Answers
According to the information, f(x) = 5(x-3)(x+5) and g(x) = (x-3)(x+5).
How to find f(x) and g(x)?To have vertical asymptotes at x = 3 and x = -5, the denominator of the function must have factors of (x-3) and (x+5), respectively. We can write the denominator as g(x) = (x-3)(x+5).
To have a horizontal asymptote of y = 5, the degree of the numerator must be the same as the degree of the denominator. Also, the leading coefficients of the numerator and denominator must be equal. Therefore, we can write the numerator as f(x) = 5(x-3)(x+5).
The function can be written as:
f(x) 5(x-3)(x+5)
y(x) = -------------- = -----------------
g(x) (x-3)(x+5)
This function satisfies all the given conditions: it has vertical asymptotes at x = 3 and x = -5, a horizontal asymptote of y = 5, and is continuous for all x ≠ 3, -5. Therefore, f(x) = 5(x-3)(x+5) and g(x) = (x-3)(x+5).
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Related Questions
Write a fraction that has a value greater than 3/8 using 3 as the numerator
Answers
Answer: 3/4
Step-by-step explanation:
Any number as the denominator of the fraction that is less than 8 (the denominator) will give you a fraction greater than 3/8.
Another different answer could be 3/5 or 3/6.
A fraction that has a value greater than 3/8 using 3 as the numerator is 3/6, 3/4, 3/2 and so on.
What is the fraction?In Mathematics, fractions are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.
The given fraction is 3/8.
Here, 3/8 = 0.375
We need write the fraction which is greater than 3/8 using 3 as the numerator is 3/6 = 0.5
Therefore, a fraction that has a value greater than 3/8 using 3 as the numerator is 3/6, 3/4, 3/2 and so on.
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Alberto gets a fish tank
Answers
Jackson is on point. Of all fish, catfish make up one-third of the total.
How much gravel does Alberto add to the tank in pounds?The United States of America, Great Britain, and other English-speaking countries predominately utilise the pound as a unit of weight.
Any quantity associated with something is assumed to contain one pound if it does. The pound is the most widely used weight unit in the English-speaking world, including Britain, America, and other nations.
Zebra danios make up half of all the fish.
More fancy guppies than catfish, by a factor of two.
x = guppies
2x = catfish
x + 2x = 1/2 of the fish population
3x = 1/2 of the fish population
3x * 2 = 6x
x = 1/6 number of fancy guppies
catfish:
2x = 2(1/6) = 2/6 = 1/3 catfish.
Jackson is on point. Of all fish, catfish make up one-third of the total.
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The complete question is,
Alberto puts 3 kinds of fish in his fish tank: zebra danios, fancy guppies, and cat fish. Half of Alberto's fish are zebra danios. He has 2 times as many catfish as he has fancy guppies. Alberto's friend Jackson says 1/3 of Alberto's fish are catfish. Is Jackson correct?
Given the following sets, find the set (AUB)' n C.
U = {1, 2, 3,...,8}
A = {1, 2, 3, 5}
B = {1, 2, 6}
C= {1, 3, 4, 5, 6}
Answers
Answer:
To find the set (AUB)' n C, we first need to find the union of sets A and B, which is the set of all elements that belong to either A or B or both:
AUB = {1, 2, 3, 5, 6}
The complement of AUB, denoted by (AUB)', is the set of all elements in U that do not belong to AUB:
(AUB)' = {4, 7, 8}
Next, we need to find the intersection of (AUB)' and C, which is the set of all elements that belong to both (AUB)' and C:
(AUB)' n C = {1}
Therefore, the set (AUB)' n C contains only the element 1.
An economy is operating with output $400 billion above its natural level, and fiscal policymakers want to close this expansionary gap. The central bank agrees to adjust the money supply to hold the interest rate constant, so there is no crowding out. The marginal propensity to consume is 3/4, and the price level is completely fixed in the short run.
to close the expansionary gap, the government would need to increase or decrease spending by $ billion.
Answers
Hence, to close the expansionary gap, the government would need to decrease spending by $100 billion.
