Find the equation of a hyperbola satisfying the given conditions. Asymptotes y = 3 X, Y = -3 x: one vertex (3.0) The equation of the hyperbola is. (Type an equation. Type your answer in standard form.
The equation of the hyperbola satisfying the given conditions with asymptotes y = 3x and y = -3x, and one vertex at (3, 0) is given by (x-3)²/9 - (y-0)²/3 = 1.
To find the equation of the hyperbola, we first note that the asymptotes of the hyperbola are given by y = mx and y = -mx, where m is the slope of the asymptotes. Here, the slopes of the asymptotes are 3 and -3, so the equations of the asymptotes are y = 3x and y = -3x.
The general equation for a hyperbola centered at the origin with asymptotes y = mx and y = -mx is given by x²/a² - y²/b² = 1, where a and b are the distances from the center to the vertices along the x-axis and y-axis, respectively.
Since the given vertex is (3, 0), we can determine that the distance from the center to the vertex along the x-axis is a = 3.
Substituting these values into the general equation, we have (x-3)²/9 - y²/b² = 1.
To determine the value of b, we consider that the distance from the center to the asymptote along the y-axis is b/3. Since the asymptotes have a slope of 3 and -3, the distance from the center to the asymptote along the y-axis is 1/3 times the distance from the center to the vertex along the x-axis. Therefore, b = 3a = 9.
Substituting the value of b into the equation, we get (x-3)²/9 - y²/9 = 1.
Simplifying, we obtain (x-3)²/9 - (y-0)²/3 = 1.
Thus, the equation of the hyperbola satisfying the given conditions is (x-3)²/9 - (y-0)²/3 = 1.
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Exercises In Exercises 1-2, use Eq. (3) to find the length of the path over the given interval, and verify your answer using geometry. 1. (3t-1,2-21), 0≤t≤5 2. (1+5t, t-5), -3≤1≤3 In Exercises 3-8, use Eq. (3) to find the length of the path over the given interval. 3. (21², 312-1), 0≤t≤4 4. (31,413/2), 0≤t≤l 5. (31²,4t³), 1≤t≤4 6. (t³+1,t²-3), 0≤t≤1 7. (sin 3t, cos 3t), 0≤t≤π 8. (sin - cos, cos 0 + 0 sin 0), 0≤0 ≤2ib ((1)(0)2) = (12 In Exercises 9 and 10, find the length of the path. The following identity should be helpful: 010 dignatus = 1 - cost n²/1/2 2 = sin² 2:20 9. (2 cost - cos 2t, 2 sin t - sin 2t), 0≤t≤ 1/ 10. (5(0 sin 0), 5(1- cos 0)), 0≤0 ≤ 2π -
In the given exercises, the length of a path over a specific interval is to be found using Equation (3). The equation represents the distance between two points in a Cartesian coordinate system. In Exercises 1-2, the length is calculated using Equation (3) and then verified using geometry. In Exercises 3-8, the length is determined solely using Equation (3). Exercises 9 and 10 involve finding the length of the path using a relevant identity and the given coordinates.
Equation (3) represents the distance between two points (x₁, y₁) and (x₂, y₂) in a Cartesian coordinate system:
d = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
In Exercises 1-2, the coordinates of the points are provided, along with the specified interval. The length of the path is found by substituting the coordinates into Equation (3) and evaluating the expression. To verify the answer geometrically, one can plot the points on a coordinate plane and measure the distance between them using a ruler or by applying geometric principles.
Exercises 3-8 involve using Equation (3) directly to find the length of the path over the given interval. The coordinates of the points are provided, and the interval specifies the range of t values to consider.
Exercises 9 and 10 require using a relevant identity to find the length of the path. The given coordinates are substituted into the identity, and the resulting expression is simplified to determine the length.
In summary, the exercises involve finding the length of paths using Equation (3) in various scenarios. The lengths are calculated by substituting the given coordinates into the equation, and in some cases, geometry or relevant identities are used to verify or simplify the expressions.
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Question 5
Read the question and type your response in the box provided. Your response will be saved automatically.
Two Grade 7 students, Layla and Camille, both solved the following math problem:
If the temperature drops from 8°F to -13°F by how much did the temperature decrease?
The students came up with different answers. Layla said the answer is 21°F, and Camille said the answer is 5°F. Who is correct? Explain y
B
X,
x?
©2020 Illuminate Education, Inc.
Simplified College Cost Worksheet
Calculate approximate monthly amount to invest at the beginning of the period. Round the final answer to two decimals. Please note that intermediate steps should be calculated to at least 4 decimals to get the answer correctly.
