I dont know how to prove the equation below...pls help prove it with clear steps, i have no clue how does sin^2x can go into 1+cosx
Answers
Answer:
Answer:
Foci: (±5, 0)
Vertex: ( ±4 , 0)
Eccentricity: e = 5/4
Step-by-step explanation:
Hyperbola:
Write the equation of hyperbola in the form,
\boxed{\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1}
a
2
x
2
−
b
2
y
2
=1
9x² - 16y² = 144
Divide the entire equation by 144,
\begin{gathered}\sf \dfrac{9x^2}{144}-\dfrac{16y^2}{144}=\dfrac{144}{144}\\\\\\ \dfrac{x^2}{16}-\dfrac{y^2}{9}=1\\\\\\\dfrac{x^2}{4^2}-\dfrac{y^2}{3^2}=1\end{gathered}
144
9x
2
−
144
16y
2
=
144
144
16
x
2
−
9
y
2
=1
4
2
x
2
−
3
2
y
2
=1
a = 4 and b =3
Eccentricity: e
\boxed{e=\sqrt{1+\dfrac{b^2}{a^2}}}
e=
1+
a
2
b
2
\begin{gathered}\sf = \sqrt{1+\dfrac{9}{16}}\\\\=\sqrt{\dfrac{16}{16}+\dfrac{9}{16}}\\\\=\sqrt{\dfrac{25}{16}}=\sqrt{\dfrac{5*5}{4*4}}\\\\\\=\dfrac{5}{4}\end{gathered}
=
1+
16
9
=
16
16
+
16
9
=
16
25
=
4∗4
5∗5
=
4
5
Foci:
(ae, 0) & (-ae , 0)
\begin{gathered}\sf \left(4*\dfrac{5}{4},0 \right) \ ; \ \left( -4*\dfrac{5}{4},0 \right)\\\\\\(5, 0) \ ; \ (-5,0)\end{gathered}
(4∗
4
5
,0) ; (−4∗
4
5
,0)
(5,0) ; (−5,0)
Vertex of hyperbola:
The vertex of the hyperbola is a point where the hyperbola cuts the axis. (-a , 0) ; (a , 0)
Vertex : (-4 , 0) ; (4 , 0)
Related Questions
which set of ordered pairs below is a function?
Answers
The first and the third set of order pairs is a function.
What is function?
A set of ordered pairs that contain only unique ordered pairs with unique x coordinates is referred to as a function. A function is defined by an equation that generates a set of ordered pairs.
Consider, the given set of order pairs,
I - {(3, 3), (6, 1), (2, 3)}
II - {(4, 7), (2, 8), (4, 2)}
III - {(9, 6), (3, 1), (7, 3)}
We have check which set of ordered pairs above is a function.
Since, a function is a set of ordered pairs in which no two different ordered pairs have the same x -coordinate.
In the given order pairs the second order pair contain same x - coordinate.
So, the second order pair is not a function.
Hence, the first and the third set of order pairs is a function.
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What is the area of right triangle with base 10 cm and height 5 cm?
Answers
Answer:25cm
Step-by-step explanation:area of triangle=1/2 × base× height
=1/2×10×5
= 25cm^2
What is the coefficient of the variable in the expression 4 − 3x − 7 6 3 points?
Answers
The coefficient of the variable x in the expression 4a+3x +a is 3
Given that,
The expression is 4a+3x +a which can be simplified as 4a+3x +a
To find : The coefficient of the variable x in the expression
The coefficient of x in 4a+3x +a is 3
A number multiplied by a variable is referred to as the coefficient of a number. For instance, the x2 coefficient in the formula 6x2+99 is 6
Coefficient is the number that is beside the variable in the equation.
As we can see that the term that is in multiplication with x is 3
Hence, the coefficient of x in the equation is 3
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can someone help me find two true statements!
Answers
1. In a triangle line parallel to one side will divides other two sides in the same ratio
2. 8/x = 12/18
x = 12
How do you find a linear equation from two points?Finding the slope between two points on a line and solving for the y-intercept in the slope-intercept equation y=mx+b allows us to formulate an equation for that line.
