Based on the calculations, the equation of this hyperbola in standard form is: A. [tex]\frac{x\;-\;7}{16} + \frac{y}{9} = 1[/tex].
How to determine the equation of a hyperbola?Mathematically, the equation of a hyperbola in standard form is given by:
[tex]\frac{x\;-\;h}{a^2} + \frac{x\;-\;k}{b^2} = 1[/tex]
Given the following data:
Center (h, k) = (7, 0)
Vertex (h+a, k) = (7, 4)
Focus = (h+c, k) = (7, 5)
Also, we can deduce that the value of a and c are 4 and 5 respectively.
For the value of b, we would apply Pythagorean's theorem:
c² = a² + b²
b² = c² - a²
b² = 5² - 4²
b² = 9.
Substituting the parameters into the standard equation, we have:
[tex]\frac{x\;-\;7}{4^2} + \frac{y\;-\;0}{3^2} = 1\\\\\frac{x\;-\;7}{16} + \frac{y}{9} = 1[/tex]
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Find equation in slope intercept form of the line passing through the points with given coordinates (-4,-2), (-2,2)
Answer:
y = 2x + 6
Step-by-step explanation:
We are given that a line runs through the points (-4,-2) and (-2,2).
We want to write the equation of this line in slope-intercept form.
Slope-intercept form is written as y=mx+b, where m is the slope and b is the value of y at the y intercept, hence the name.
First, we need to find the slope of the line.
The slope can be found using the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points.
Even though we have two points, let's label the values of the points to avoid confusion & mistakes.
[tex]x_1=-4\\y_1=-2\\x_2=-2\\y_2=2[/tex]
Now let's find the slope (remember: the formula has subtraction):
m=[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
m=[tex]\frac{2--2}{-2--4}[/tex]
This can be simplified:
m=[tex]\frac{2+2}{-2+4}[/tex]
Add the numbers together.
m=[tex]\frac{4}{2}[/tex]
Divide.
m = 2
The slope of the line is 2.
We can substitute this into the equation for m.
Here is the equation of the line so far:
y = 2x + b
We now need to find b.
As the equation passes through both (-4, -2) and (-2, 2), we can use either one of them to help solve for b.
Taking (-4, -2) for instance:
Substitute -4 as x and -2 as y in the equation we have.
-2 = 2(-4) + b
Multiply.
-2 = -8 + b
Add 8 to both sides.
6 = b
Substitute 6 as b into the equation.
y = 2x + 6
Topic: finding the equation of the line (slope-intercept form)
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Which of the following represents a geometric sequence with a common ratio r = 6?
1, 6, 36, 216, 1,296
3, 18, 24, 144, 864
7, 13, 19, 25, 31
648, 108, 18, 3, 0.5
Answer:
The first sequence is correct being (6)^k
Step-by-step explanation:
To find a ratio divide a term by its previous member such as 36/6 to give you r=6.
Raj’s bathtub is clogged and is draining at a rate of 1.5 gallons of water per minute. The table shows that the amount of water remaining in the bathtub, y, is a function of the time in minutes, x, that it has been draining. what is the value of this function
The range of y will be all real numbers such that 0≤y≤40
The complete question is:
Raj’s bathtub is clogged and is draining at a rate of 1.5 gallons of water per minute. The table shows that the amount of water remaining in the bathtub, y, is a function of the time in minutes, x, that it has been draining. What is the range of this function? all real numbers such that y ≤ 40 all real numbers such that y ≥ 0 all real numbers such that 0 ≤ y ≤ 40 all real numbers such that 37.75 ≤ y ≤ 40
What is inequality?The relation between two expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than, or < ‘less than.
Raj’s bathtub is clogged and is draining at a rate of 1.5 gallons of water per minute.
The amount of water remaining in the bathtub = y
The function of time in minutes, that it has been draining = x
At 0 minutes the amount of water is 40 gallons.
The highest volume of water is 40 which is decreasing at the rate of 1.5 gallons per minute.
The given function is a linear function
y = 0
However, the volume of water can be 0 but cannot ever be negative.
Therefore the range of y will be all real numbers such that 0≤y≤40
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The following excerpt comes from the International Bottled Water Association
"In 2012, total U.S. bottled water consumption increased to 9.67 billion gallons, up from 9.1 billion gallons in 2011. In fact, 2012's consumption growth was the strongest it has been in five years. In addition,
per-capita consumption is up 5.3 percent in 2012, with every person in America drinking an average of 30.8 gallons of bottled water last year. Bottled water increased in absolute volume more than any other
beverage category in the U.S. Bottled water sales increased by 6.7 percent in 2012, and now total $11.8 billion."