Assume the following information for your client
Step 1 Child's AGE 5
2 Years to college 18-(Step 1)
3 Annual College Costs (current dollars) $65,643
4 Assumed College Inflation rate 3.2%
5 College Inflation Factor. Calculate as (1+Step 4)^(Step 2) (1+Step 4)^(Step 2)
6 Estimate Future Annual Costs Calculate as (Step 3* Step 5) or use FV formula in Excel or financial calculator
7 Estimated Total Future Costs (Step 6*) years in college 4 Step 6*years in college
Funding requirements 8 Assumed after-tax return 10.3%
9 PV Factor 1/(1+Step 8)^Step 2
10 Total Lump-Sum Investment Currently Required Step 7*Step 9 or use PV formula in Excel or financial calculator
11 Amount already earmarked for education $27,231
12 Additional Limp-Sum funding required Step 10-Step 11
13 Years of funding 7
14 PVIFA Due Move to Step 15 to use Excel or financial calculator
15 Annual target amount to invest (Step 12*Step 14) or use formula PMT in Excel or financial calculator
16 Approximate monthly amount to invest (Step 15)/12 months
The client wants to fund the estimated cost of his child's education.The total future costs would be $97,621.34 x 4 = $390,485.36.The approximate monthly amount to invest is $42,798.40.
Given the age of the child, 5 years, and the years to college which is 18-5=13 years.The annual college cost of the child is $65,643, and the assumed college inflation rate is 3.2%.
Therefore the college inflation factor can be calculated as follows;(1+3.2%)^(13) = 1.48774963Estimate future annual costs (Step 6*):$65,643 x 1.48774963 = $97,621.34.
The total future costs (Step 7*) would be $97,621.34 x 4 = $390,485.36The client's assumed after-tax return is 10.3%.
Therefore the present value factor (PV Factor) can be calculated as follows;1/(1+10.3%)^13 = 0.28936396.
Total Lump-Sum Investment currently required (Step 10):$390,485.36 x 0.28936396 = $112,902.19The amount already earmarked for education is $27,231.
Therefore the additional Lump-Sum funding required would be; $112,902.19 - $27,231 = $85,671.19The client wants to fund the child's education for seven years.
Therefore the Present Value Interest Factor of Annuity Due (PVIFA Due) can be calculated as follows;PVIFA Due (Step 14) = ((1 - (1 + 10.3%)^(-7))) / (10.3% * (1 + 10.3%)^(-7))= 5.98297324.
Annual target amount to invest (Step 15):$85,671.19 x 5.98297324 = $513,580.78.
Approximate monthly amount to invest $513,580.78 / 12 = $42,798.40Rounding this to two decimal places will give $42,798.40.
The approximate monthly amount to invest is $42,798.40.
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Annie can walk 1/2
of a mile in 1/4
of an hour. How many miles can she walk in 2 1/2hours if she walks at the same constant speed?
5 miles
2 1/2 hrs ÷ 1/4 hr = 10
1/2 mile × 10 = 5
Answer: The answer would be 5 miles
Step-by-step explanation:
There are 4 1/4 in a hour, 4 1/2 which would equal 2.
Then since you got 2 hours you would multiply 2 x 2 and get 4, and you got another half so that would equal 2/4 which would be plus 1 mile. =5
CAN SOMEONE HELP THIS IS DUE TODAY PLSS AND THANK YOU!!!
Answer:
YEA MAYBE STOP YELLING theres my advice like wht r u doing?
Step-by-step explanation:
Where is the f(x) negative
a) (0,-6)
b) (-10,7)
c) (-10,0)
d (-7,10)
F(0)=
a) -2
b) 10
c) 7
d) -6
If F(x)=0, find x
a) -2
b) -4
c) 0
d) 7
What is the domain of F(x)?
a) (-10,8)
b) (-6,2)
c) (-10,-7,-2,-1,4,8)
d) (-7,-6,-2,2)
Answer:
1. a) (0,-6)
2. d) f(0)=-6
3. d
4. a (?)
Step-by-step explanation:
1. f(0)=-6, which is negative
2. f(0)=-6
3. when x=7, f(7)=0
4. domain: [-10,8] because x=-10 and 8 are included in the domain
Hopefully, this helps.