A linear equation has the slope-intercept form y = mx + b. Variables in the equation are x and y. When x is 0, the integers m and b provide the line's slope (m) and the value of y. (b).
Because (0,y) is the location where the line crosses the y-axis, the value of y when x is 0 is referred to as the y-intercept.
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2x-(x+5)=-1
Solve for x.
Answers
Answer:
x=4
Step-by-step explanation:
2x-(x+5)=-1
Solve for x
Distribute the minus sign
2x-x-5=-1
Combine like terms
x -5 = -1
Add 5 to each side
x-5+5 = -1+5
x=4
[tex]2x-(x+5)=-1[/tex]
Distribute negative sign:
[tex]2x+-1(x+5)=-1[/tex]
[tex]2x+-x+-5=-1[/tex]
Combine Like Terms:
[tex](2x+-x)+(-5)=-11[/tex]
[tex]x-5=-1[/tex]
Add 5 to both sides:
[tex]x-5+5=-1+5[/tex]
[tex]\fbox{x = 4}[/tex]
A town has a population of 4000 and grows at 2% every year. To the nearest year, how long will it be until the population will reach 4500?
Answers
=========================================================
Work Shown:
x = number of years
y = population
y = 4000*1.02^x
4500 = 4000*1.02^x
4500/4000 = 1.02^x
1.125 = 1.02^x
log(1.125) = log(1.02^x)
log(1.125) = x*log(1.02)
x = log(1.125)/log(1.02)
x = 5.94784893412128
Rounding to the nearest year, it takes about 6 years for the population to reach 4500 people.
---------------
Check:
Plug in x = 5
y = 4000*1.02^x
y = 4000*1.02^5
y = 4416.3232128
y = 4416
We come up a bit short. Now try x = 6
y = 4000*1.02^x
y = 4000*1.02^6
y = 4504.649677056
y = 4505
We go a bit over the target, but it's better to go over rather than come up short.
This confirms the answer is approximately 6 years.
Can you make a triangle with 3 different length sides?
Answers
Yes, you can make a triangle with three different length sides.
1. Draw three lines of different lengths on a piece of paper.
2. Connect the ends of the lines to form a triangle.
3. The triangle will have three different length sides. The lengths of the sides depend on the lengths of the three lines you drew.
4. To make sure the triangle is valid, use the triangle inequality theorem. This theorem states that the sum of any two sides of a triangle must be greater than the third side.
5. Measure each side of the triangle with a ruler and add them together.
6. Compare the sum of the two sides to the third side. If they are not equal or greater, adjust the lines to create a valid triangle.
7. Once the triangle is valid, it will be a triangle with three different length sides.
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Which word or phrase describes the value of the function over the interval [ -6, 0]
Answers
Answer:
decreasing is the answer
last month greg arrived late to work 3 out of 21 days his boss used this information to determine the probability of greg being late again. which pair of probabilites is correct
Answers
The probability of Greg being late again, along with the type of probability, is given as follows:
1/7, experimental probability.
How to obtain a probability?A probability is calculated as the division of the number of desired outcomes by the number of total outcomes.
In the context of this problem, the outcomes are given as follows:
Desired: Greg being late, 3 times.Total outcomes: 21 times that Greg went to work.The probability of Greg being late is then calculated as follows:
3/21 = 1/7.
Which is an experimental probability, as the number of outcomes are taken from previous "experiments".
The other type of probability is theoretical, for example, a fair coin has a 50% chance of falling heads and 50% change of falling tails.
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prove that (4x-5)2 -5x(3x-8) is a positive for all values of x
Answers
To prove that (4x - 5)^2 - 5x(3x - 8) is positive for all values of x, we can start by expanding the expression:
(4x - 5)^2 - 5x(3x - 8) = 16x^2 - 40x + 25 - 15x^2 + 40x = -14x^2 + 80x - 25
Now, we can complete the square to find that this expression is always positive:
-14x^2 + 80x - 25 = (-14x^2 + 80x) - 25
= (-14x^2 + 80x + (80/2)^2 - (80/2)^2) - 25
= (-14x + 40)^2 - 25 - 40^2 + 40^2
= 15^2 - 25
= (15 - 5)(15 + 5)
= 10(20)
= 200
Since this expression is equal to 200, which is a positive number, we can conclude that (4x - 5)^2 - 5x(3x - 8) is positive for all values of x.