(i) Estimate the per capita consumption from 2011 from data in the statement. Express your answer rounded correctly to the nearest tenth of a gallon.
Using proportions, the estimate of the per capita consumption from 2011 is of 29.2 gallons.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
From the data, the 2012 consumption was 5.3% percent higher than in 2011, that is 105.3% of 1.053x. This consumption was of 30.8 gallons, hence:
1.053x = 30.8.
x = 30.8/1.053
x = 29.2.
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Please help! thank youuuu
Use polynomial long division to divide. Determine whether the divisor evenly divides into the dividend.
(8x-5)/(2x+1)
Answer:
The solution is 4 - 9/(2x + 1)
No, the divisor does not evenly divide into the dividend
Step-by-step explanation:
Please see the attached image
What is the equation for a cosecant function with vertical asymptotes found at x equals pi over 3 plus pi over 3 times n comma such that n is an integer?
f (x) = 3csc4x
g(x) = 3csc2x
h(x) = 4cscx
j (x) = 4csc3x
Despite not being a part of the function graph, a vertical asymptote is a vertical line that serves as a guide f (x) = 3csc4x.
How do you find a vertical asymptote?The denominator of the function, n(x), is the place where vertical asymptotes can be located.
If the numerator, t(x), is not zero for the same value of x, then the vertical asymptotes can be determined by solving the equation n(x) = 0 where n(x) is the denominator.
f (x) = 3csc4x.
Periodicity of 3csc(4x) : π / 2
Domain of 3csc(4x) :
Solution : π / 2 n<x < π / 4 + π / 2 n or π /4 + π / 2 n< x < π /2 + π /2 n
Interval Notation : ( π / 2 n,π /4 + π /2)n) ∪ (π / 4 + π /2 n, π / 2 + π /2 n)
Range of 3csc(4x) :
Solution : f(x) ≤ -3 or f(x) ≥ 3
Interval notation : ( - ∝ , -3 ) ∪ (3,∝)
Axis interception points of 3csc(4x) : None
Asymptotes of 3csc(4x) : vertical: x = π / 2 n, x = π / 4 + π/2 n
extreme points of 3csc(4x) :
Minimum (π /8 + π / 2 n,3 ),
Maximum ( 3π / 8 + π/2n, -3).
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Despite not being a part of the function graph, a vertical asymptote is a vertical line that serves as a guide f (x) = 3csc4x.
How do you find a vertical asymptote?The denominator of the function, n(x), is the place where vertical asymptotes can be located.
If the numerator, t(x), is not zero for the same value of x, then the vertical asymptotes can be determined by solving the equation n(x) = 0 where n(x) is the denominator.
f (x) = 3csc4x.
Periodicity of 3csc(4x) : π / 2
Domain of 3csc(4x) :
Solution : π / 2 n<x < π / 4 + π / 2 n or π /4 + π / 2 n< x < π /2 + π /2 n
Interval Notation : ( π / 2 n,π /4 + π /2)n) ∪ (π / 4 + π /2 n, π / 2 + π /2 n)
Range of 3csc(4x) :
Solution : f(x) ≤ -3 or f(x) ≥ 3
Interval notation : ( - ∝ , -3 ) ∪ (3,∝)
Axis interception points of 3csc(4x) : None
Asymptotes of 3csc(4x) : vertical: x = π / 2 n, x = π / 4 + π/2 n
extreme points of 3csc(4x) :
Minimum (π /8 + π / 2 n,3 ),
Maximum ( 3π / 8 + π/2n, -3).
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How do you do this?
a) The ratio of the perimeters is the same as the scale factor, so the ratio is the perimeters is 2/3.
b) The perimeter of triangle ABC is [tex]36\left(\frac{3}{2} \right)=\boxed{54 \text{ ft}}[/tex]
c) The ratio of the areas is the scale of the scale factor. In this case, the ratio is [tex](2/3)^{2}=\boxed{4/9}[/tex]
d) The area of FDE is [tex]36(4/9)=\boxed{16 \text{ ft}^{2}}[/tex]
In 2010, the number of clown costumes sold at a single costume shop was 11. By 2015, that number had grown to 23. Assuming a constant increase in clown costume sales, calculate the average rate of change (the slope) from 2010 to 2015.