To close the expansionary gap, the government would need to decrease spending by $100 billion.
The formula to calculate the change in equilibrium output due to a change in government spending is:
∆Y = (∆G / (1 - MPC))
Where:
∆Y = change in equilibrium output
∆G = change in government spending
MPC = marginal propensity to consume
Here, the output is $400 billion above its natural level, and the central bank agrees to adjust the money supply to hold the interest rate constant, so there is no crowding out. Therefore, we can assume that the change in government spending (∆G) would have a one-to-one effect on the equilibrium output (∆Y).
Given that MPC = 3/4, we can plug in the values into the formula:
400 = (∆G / (1 - 3/4))
∆G = (1 - 3/4) * 400
∆G = (1/4) * 400
∆G = 100
Hence, to close the expansionary gap, the government would need to decrease spending by $100 billion.
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what is the percent of decrease from 8 to 6
Answers
The calculated value of the percent of decrease from 8 to 6 is 25%.
What is the percent of decrease from 8 to 6To find the percent of decrease from 8 to 6, we need to find the difference between the two numbers and express it as a percentage of the original number (8).
The difference between 8 and 6 is 8 - 6 = 2.
To express this difference as a percentage of 8, we can use the formula:
percent decrease = (difference / original number) x 100%
Substituting the values we get:
percent decrease = (2 / 8) x 100% = 0.25 x 100% = 25%
Therefore, the percent of decrease from 8 to 6 is 25%.
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A coordinate plane with 2 lines drawn. The first line is labeled f(x) and passes through the points (0, negative 2) and (1, 1). The second line is labeled g(x) and passes through the points (negative 4, 0) and (0, 2). The lines intersect at about (2.5, 3.2)
How does the slope of g(x) compare to the slope of f(x)?
The slope of g(x) is the opposite of the slope of f(x).
The slope of g(x) is less than the slope of f(x).
The slope of g(x) is greater than the slope of f(x).
The slope of g(x) is equal to the slope of f(x)
Answers
Therefore, the correct answer is: The slope of g(x) is less than the slope of f(x).
Where do the X and Y axes intersect on the coordinate plane, at position 0 0?The origin is the location where the two axes meet. On both the x- and y-axes, the origin is at 0. The coordinate plane is divided into four portions by the intersection of the x- and y-axes. The term "quadrant" refers to these four divisions.
We can use the slope formula to get the slopes of the lines f(x) and g(x):
slope of f(x) = (change in y)/(change in x) = (1 - (-2))/(1 - 0) = 3/1 = 3
slope of g(x) = (change in y)/(change in x) = (2 - 0)/(0 - (-4)) = 2/4 = 1/2
The slope of g(x) is 1/2, which is less than the slope of f(x), which is 3.
Therefore, the correct answer is: The slope of g(x) is less than the slope of f(x).
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Maria 400 mL in every hour how many hours will it take her to drink 2 L of water
Answers
Answer:
5 hours
Step-by-step explanation:
Every liter is 1000 ml, and 2000 ml divided by 400 ml would be 5 hours.
A local department store can sell 100 gaming systems at a price of $180 each. Research
shows that for each $18 price increase, the store will sell 4 fewer systems, and vice versa.
a. Find a linear price equation for the gaming system. That is, express the price, p, as a
function of the quantity, q. (2 points)
b. Use your answer from (a) to find the revenue equation.
c. Use your knowledge of quadratic functions to find the price and quantity sold that will maximize revenue.
Answers
Therefore, the price that maximizes revenue is $135, and the quantity sold at that price is q0/2.
What is the unit price?
In order to find the unit price of a certain item, we just need to divide the total price paid (or total cost) by the number of items bought.