560Steve can paint his greenhouse in four hours working alone; his ten-year-old son can paint the greenhouse alone in nine hours. Working together, how long, to the nearest minute, would it take them
Answers
By working together, they can make the work done in two hours and forty-six minutes.
How do you calculate work done together?The equation below may be used to compute work: Work is equal to force times distance. The Newton metre is the SI unit of work (N m). When an item is moved across a distance of 1 m with 1 N of force, one joule is equal to the amount of work that is accomplished.Every time a force pushes anything over a distance, work is done. By multiplying the force by the distance travelled in the direction of the force, you may determine the amount of energy transferred, or work done.Therefore, the formula for work is W = F d s, which stands for work done equal to force times displacement.Given data :
Think of these work rates as "greenhouses per hour", not "hours per greenhouse". Then Steve's work rate is one fourth of a greenhouse per hour, and his son's is one ninth of a greenhouse per hour. Now, multiply each rate by the time t to get the portion of the greenhouse each paints, and add the products up to get one greenhouse. The work equation becomes:
1 / 4 t + 1 / 9 t = 1
9 / 36 . t + 4 / 36 t = 1
13 / 36 t = 1
36 / 13. 13 / 36 t = 36 / 13 .1
t = 36 / 13 hrs
or 36 / 13. 60 = 166 min
This is two hours and forty-six minutes.
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For a hypothesis test of H0: p = 0. 65 against the alternative Ha: p < 0. 65, the z test statistic is found to be -2. 35. What can be said about his finding?
Answers
The finding is significant at both α = 0.05 and α = 0.01.
What is hypothesis ?
A hypothesis (plural: hypotheses), in a scientific context, is a testable statement about the relationship between two or more variables or a proposed explanation for some observed phenomenon.
Have given ,
H0: p = 0. 65 , Ha: p < 0. 65 and the z test statistic is found to be -2. 35.
From the test statistic value, we can obtain the p value of the test ,
To determine if the claim is significant at both α = 0.05 and α = 0.01
We compare the Z statistic and Z critical values
If Z statistic < Z critical ; then we reject H0; if otherwise, we fail to reject H0
Z critical at 95% one tailed = - 1.645 (left tail)
Z critical at 95% one tailed = -2.326 (left tail)
- 2.35 < - 1.645
-2. 35 < 2.326
Since, Z statistic < Z critical;
The finding is significant at both α = 0.05 and α = 0.01.
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Joanne works two jobs to pay for college. She tutors for $25 per hour and also works as a receptionist for $12 per hour. Due to her class and study schedule, Joanne is only able to work up to 25 hours per week, but must earn at least $200 per week. If t represents the number of hours Joanne tutors and r represents the number of hours she works as a receptionist, which system of inequalities represents this scenario?
A. t + r >= 25
25t + 12r = 200
B. t + r <= 25
25t + 12r <= 200
C. t + r <= 25
25t + 12r >= 200
D. None of the systems shown represent this scenario.
Answers
The required system of inequality will be
t + r ≤ 25
25t + 12r ≥ 200
What is system of equations?
A set or group of equations that are solved together is referred to as a system of equations. Both algebraic and graphic solutions are possible for these equations. The intersection of two lines represents the system of equations' solution.
If 't' represents the number of hours Joanne tutors and 'r' represents the number of hours she works as a receptionist.
Then the total hours she can work = t + r
Since she can work up to 25 hours.
Thus t + r ≤ 25
If she earns $25 per hour by tuition and $ 12 per hour by working as a receptionist .
Then her total earning = $25t + 12r
Since she must earn $200.
Then 25t + 12r ≥ 200
Hence, the required system of inequality will be
t + r ≤ 25
25t + 12r ≥ 200
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The annual cost of Randall coverage is 4900. The college pays 62% of the cost. How much is Devi red from his biweekly paycheck for medical insurance?
Answers
$1862 is Devi red from his biweekly paycheck for medical insurance
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
The annual cost of Randall coverage is 4900.
The college pays 62% of the cost.