A square chunk of plastic has a length of 5 cm, width of 5 cm and height of 5 cm. It has a mass of 200 g. What is it’s density
Answer: isn’t it 1.6 because you multiply 5 x 5 x 5 = 125 and then you divide 200\125 =1.6
What is the intermediate step in the form
(�+�)2=�(x+a) 2=b as a result of completing the square for the following equation?
�2+32�+116=x 2 +32x+116=12�12x
The intermediate step in the form (a + b)2 = c (x + 4)2 = 8 as a result of completing the Square for the given equation.
The intermediate step in the form (a + b)2 = c (x + a)2 = b as a result of completing the square for the following equation is (x + 4)2 = 8.
The given quadratic equation is x2 + 32x + 116 = 12×12xTo solve this equation using the method of completing the square,
we need to express the quadratic equation in the form of (x + a)2 = b where a and b are real numbers. Hence, let's follow the steps given below:
Step 1: Rearrange the termsx2 + 32x + 116 = 12×12x becomes x2 + 32x + 116 - 144x = 0x2 - 112x + 116 = 0
Step 2: Find the value of a by dividing the coefficient of x by 2.a = (-112)/2 = -56
Step 3: Add and subtract the square of (a) to the left side of the equation.x2 - 112x + 116 = (x - 56)2 - 56(56) + 116x2 - 112x + 116 = (x - 56)2 - 3080
Step 4: Simplify(x - 56)2 - 3080 = 0(x - 56)2 = 3080
Now, comparing with (a + b)2 = c, we can say that the intermediate step in the form (a + b)2 = c (x + 4)2 = 8 as a result of completing the square for the given equation.
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ayo i need help this is vv hard or i’m jus dumb.
Kyle can type 270 words in 4 minutes. How many words per minute can Kyle type?
(ratio form)
Answer:
67.5
Step-by-step explanation:
true or false: in large samples you do not have to worry about omitted variable bias. true false suppose you take a sample of student test scores and develop the following frequency distribution. the histogram has ten bins, with the height in each bin being the proportion of observations falling within the corresponding score interval. (note: in this case, the students who studied did well on the exam and those who didn't scored poorly.) 0 10 20 30 40 50 60 70 80 90 100 0.5 0.4 0.3 0.2 0.1 0 proportion in cell student scores (in percentage form) true or false: the data appear to be normally distributed. true false
Answer:
Step-by-step explanation:
false
The average sea level in 1920 at the London bridge was 33 feet. in 1925 it was 33.08 feet.
Use linear interpolation or extrapolation to find what the average sea level was in 1928.
The average sea level at the London bridge in 1928 was approximately 33.16 feet.
Linear interpolation is the process of determining the value of a variable between two points by using the data in between the two points. To find out the average sea level in 1928, we can use linear interpolation or extrapolation.
Extrapolation is the method of estimating a value outside the range of a known set of values. Here are the steps to find out the average sea level in 1928 using linear interpolation:
Step 1: Write down the given data:
In 1920, the average sea level at the London bridge was 33 feet.In 1925, the average sea level at the London bridge was 33.08 feet.
Step 2: Calculate the rate of change:
Step 3: Write the linear equation:
Using the point-slope form of a line, we can write the equation of the line passing through the points (1920, 33) and (1925, 33.08) as follows:y - 33 = 0.016(x - 1920)Simplify the equation: y = 0.016x - 31.872
Step 4: Find the average sea level in 1928:The average sea level in 1928 can be found by substituting x = 1928 in the equation derived in step 3:
y = 0.016(1928) - 31.872y = 0.16 feet
The average sea level at the London bridge in 1928 was approximately 33.16 feet.
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a sauce recipe calls for 4 pounds of ground beef and you want to halve it how many ounces of ground beef do you need?
If you want to halve the recipe, you will need 32 ounces of ground beef. We can find it by multiplying.
To determine the amount of ground beef needed when halving the recipe, we need to convert pounds to ounces. There are 16 ounces in 1 pound.
Given that the original recipe calls for 4 pounds of ground beef, we can calculate the amount needed when halving the recipe by dividing it by 2.
Amount needed when halving = 4 pounds / 2 = 2 pounds.
To convert 2 pounds to ounces, we multiply it by 16:
2 pounds * 16 ounces/pound = 32 ounces.
Therefore, when you halve the recipe, you will need 32 ounces of ground beef.
In conclusion, to make half of the original recipe that calls for 4 pounds of ground beef, you will need 32 ounces of ground beef.
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3^3 times 3^4 find the exponent for the 3
Answer:
3^3= 27
3^4= 81
Step-by-step explanation:
Hope that help
a football team ran two plays. on the first play, the quarterback was tackled for -8 yards. on the second play, the running back ran for 10 1/2 yards. on which play did the team gain more yards?