According to given information
a. Let x be the number of $18 price increases from the original price of $180, and let y be the corresponding decrease in the number of gaming systems sold. Then we have the data point (x, y) = (0, 0) and the slope of the line is -4/18 = -2/9. Using the point-slope form of a linear equation, we have:
y - 0 = (-2/9)x
y = (-2/9)x
Substituting x = 18n, where n is the number of $18 price increases, we have:
y = (-2/9)(18n) = -4n
Let q be the quantity of gaming systems sold at a price of $180. Then the equation for the price of the gaming system as a function of the quantity sold is:
p(q) = 180 + 18n = 180 + 18(q - q0)/4 = 180 + (9/2)(q - q0),
where q0 is the quantity of gaming systems sold at the original price of $180.
b. The revenue equation is given by R(q) = q*p(q). Substituting the expression for p(q) found in part (a), we have:
R(q) = q*(180 + (9/2)(q - q0)) = (9/2)q^2 + (180 - 9q0/2)q
c. To find the price and quantity sold that will maximize revenue, we can differentiate the revenue equation with respect to q, set the derivative equal to zero, and solve for q:
R'(q) = 9q - 9q0/2 = 0
q = q0/2
Substituting q = q0/2 into the revenue equation, we have:
R(q0/2) = (9/2)(q0/2)^2 + (180 - 9q0/2)(q0/2)
R(q0/2) = (9/8)*q0^2 + (45/2)*q0
To find the corresponding price, we can use the expression for p(q) found in part (a):
p(q0/2) = 180 + (9/2)(q0/2 - q0)
p(q0/2) = 135
Therefore, the price that maximizes revenue is $135, and the quantity sold at that price is q0/2.
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So I tried solving this problem with the population growth formula,
· Population Growth: =^; a=initial amount, r=growth rate as a decimal; t=time in years; y=resulting population
My equation looked like this but I got this question wrong so any help will be appreciated
9667=11211e^(.418)(t)
Answers
The number of years it would take is approximately equal to 53 years.
How to determine the population after a number of year?In Mathematics, a population that increases at a specific period of time represent an exponential growth. This ultimately implies that, a mathematical model for any population that increases by r percent per unit of time is an exponential function of this form:
P(t) = I(1 + r)^t
Where:
P(t ) represent the population.t represent the time or number of years.I represent the initial number of persons.r represent the exponential growth rate.By substituting given parameters, we have the following:
96627 = 11211(1 + 0.0418)^t
8.61894567835 = (1.0418)^t
By taking the ln of both sides, we have:
Time, t = ln(8.61894567835)/ln(1.0418)
Time, t = 52.60 ≈ 53 years.
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Which is the closest to the area of the shaded region in the given square containing a circle? (Use ≈ 3.14.)
10 m
21.5 square meters
50 square meters
O 78.5 square meters
O 100 square meters
5m
Answers
The area of the shaded region in the specified figure of the square containing a circle is 21.5 m², which is the first option
21.5 square metersWhat is a square?A square is a quadrilateral with all sides of the same length and four interior angles which are right angles.
The figure is a composite figure comprising of a square and a circle
The radius of the circle, r = 5 meters
The side length of the square = 10 meters
The value of π = 3.14
The area of the square = 10 m × 10 m = 100 m²
The area of the circle = π × (5 m)² = 25·π m²
The area of the shaded region is the difference between the area of the square and the area of the circle, therefore;
The area of the shaded region = (100 - 25·π) m²
The area of the shaded region is therefore; (100 - 25 × 3.14) m² = 21.5 m²
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rachel turned 62 in 2022.
estimate her full social security retirement benefit if she were 67 and if her average monthly salary over her 35 highest paying years was 3,100
if she chooses, to start collecting her benefit now at age 62, estimate what will her reduced benefit would be
Answers
She reduced benefit would be approximately: $2,293.52 x (1 - 0.30) = $1,605.46 per month.
How can I calculate average?Average This is an arithmetic mean calculated by adding a set of numbers and then dividing by the number of those numbers. For example, the average of the numbers 2, 3, 3, 5, 7 and 10 is 30 divided by 6, which is 5.