We need to find How much is Devi red from his biweekly paycheck for medical insurance
Let us convert 62% to decimal value by dividing with 100
62/100=0.62
Now multiply 0.62 with 4900
0.62×4900
3038
Now subtract 3038 from 4900
4900-3038
1862
Hence, $1862 is Devi red from his biweekly paycheck for medical insurance
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1. Cameron ran 3 miles in 24 minutes. Identify the equivalent ratio that represents Cameron's run. *
Answers
The equivalent ratio that represents Cameron's run is 1 : 8
Equivalent Ration:
In mathematics, the definition of equivalence ratio says: The equality of two ratios is also called a ratio. Front and follow have different values, but reducing them to their simplest form gives the same value.
For example, to find out if 2:3 and 16:24 are equivalent ratios, we need to reduce both ratios to their simplest form. The HCF of 2 and 3 is 1, so 2:3 is already in the simplest form. The HCF for 16 and 24 is 8. Divide these two numbers by 8 to get the reduced form. This means (16÷8):(24÷8) = 2:3. It's clear that 2:3 and 16:24 are the same value, equivalent ratios.
Based on the question :
Cameron ran 3 miles in 24 minutes.
Now,
Therefore we can say that :
Cameron ran 1 miles in 8 minutes;
Therefore, the equivalent ration is 1 : 8.
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the swim team trains 4 days every week. Mon, Tues, and Thurs, the swim tea spends 1/2h in the gym followed by 1 1/4 in the pool. On Saturday there is no gym work but double the swim team time. how many hours per week does the team train?
Answers
The number of hours that the team trains per week is 17 hours.
How to illustrate the fraction?A fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.
Since the swim tea spends 1/2h in the gym followed by 1 1/4 in the pool. On Saturday there is no gym work but double the swim team time. The number of hours will be:
( 1 1/4 + 1/2 + 2 1/2) × 4
= 4.25 × 4
= 17 hours.
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If a coin is tossed 4 times, the odds of it coming up either heads every time or tails every time are 1:7.
What is the probability the coin comes up all heads or all tails after 4 tosses?
Answers
The probability the coin comes up all heads or all tails after 4 tosses is 12.5%.
What is probability?Probability refers to the ratio, chance, likelihood, or odds that an expected outcome is achieved when there are many possible outcomes.
Probability is a fractional value, especially when the odds of the occurrence of the expected event are not 100% uncertain or certain.
In this case, probability lies between zero and one.
Odds of a coin coming up either heads or tails every time = 1:7
The sum of ratios = 8 (1 + 7)
Thus, the probability of the coin coming up with all heads or all tails = 12.5% (1/8 x 100)
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Mario has 3/4 cups of chocolate chips needs 1/3 cup to make a batch of cookies how many batches can Mario make if he uses all of his chocolate chips
Answers
In the given equation, Mario can make 2.25 batches of cookies using all of his chocolate chips.
How to find the given equation?First, you can find out how many total cups of chocolate chips Mario has by multiplying the number of cups per unit (3/4) by the number of units (1).
[tex]3/4 cups * 1 = 3/4 cups\\[/tex]
Then you can divide the total amount of chocolate chips Mario has by the amount of chocolate chips needed per batch:
[tex]3/4 cups / 1/3 cup/batch = 3/4 cups * 3/1 cup/batch = 9/4 cups/batch[/tex]
Finally, you can convert this to a whole number by dividing both the numerator and denominator by their greatest common factor, which is 1:
[tex]9/4 cups/batch = (9/1) / (4/1) = 9/4 = 2.25 batches[/tex]
Mario can make 2.25 batches of cookies using all of his chocolate chips.
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In the figure, JKLM-EFGH. Complete the following table,
type your answer
Value of x
Value of y
Scale factor of JKLM to EFGH
type your answer.
type your answer
Value of z
type your answer.
Answers
The values of x , y and z are 80,2,110 respectively from the given figure where JKLM ~ EFGH
What is Quadrilateral ?
Quadrilateral can be defined as the type of the polygon in which it contains four sides.