The average rate of change = 109%
What is the meaning of Rate of Change ?Rate of Change is the measure of the increase or decrease in the value with respect to the initial value.
It is given that
No of clown sold in 2010 = 11
No of clown sold in 2015 = 23
The average rate of change = ((23-11)/11 ) *100 %
The average rate of change = 109%
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What is the equation of the line that has a slope of 3 and passes through the point (1,-2)?
Y=3x+5
Y=3x-5
Y=3x+1
Answer:
y = 3x - 5
Step-by-step explanation:
y = mx + b
y - b = mx
-b = mx - y
b = -mx + y
b = -3(1) -2
b = -5
Ahmed is three times as old as Suad. Deeqa is three times o Ahmed. If Deeqa is 39 years old. How old is Suad?
Answer:
Suad is 4.333 years old.
Step-by-Step explanation:
let x = Suad's age
let y = Deeqa's age
let z = Ahmed's age
y = 3x
z = 3y = 3(3x)
but z = 39
3(3x) = 39
x = 4.33333
Graph the inequality.
y > |x-2| -5
Here is the graph! I hope this helps!
Answer:
^this means graph C.
Step-by-step explanation:
Solve: 12x- +5x-4-_ = 172x+6 O x=2 0 x==5 O x=2 x=-5 no solution
The solution to the equation 12x + 5x - 4 = 172x + 6 is D. No solution.
How to calculate the equation?The information given is 12x + 5x - 4 = 172x + 6 and we are told to find the value of x.
12x + 5x - 4 = 172x + 6
Collect like terms
12x + 5x - 4 = 172x + 6
12x + 5x - 172x = 6 + 4
-155x = 10
x = 10/-155
x = -0.0645
Therefore, the correct option is no solution as the correct answer isn't given.
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Which statements are true from the graph ? Check all that apply:
the line is horizontal the line is vertical
The statements that are true from the graph are:
The line is horizontalThe line goes through (4, 4 )The y- values are all 4Calculations and ParametersGiven the graph, we can note some things:
There is a line with a slope of zero.This is a horizontal line parallel to the x-axis.The equation of the line is y = cc is the value of the y- coordinates the line passes through.
The line passes through (0, 4 ) with a y- coordinate of 4
Therefore, the equation of the line of y= 4
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Clara is building a triangular garden. She wants the length of the longest side to be three more than twice as long as the length of the shortest side, and the third side will be twelve feet long.
What expression could she write to determine the perimeter of the triangle if s represents the length of the shortest side?
The garden's perimeter = 3+2s+12+s.
The simpler version of the expression = 15+3s.
The variable's frequency in the abbreviated statement is 3.
Calculation of a triangle's perimeter:The triangle's perimeter is equivalent to the sum of its edges, where a, b, and c are its sides.
Let the triangle's longest edge be 'a'shorter one to be 's'the opposite side 'b'With the information provided, the formula will become 'a = 3+2s'
and 'b' is also 12 ft.
Triangle perimeter =the total of its edges
= a + b + s
= 3 + 2s + 12 + s
= 15 + 3s
As a result, the garden's perimeter = 3+2s+12+s.
Therefore,the simpler version of the formula is 15+3s.
The variable's coefficient in the abbreviated statement is 3.
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100 adults were asked how they keep fit.
Each adult goes to the gym, runs or cycles.
45 of these adults are female.
30 of the 52 adults who go to the gym are female.
35 adults run.
9 males cycle.
One of the females is chosen at random.
Write down the probability that she runs.
The probability that a female chosen runs = 0.24
How to find the probability?We are given;
Number of adults surveyed = 100
Total number of females = 45
Total Number of Males = 55
Number of female adults that go to gym = 30
Number of adults that run = 35
Number of males that cycle = 9
Thus;
From the probability table attached, we can say that;
Number of females that can run = 11
Probability that she runs = 11/45
Probability that she runs = 0.24
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What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point (−3, 1)?
y – 1=-3/2(x + 3)
y – 1=-2/3(x + 3)
y – 1=2/3(x + 3)
y – 1=3/2(x + 3)
The equation of the line parallel to given line and passing through (-3, 1) will be y – 1 = (3/2)(x + 3). Then the correct option is D.
What is the equation of line?The equation of line is given as
y = mx + c
Where m is the slope and c is the y-intercept.
