 0.253
Therefore, probability of drawn ticket at random with the condition of first ticket drawn is red and for second ticket drawing blue is equal to 0.253.
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Determine the domain and range of the following parabola. f(x)=−3x2−30x−74
Answers
Step-by-step explanation:
1. if to rewrite the given equation, then
y= -3(x+5)²+1;
2. the vertex of the parabola is (-5;1) and it is the highest point. Then
3. domain: x is any value, x∈(-∞;+∞);
range: y not greater than 1, y∈(-∞;1].
For the parabola f(x) = -3x² - 30x - 74, the domain is all real numbers (-∞, ∞), and the range is also all real numbers (-∞, ∞).
To determine the domain and range of the given parabola f(x) = -3x² - 30x - 74, we need to understand the behavior of parabolas.
The domain of a function represents all the possible x-values for which the function is defined. In this case, there are no restrictions on the x-values for the given quadratic function. Therefore, the domain is all real numbers, expressed as (-∞, ∞).
The range of a function represents all the possible y-values that the function can produce. For a parabola, if the coefficient of the squared term (in this case, -3) is negative, the parabola opens downward. Since the coefficient is negative, the parabola will have a maximum value. In this example, there is no minimum or maximum specified; thus, the range will also be all real numbers.
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f(x)=−x 2 −2x+3 \text{Find }f(1) Find f(1)
Answers
f(1) = -1
Find the y-coordinate of the y-intercept of the polynomial function defined below.
f(x) = 3 + 5x² + 4x^4 - 3x³ - 6x
Answers
Answer:
3
Step-by-step explanation:
When you substitute 0 in for x in the equation. You are left with f(0) = 3.
Joseph invested $7,850 in an account that pays 7.5% interest per year. How much interest will Juan have earned at the end of 36 months?
Answers
Juan have earned of interest after 36 months at the rate of 7.5% interest per year.
What is Simple Interest?Simple interest is a quick and simple formula for figuring out how much interest will be charged on a loan. The daily interest rate, the principle, and the number of days between payments are multiplied to calculate simple interest. The formula of simple interest is as follows:
S.I = p×r×t
Where p is principle value, r is rate and t is time in years.
Here, the principle amount is $7,850 and rate is 7.5% i.e [tex]\frac{7.5}{100}=0.075[/tex] and time is 36 months i.e 3 years.
Now, putting the values,
SI = 7850×0.075×3
= $ 1766
Therefore, the interest Juan have earned in 36 months is $1,766.
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4
The height of a cone is two times its base diameter. What is the volume of the cone in terms of its base radius r
Answers
The volume of cone will be [tex]\frac{88r^{3} }{21}[/tex]
Cone:
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base.
Given,
The height of a cone is two times its base diameter.
we have to find the volume of cone in terms of base radius.
let us assume the base radius of cone is equal to r units
so diameter(d) of cone will be, d= 2r units
so,
height(h) of cone will be equal to, h = 2d units
h = 2x2r units
h = 4r units
We know,
Volume of the cone(V) = [tex]\pi r^{2} h/3[/tex]
Where,
r = radius of base
h = height of cone
π = 22/7
Thus by using formula,
Volume of cone(V) = [tex]\pi r^{2} h/3[/tex]
= 22/7 * [tex]r^{2}[/tex] * 4r/3
= 88[tex]r^{3}[/tex]/21
= [tex]\frac{88r^{3} }{21}[/tex]
Hence,
The volume of cone will be [tex]\frac{88r^{3} }{21}[/tex]
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Does anyone know what the Answer is...
45 ÷ ? = 8
Answers
Answer:
Step-by-step explanation:
Instead you do 45 divided by 8
And you will get 5.625
2 market women shared a basket full of oranges in order to sell them.the share of the first woman is twice the other if the are 300 oranges in the basket what is the share of each women
Answers
Answer:
Step-by-step explanation:
no idea
The number of oranges with first women are 200 and the number of oranges with other women are 100.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Given that, 2 market women shared 300 oranges.
The share of first woman is twice the other.
Let the number of oranges with other women be x.
Then, the number of oranges with first women will be 2x.
Now, x+2x=300
3x=300
x=100
So, 2x=200
Therefore, the number of oranges with first women are 200 and the number of oranges with other women are 100.
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Find the midpoint of the line segment with end coordinates of:
(0, 2) and (8,8)
Answers
The mid point of the line segment with the end coordinates of:
(0, 2) and (8,8) is (4,5).
Define the term mid point?A mathematical expression known as the midpoint formula is used to determine where two data points meet midway. This formula is also used in the study of economics to determine this same coefficient of elasticity, etc.The midpoint is the point on the center of the line that is equally distant as from 2 points if a line is created connecting the two points.For the given question;
The given coordinates are-
(x1, y1) = (0, 2)
(x2, y2) = (8,8)
The mid point is found by the formula;
x = (x1 + x2 )/2
y = (y1 + y2)/2
Put teh values,
x = (0 + 8)/2
x = 4
y = (2 + 8)/2
y = 5
Thus, the mid point of the line segment with the end coordinates of:
(0, 2) and (8,8) is (4,5).
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