Given ,
JKLM ~ EFGH
Since , JKLM ~ EFGH
3/30 = 11/z
z = 30*11/3
z =110
3 / 30 = 8/ x
3 * x = 8 * 30
x = 80
3 / 30 = y/ 20
y = 20 * 3 / 30
y = 2
Hence the values of x , y and z are 80,2,110 respectively.
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Stefanie has 18 grams of 20% sugar syrup and john 30 of 25% what is the concentration in the new mixture
Answers
The concentration in the new mixture will be 11.1 grams.
What is mean by Percentage?A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.
Given that;
Stefanie has 18 grams of 20% sugar syrup.
And, john has 30 grams of 25%.
Here,
Stefanie has 18 grams of 20% sugar syrup.
Hence, The amount of sugar = 18 × 0.2
= 3.6 grams
And, john has 30 grams of 25%.
Hence, The amount of sugar = 30 × 0.25
= 7.5 grams
Thus, The concentration in the new mixture = 3.6 + 7.5 grams
= 11.1 grams
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What is a 3 sided called?
Answers
A triangle is a polygon with 3 sides, and the sum of its interior angles is 180 degrees.
The properties of a triangle are:
A triangle has three sides, three angles, and three vertices.The sum of the length of any two sides of a triangle is greater than the length of the third side.The side opposite the largest angle of a triangle is the largest side.There are several types of triangles, including:
Equilateral triangle: This is a triangle with three sides of equal length and three angles of 60 degrees.
Isosceles triangle: This is a triangle with two sides of equal length and two angles of equal measure.
Scalene triangle: This is a triangle with three sides of different lengths and three angles of different measure.
Right triangle: This is a triangle with one 90-degree angle.
Acute triangle: This is a triangle with three angles that are all less than 90 degrees.
Obtuse triangle: This is a triangle with one angle that is greater than 90 degrees.
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What is the value of x?
3/4(x−8)= −1/2
Answers:
x = -26/3
x = 10
x = 22/3
Answers
the value of x would be 22/3 after simplification.
What is simplification?The acts or process of simplifying something such that it is simpler to accomplish or understand, or the item that is the consequence of such simplifying: The group offers suggestions on how to make trade procedures simpler. In mathematics, we generally use the PEMDAS method to simplify the values.
Given, the equation in the form of x is 3/4(x−8)= −1/2
For the value of x
3/4(x−8)= −1/2
multiply by 4 on both sides
3(x - 8) = -2
divide by 3 into both the sides
x - 8 = -2/3
add 8 on both sides
x = -2/3 + 8
x = 22/3
therefore, the value of x for the given equation 3/4(x−8)= −1/2 is 22/3.
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what is 20% of 14000
Answers
Answer:
2800
Step-by-step explanation:
20% of 14,000 is 2800
Multiply both sides by 14,000, then divide the left side to get the result.
Answer:
2800
is what the calculator says
What is the value of x? (HINT: find the sum of the
interior angles of the pentagon first)
Answers
Answer:
(2x+10°) + (x°) +(2x-20°) = 540°
540 is the sum angle of the polygon
collect like terms
2x+x+2x+10°-20°=540°
5x-10°=540°
5x=540°+10°
5x=550°
x=550°/5
x=110°
(2(110)+10)+(110)+(2(110)-20)=540°
How do you graph the transformation of an absolute value function?
Answers
By translating, stretching and contracting, or reflecting, two functions from the same function family can become one another. Any function from a function family can be obtained by applying one or more transformations to a parent function.
By altering the graph of the parent absolute value function, many other absolute value functions can be created. The resulting absolute value functions take the following form:
[tex]y = |x-h| + k[/tex], where h and k are real numbers.
Absolute value functions can be created using the same kinds of operations that produce new linear functions. They also have a similar impact on absolute value functions. The transformations may, however, appear differently graphically due to some important discrepancies between linear functions and absolute value functions.
Here are some of the transformations:
Vertical Transformations
Translation up k units, k > 0; [tex]y = |x|+k[/tex]Translation down k units, k < 0; [tex]y = |x| + k[/tex]Horizontal Transformations
Translation to the right h units, h > 0; [tex]y = |x - h|[/tex]Translation to the left h units, h < 0; [tex]y = |x - h|[/tex]This is how we graph the transformation of an absolute value function.