The slope of the parallel lines are equal.
m = (2 + 4) / (2 + 2)
m = 6/4
m = 3/2
Then the equation of the line will be
y = (3/2)x + c
The line is passing through (-3, 1). Then the value of c will be
1 = (3/2)(-3) + c
c = 9/2 + 1
Then the equation will be
y = (3/2)x + 9/2 + 1
y – 1 = (3/2)(x + 3)
Then the correct option is D.
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This year, a teacher graded 83 more final exams than last year. If the teacher graded 625 final exams this year, how many final exams did the teacher grade last year?
Answer:
542
Step-by-step explanation:
So X is the number of finals graded by teacher last year
she graded 83 more finals than the last year
and the total number of finals she graded this year is 625 so we have an equation
X + 83 = 625
and as you solve it for X
X = 625-83
x = 542
Answer:
542,
Step-by-step explanation:
You could solve these types of problems with inverse operations, so if you subtract 83 from 625, you'd get 542.
f(x) = x². What is g(x)?
g(x) is the inverse of f(x)
[tex]f(x) = {x}^{2} [/tex]
[tex]g(x) = \sqrt{x} [/tex]
Assuming two hikers begin at a trail start, which scenarios must be true based on the table? Check all that apply.
Melissa and Corey started on different trails.
Melissa hiked up the trail for a longer period of time than Corey.
Melissa and Corey crossed paths between 60 and 90 minutes.
Corey began his descent before Melissa.
As Melissa’s time increased from 0 to 60 minutes, her elevation decreased.
The true statements are (A), (B) and (D)
What is data?Data is information that has been translated into a form that is efficient for movement or processing.
As, from the given table we can see that
Melissa and Corey started on different trails.Melissa hiked up the trail for a longer period of time than Corey.Corey began his descent before Melissa.Learn more about this concept here:
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A manufacturing company regularly conducts quality control checks at specified periods on the products it manufactures. Historically, the failure rate for LED light bulbs that the company
manufactures is 14%. Suppose a random sample of 10 LED light bulbs is selected. Complete parts (a) through (d) below.
a. What is the probability that none of the LED light bulbs are defective?
The probability that none of the LED light bulbs are defective is
(Type an integer or a decint. Round to four decimal places as needed.)
A common discrete distribution is used in statistics, as opposed to a continuous distribution is called a Binomial distribution. The probability that none of the LED light bulbs are defective is 0.2213.
What is Binomial distribution?A common discrete distribution is used in statistics, as opposed to a continuous distribution is called a Binomial distribution. It is given by the formula,
P(x) = ⁿCₓ (pˣ) (q⁽ⁿ⁻ˣ⁾)
Where,
x is the number of successes needed,
n is the number of trials or sample size,
p is the probability of a single success, and
q is the probability of a single failure.
Given the probability of a bulb being a failure is 14%, therefore, the probability of bulb passing the quality check is 86%.
Let the probability of success be defined as 0.14 for binomial distribution, and the probability for failure be 0.86. Therefore, The probability that none of the LED light bulbs are defective is
P(x=0) = ¹⁰C₀ (0.14)⁰ (0.86)¹⁰ = 0.2213
Hence, The probability that none of the LED light bulbs are defective is 0.2213.
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Sasha's bank account earns 4% simple interest. How much must she deposit in the account today if she wants it to be worth $1,000 in 5 years?
If Sasha wants her account to be worth $1,000 in 5 years, then she must deposit $833.34.
What is simple interest?Simple interest is a method of calculating interest on an amount for n period of time with a rate of interest of r. It is calculated with the help of the formula,
SI = PRT
where SI is the simple interest, P is the principal amount, R is the rate of interest, and T is the time period.
Sasha's bank account earns 4% simple interest. Therefore, if she wants it to be worth $1,000 in 5 years then the principal amount must be,
Account balance = PRT + P
$1,000 = P[(0.04×5)+1]
$1,000 = 1.2 P
P = $833.34
Hence, if Sasha wants her account to be worth $1,000 in 5 years, then she must deposit $833.34.
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Comparing Fractions Choose the correct symbol to compare the fractions please see attached image
Answer:
>
Step-by-step explanation:
Given: [tex]\frac{1}{2} ? \frac{1}{3}[/tex]
Want: Which fraction is larger?
To identify the size of a fraction, converting it into a fraction is easier
[tex]\frac{1}{2}[/tex] = 0.5
[tex]\frac{1}{3}[/tex] = 0.33
where 0.5 is greater than 0.33
Choices:
< is less than symbol
> is greater than symbol
= is equal to
[tex]\frac{1}{2} > \frac{1}{3}[/tex]
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Visual Representation:
Which expression is equivalent to 36x² - 25?