Answer:
both are 2cnd play
Step-by-step explanation:
What are the coefficients of the terms?
Consider the expression
x + 5x^(6) - x^(7)
Answer:
A term is a single mathematical expression.
It may be a single number (positive or negative), a single variable ( a letter ), several variables
multiplied but never added or subtracted. Some terms contain variables with a number in front
of them. The number in front of a term is called a coefficient.
Examples of single terms:
3x is a single term. The "3" is a coefficient. The "x" is the variable.
-5x2
y
3
z
4 is also a single term.
The "-5" is the coefficient. " x
2
y
3
z
4
" are the variables comprising the term.
12 is a single term.
It is simply a numerical term called a constant.
w is a single term too.
It is a single variable term and since there is only one "w" it has an implied coefficient of 1 .
We join terms together with addition to create expressions.
5x + 7y is a two termed expression.
The 5x is one term and the 7y is the second term. The two terms are separated by a plus sign.
7x2
y
3
- 2xy
4
+ 7 is a three termed expression.
7x2
y
3
is one term, - 2xy
4
is the second term, and 7 is the third term.
The coefficient of the first term is 7.
The coefficient of the second term is negative 2.
The third term is a constant, so it has no coefficient.
Like Terms
"Like terms" are terms which have variables which look EXACTLY alike.
3x2
and -7x2
are "like terms" because both terms have the variable x
2
5x2
y
6
z
5
, 13x2
y
6
z
5
, and -21x2
y
6
z
5
are "like terms" because all of the terms have the variables "
x
2
y
6
z
5
" in them.
Combining Like Terms
To combine "like terms", combine the coefficients of terms which look exactly alike.
If you start with several terms separated by addition and subtraction signs, your answer should
also contain addition and subtraction signs based on the signs obtained from doing the
arithmetic.
Combine like terms:
3x2
+ 2x + 9 - 2x
2
+ 5x - 3
3x2
+ 2x - 9 - 2x
2
+ 5x - 3
(3 - 2 )x
2
+ (2 + 5 )x + (-9 - 3 )
1x2
+ 7x + (-12)
1x2
+ 7x - 12
You may be required to perform some other algebraic operations before you are ready to
combine like terms.
Simplify. Combine like terms:
-5( 2x3
- 3x2
+ 1) + 2(-2x
3
+ 5x2
- 1)
Distribute the -5 into the first parentheses and multiply.
Do the same with the 2 in front of the second parentheses.
-5(2x
3
) -5(-3x
2
) -5(1) + 2(-2x
3
) + 2(5x2
) + 2(-1)
-10x
3
+ 15x
2
-5 -4x3
+10x2
-2
Now combine like terms,
(-10 - 4 )x
3
+ (15 + 10)x
2
+ (-5 - 2)
-14x3
+25x2
-7
Step-by-step explanation:
A key joint in a precision machining process has a lower specification limit of a width of 1.16 mm and an upper specification limit of 1.24 mm. The standard deviation is 0.03 mm and the mean is 1.2 mm
What is the process capability index for the precision machining process? (Round off your answer to the SECOND decimal place)
the process capability index for the precision machining process is 0.44 (rounded off to the second decimal place). Since the process capability index is less than 1, the process is considered incapable of meeting the specifications.
Given the following data:
Lower Specification Limit (LSL) = 1.16 mm
Upper Specification Limit (USL) = 1.24 mm
Standard deviation = 0.03 mm
Mean = 1.2 mm
Process Capability Index, Cp, is defined as the ratio of the specification range to the process range. It can be calculated using the following formula:
Cp = (USL - LSL) / (6σ)
where σ is the standard deviation and 6 is a constant that relates to the normal distribution of data.
For a process to be considered capable, Cp must be greater than or equal to 1.
Cp = (1.24 - 1.16) / (6 × 0.03)
= 0.08 / 0.18= 0.44 (rounded off to the second decimal place)
Therefore, the process capability index for the precision machining process is 0.44 (rounded off to the second decimal place). Since the process capability index is less than 1, the process is considered incapable of meeting the specifications.
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Find the equation for
each problem.