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Carolina is mowing lawns for a summer job.for every mowing job,she charges an initial fee plus $6 for each hour of work.her total fee for a 4-hour job,for instance is $32
Answers
The function Formula is, f(t) = 6t + 8.
What is Function?In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The collection of values that could result in a function's outputs is known as the co-domain.
Given:
Carolina charges an initial fee plus $6 for each hour of work
Her total fee for a 4-hour job = $32
So, if she work for 't' hours then linear equation will be of the form
f(t) = mt + b
Here m is the constant rate = 6
Now we plug in the values and find out b, m=6
f(t) = mt + b
32 = 6(4) + b
32= 24 + b
8 = b
So, the function is f(t) = 6 t+8.
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BRAINLY GRANTED FOR CORRECT ANS PLEASE HELP
Answers
P (X)= [tex]x^3+4 x^2-3x+2[/tex] moreover, [tex]\frac{P(x)}{x - 1} = x^{2} + 5x +2[/tex] as a result, the actual claim is [tex]$p(x)=(x-1)\left(x^2+5 x+2\right)+\frac{4}{x-1}$[/tex]
How do you find the probability equation?The likelihood of an event occurring is determined by probability: P(A) = f / N. Probability and odds are related, but probability is necessary for odds to exist. Prior to calculating the likelihood that an event will occur, probability is required.
The steps listed below can be used to determine an event's likelihood: Step 1: Identify an event that has only one outcome. Step 2: Compile a list of all potential outcomes, including any positive ones. Step 3: Subtract the number of favourable outcomes from the total number of outcomes that are feasible. The probability of A or B is equal to p(A or B) = p(A) + p(B) if occurrences A and B are mutually exclusive (B). A probability is a number that expresses the possibility or likelihood that a specific event will take place.
Therefore the correct answer is option A ) [tex]$p(x)=(x-1)\left(x^2+5 x+2\right)+\frac{4}{x-1}$[/tex]
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Emily's least favorite weekly chore is mowing the lawn, even though it only takes her about 30 minutes. Her family's lawn is 1,000 square yards, but her parents are thinking about moving to the country, where they would have 5 acres of land. If the time it takes Emily to mow a lawn is proportional to its area, how many hours would it take her to mow 5 acres?
Answers
It will take Emily 12.1 hours to mow 5 acres
How determine how many hours it will take to mow 5 acres?
Variation is the relation between a set of values of one variable and a set of values of other variables
Let T and A represent the time and area respectively
Since the time it takes Emily to mow a lawn is proportional to its area. We can write:
T α A
T = kA (where k is constant of proportionality)
It takes her about 30 minutes to mow 1,000 square yards. That is:
T = 30 and A = 1,000
T = kA
30 = 1000k
k = 30/1000
k = 0.03
Time for 5 acres:
5 acres = 5 × 4840 = 24200 square yards (1 acre = 4840 square yards)
T = kA
T = 0.03 × 24200
T = 726 minutes (divide by 60 to get time in hours)
T = 12.1 hours
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Similar Figures / Proportion (Level 1)
Triangle OPQ is similar to triangle RST. Find the measure of side ST. Round your
answer to the nearest tenth.
Answers
Rounded to the nearest tenth, the measure of side ST is 8.0.
How to find measure of sides?In order to find the measure of side ST, you will need to use the concept of similarity. If two triangles are similar, then their corresponding sides are in proportion. If we let x be the measure of side ST, then we can set up a proportion using the given information:
[tex]OP/PQ = ST/RQ[/tex]
We also know that
[tex]OP/PQ = 2/3[/tex]
and
[tex]RQ = 12[/tex]
So we can substitute these values into the proportion and solve for x:
[tex](2/3) = ST/12[/tex]
[tex]ST = (2/3) * 12[/tex]
[tex]ST = 8[/tex]
Rounded to the nearest tenth, the measure of side ST is 8.0.
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Donna reads 56 chapters of a book in 7 hours. What is her rate in chapters per hour?
Answers
Answer:
To find what her rate for how many chapters she reads per hour, you want to find out what times 7 gives you 56, or you can simply divide:
56/7 = 8
Donna reads 8 pages per hour.
Step-by-step explanation:
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