O (18x)²+(-5)²
O (18x)²-(5)²
O (6x)²+(-5)²
O (6x)²-(5)²
Answer:
[tex] \boxed{ \rm(6x)²-(5)²}[/tex]
Step-by-step explanation:
Given expression/polynomial :
[tex]3 6{x}^{2} - 25[/tex]
Let's evaluate.
We can re-write the polynomial as
[tex]6 {}^{2} x {}^{2} - 5 {}^{2} [/tex]Further
[tex]{(6x) {}^{2} - 5 {}^{2} }[/tex]Hence, the expression is equivalent to last Choice, (6x)²-(5)².
-Regards, Hannah :)
a triangle has wo sides of lengths 6 and 9. what value could the length of the third side be?
knowed that the 3rd side of a triangle need being greater than the difference of the given two sides length and less that the sum of the given two sides length
so with these knowledge we get
x > (9-6) so x > 3
and
x < (9+6) so x < 15
from these result that the 3rd side length can being choice B. 4 ,C. 7,F. 10,
E. 12,
hope this will help you .
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Which shows the first step in the solution to the
equation log₂x + log₂(x - 6) = 4?
log was used calculate big numbers before calculators
log is a re-arranged way to show a number with an exponent
example
log₂ 16 = 4 means 2^4 = 16
logx(Z) = y means x^y=Z
log(x-6)/log(2) + log(x)/log(2) = 4
(log(x-6)+ log(x))/log(2) = 4
(log(x-6)+ log(x)) = 4log(2)
(log(x-6)x) = log(16)
x=8
John needs to put the set of numbers below in ascending order.
3, -1, √5, 13, -11
Which number should John write first?
A. -1
B. -11
C. √5
D. 13
Based on statistics from a worldwide health organization, in there were million people worldwide living with a certain disease, and million deaths from the disease. By , the number of people living with the disease had grown to million, and million deaths were reported. Find the percent change for each statistic, and write any conclusions you can draw.
Based on the statistics from the worldwide health organization, the percentage change in those living with the disease is 19.7% and the change in those who have died is 70%.
This shows that the disease is getting worse and not being treated effectively.
What do the statistics on the disease show?The number of those living with the disease went from 71 million to 40 million:
= (85 - 71) / 71
= 19.7% increase
The percent change in those who have died is:
= (6.8 - 4) / 4
= 70% increase
This shows that the measures being implemented against the disease are not working because both the number of people who have the disease and have died from it has increased.
Full question is:
Based on statistics from a worldwide health organization, in 1999 there were 71 million people worldwide living with a certain disease, and 4 million deaths from the disease. By 2015, the number of people living with the disease had grown to 85 million, and 6.8 million deaths were reported. Find the percent change for each statistic, and write any conclusions you can draw.
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URGENT: Click on the graph to choose the correct answer to the equation. x > 2
The graph of the inequality, x > 2 is the graph attached below.
How to Find the Graph of Inequality?Given the inequality as, x > 2, it means all possible values of x must be greater than 2.
Thus, the graph that will show all possible values of x that would be greater than 2 would be a vertical line indicating the values are over 2 and upwards.
Therefore, the graph that represents x > 2 is shown in the image attached below.
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Please help me solve this
A function assigns the values. The value of (p·p)(x) is x⁴ + 10x³ + 30x² + 25x.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
Given p(x)= x²+5x, therefore, the value of (p·p)(x) can be written as,
(p·p)(x) = p(p(x))
= (x²+5x)²+5(x²+5x)
= x⁴ + 25x² + 10x³ + 5x² + 25x
= x⁴ + 10x³ + 30x² + 25x
Hence, the value of (p·p)(x) is x⁴ + 10x³ + 30x² + 25x.
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Write the equation for the function graphed below.
Answer:
f(x) = (x + 1)^2 + 2
Step-by-step explanation:
A parabola function has different forms:
standard form: f(x) = ax + bx + c^2
vertex form: f(x) = a (x − h)^2 + k
Our equation that includes a horizontal translation is h
- negative h shifts right
- positive h shifts left
this graph shows a shift to the left by 1 so h = 1
Our equation that includes a vertical translation is k
- negative k shifts down
- positive k shifts up
this graph shows a shift to the up by 2 so k = 2
- a represents a compression / stretch which seems not to be in this graph
combine, f(x) = (x + 1)^2 + 2