Answer:
-2, 3
Step-by-step explanation:
slope=-4
8th grade math please help
Answer:
22.5 for both L and M
Step-by-step explanation:
What is the area of this triangle? A=bh2 a-54 cm² b-90 cm² c-108 m² d-216 m² 15 cm 12 cm Cm^2
Answer: To find the area of a triangle, we can use the formula:
Area = (base * height) / 2
Given the measurements:
Base (b) = 15 cm
Height (h) = 12 cm
Substituting these values into the formula:
Area = (15 cm * 12 cm) / 2
Area = 180 cm² / 2
Area = 90 cm²
Therefore, the area of the triangle is 90 cm².
What would the answer be (: ASAP plsss
Answer:
t = -3
Step-by-step explanation:
Complete the following statement.
(-4) (?) = -20
(-4) (5)= -20
I hope it's helpful
A computer company makes a rectangular screen with a The dimensions of the computer screen are modeled by diagonal of 20 inches. The width of the screen is 4 inches less than its length. the equation x (x 4) 20. What is the value of x, the length of the screen? Ox-16 O X12 Ox 12 O x 16 20 in. (x-4) in. Xin. Not drawn to scale
Answer:
x = 16
Step-by-step explanation:
The equation shown can be solved this way.
x^2 +(x -4)^2 = 20^2
x^2 +x^2 -8x +16 = 400
2x^2 -8x -384 = 0
x^2 -4x -192 = 0
(x -16)(x +12) = 0
The positive solution is ...
x = 16
_____
Since this makes use of the Pythagorean theorem, you've probably run across the 3-4-5 triangle. You often find scaled versions of it in algebra and geometry problems. Here, it is scaled by a factor of 4 to give a 12-16-20 triangle as half of the display screen.
The 3-4-5 triple is the only Pythagorean triple that is an arithmetic sequence. So, if the difference in side lengths is 4 and the diagonal is 5×4, you can be pretty certain that x = 4×4 and x-4 = 4×3.
I got CONFUSED PLZ HELP?
Answer:
A, B, F
Step-by-step explanation:
To solve this, you look at the powers of x and take either the square root (if it is to a power of 2) or the cube root (if a power of 3) of both sides to isolate x. For the first equation, you take the square root of 48, so A is correct. For the second one, you take the square root of 63, so B is correct. For the final equation, you take the cube root of 73, so F is correct.
Answer:
[tex]x = \sqrt{48} \\\\x = \sqrt{63} \\\\x = \sqrt[3]{73}[/tex]
Step-by-step explanation:
[tex]x^2 = 48\\x^2 = 63\\x^3 = 73[/tex]
Solve for x
Equation 1 ;
[tex]x^2 =48\\\\Square\:root\:both\:sides\\\\\sqrt{x^2} =\sqrt{48} \\\\Simplified \:form;\\x =4\sqrt{3}[/tex]
Equation 2;
[tex]x^2 =63\\\\Square\:root\:both\:sides\\\\\sqrt{x^2} =\sqrt{63} \\\\Simplified \:form;\\x =3\sqrt{7}[/tex]
Equation 3 ;
[tex]x^3 =73\\\\Cube\:root\:both\:sides\\\\\sqrt[3]{x^3}= \sqrt[3]{73} \\\\x = \sqrt[3]{73}[/tex]
what quadrant is (-2,-3) in?
Answer:
It would be quadrant 3
Step-by-step explanation:
Answer:
The point is located in the second quadrant because x is negative and y is positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.
shop sells bagels for $5 per dozen.
You can buy bagels for $50.
Answer: You can buy at most, 120 bagels.
Plz mark brainliest:)
Answer:
120 bagels:) have a great day!!!!!!!!!!!!!!!
Step-by-step explanation:
Find the total surface area of a cubical dice having length of each side 2 cm.
Answer:
24
Step-by-step explanation:
A cube has six faces.
each face has a side length of 2 cm
Since each face is a square, the are of one face is 2*2 = 4 cm^2
So the total surface area is 6 * 4 cm^2 = 24 cm^2
PLZ SHOW WORK ASAP!!!!
A photograph of an elephant has a scale of 1 cm representing 0.5 m. The elephant's trunk in the photograph measures 4.1 cm.
What is the actual length of the elephant's trunk?
1. 8.2 m
2. 2.04 m
3. 2.05 m
4. 2.08 m
Answer:
2.05
Step-by-step explanation:
You need to multiply the 0.5m with the 4.1 cm !!
0.5m x 4.1 = 2.05
I hope I helped!!
Owen had 18 guitars. He sold 5 guitars and kept the rest. What is the ratio of guitars he kept to guitars he sold?
Answer:
to get what he kept subtract 5 from 18, to get 13 so 13 was kept. the ratio will be kept:sold which is 13:5