If Rachel retires at age 67, her total Social Security pension can be estimated using the following steps:
Determine Rachel's average indexed monthly income (AIME) by taking the sum of her 35 highest earnings (in today's dollars) and dividing it by 420 (the number of months over 35 years). GOAL = $3100 x 12 x 35 / 420 = $3100
Use the formula to determine the Primary Sum Assured (PIA), which is the monthly benefit amount payable under AIME at full retirement age. PIA = 90% x $996 32% x ($3100 - $996) = $2293.52
So, if Rachel were to retire at age 67, her Social Security pension would be approximately $2,293.52 per month. If Rachel decides to start collecting her benefits now at age 62, her benefit amount will be reduced by a certain percentage based on the longer period she receives benefits. The reduction factor is based on how many months before full retirement age he begins receiving benefits.
In 2022, a person turning 62 will have a full retirement age of 67, which means the discount factor for starting benefits at age 62 is 30%. So, if Rachel started collecting Social Security now at age 62, her reduced benefit would be approximately:
$2,293.52 x (1 - 0.30) = $1,605.46 per month.
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Calculate the value for each determinant:
Answers
Answer:
step by step explanation:
1. 9 4. -12
2. -48 5. 0
3. 17 6. 1
At a train station in the morning rush hour, the Westbound train arrives every 9 minutes, and the Eastbound train arrives every 15 minutes. If both trains were just in the station, after how many minutes will both trains again be in the station at the same time?
Answers
Step-by-step explanation:
Find the LCM of 9 and 15
West 9 18 27 36 45
East 15 30 45 In 45 minutes
Use the slip and y-intercept to identify the equation of this line
Answers
Answer:
y=2x
Step-by-step explanation:
equation formula is y=mx+b
m is slope
b is y-intercept
you know that slope is 2
y intercept is 0
so plug it into the equation
y=2x+0 or just y=2x
4. Tamara claims that Sweet Grove apple juice tastes better than Bushels apple juice. Isaac claims that there is no difference between the 2 types of apple juice. Tamara and Isaac would like to find the answer to the following question: Do more 6th graders prefer Sweet Grove apple juice or Bushels apple juice? a. Is this a statistical question? Explain your reasoning. 5. Explain how this question can be answered with an experiment
Answers
Answer: a. Yes, this is a statistical question since it includes collecting information from a test of 6th graders and utilizing factual strategies to analyze the comes about and draw a conclusion approximately the inclinations for Sweet Grove and Bushels apple juice.
b. Draw a conclusion based on the results of the experiment, and report any limitations or assumptions made during the experiment.
Step-by-step explanation:
The width of the set of the shelves needs to be 5 times the height of the set of shelves and the set of shelves must have three shelves If there are 80 feet of wood to use to build the set of shelves what should the dimensions the set of shelves be?
Answers
The dimensions of the set of shelves should be approximately 4.21 feet in height and 21.05 feet in width.
What is a linear equation?When plotted on a coordinate plane, an algebraic equation known as a linear equation looks like a straight line. The variable (or variables) in this equation always have a greatest power of 1, making it a constant. In other words, a linear equation simply makes use of the fundamental operations of addition, subtraction, and multiplication of variables to the first power, as well as constants.
Given:
Width of the set of shelves = 5 times the height of the set of shelves
Number of shelves = 3
Total amount of wood available = 80 feet
Let's denote the height of the set of shelves as "h" and the width of the set of shelves as "w".
Using the first given information, we can write the equation:
Width (w) = 5 times the height (h) ==> w = 5h
Next, we know that the set of shelves has 3 shelves, so the number of shelves is fixed at 3.
Finally, we are given that we have 80 feet of wood available to build the shelves.
Using these equations, we can substitute w = 5h into the equation for the total amount of wood:
3w + 4h = 80
Substituting w = 5h, we get:
3(5h) + 4h = 80
15h + 4h = 80
19h = 80
h = 80/19
So, the height of the set of shelves is approximately 4.21 feet.
Now, substituting this value of h back into w = 5h, we get:
w = 5(4.21)
w = 21.05
So, the width of the set of shelves is approximately 21.05 feet.
Therefore, the dimensions of the set of shelves should be approximately 4.21 feet in height and 21.05 feet in width.
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simplify the expression ^5√-32x^5y^30
Answers
The fifth root of -32x^5y^30 can be simplified as follows:
First, we can write -32 as (-2)^5.
Next, we can rewrite the expression as follows:
^5√-32x^5y^30 = ^5√((-2)^5 * x^5 * y^30)
Now we can split the root:
^5√((-2)^5 * x^5 * y^30) = (^5√(-2)^5) * (^5√x^5) * (^5√y^30)
Simplifying each part separately:
^5√(-2)^5 = -2
^5√x^5 = x
^5√y^30 = y^6
So the simplified expression is:
-2xy^6
What is the area of the parallelogram shown below?
17
8
units2
9
17
4
Answers
======================================================
Explanation:
The triangle on the left has a = 8 as one of the legs, and c = 17 as the hypotenuse. Use the pythagorean theorem to find the missing side.
[tex]a^2+b^2 = c^2\\\\8^2+b^2 = 17^2\\\\64 + b^2 = 289\\\\b^2 = 289-64\\\\b^2 = 225\\\\b = \sqrt{225}\\\\b = 15\\\\[/tex]
This is the height of the parallelogram as marked by the dashed line.
The base is 9 units across the bottom. We won't use 8 as the base because it's too short along the top.
area = base*height
area = 9*15
area = 135 square units
Can someone please help me with this problem? I'm really struggling and the assignment is already late :(
Answers
The simplified exponential expression is given as follows:
[tex]21^6[/tex]
How to simplify the exponential expression?The exponential expression in the context of this problem is defined as follows:
[tex]\frac{21^3 \times 21^5}{21^2}[/tex]
First we look at the numerator, when we have two terms with the same base and different exponents multiplying, we keep the base and add the exponents, hence:
21^3 x 21^5 = 21^8
Now we have the following division operation:
21^8/21^2.
When we have two terms with the same base and different exponents dividing themselves, we keep the base and subtract the exponents, hence:
21^(8 - 2) = 21^6.
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How do I solve my problem?
Answers
Answer:
mAngleABC = 76°
Step-by-step explanation:
AD is a diameter. So arcAD is half of the circle, that's 180°.
arcCD = 28°
arcAC + arcCD = 180°
arcAC + 28° = 180°
subtract 28°
arcAC = 152°
AngleABC is an inscribed angle (vertex is on the circle) so AngleABC is half of arcAC.
AngleABC
= 152 ÷ 2
= 76
AngleABC = 76°
What is the range of the relation whose graph is shown?
Answers
The range of the relation whose graph is shown include the following: C. -1 ≤ y ≤ 1.
What is a domain?In Mathematics and Geometry, a domain is the set of all real numbers for which a particular function is defined.
Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = {-4, 4} or -4 ≤ x ≤ 4.
Range = {-1, 1} or -1 ≤ y ≤ 1.
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a right triangle has a hypotenuse of length 7 inches. If one angle is 38 degrees, find the length of each leg.
Answers
Check the picture below.
[tex]\sin(38^o )=\cfrac{\stackrel{opposite}{x}}{\underset{hypotenuse}{7}}\implies 7\sin(38^o)=x\implies 4.31\approx x \\\\[-0.35em] ~\dotfill\\\\ \cos(38^o )=\cfrac{\stackrel{adjacent}{y}}{\underset{hypotenuse}{7}}\implies 7\cos(38^o)=y\implies 5.52\approx y[/tex]
Make sure your calculator is in Degree mode.
Answer:
4.31 in5.52 inStep-by-step explanation:
You want the leg measures in a right triangle with a hypotenuse of 7 inches and an angle of 38°.
Trig functionsThe mnemonic SOH CAH TOA reminds you of the relations between side lengths and trig functions:
Sin = Opposite/Hypotenuse ⇒ Opposite = Hypotenuse×Sin
Cos = Adjacent/Hypotenuse ⇒ Adjacent = Hypotenuse×Cos
ApplicationThe given triangle will have opposite and adjacent sides of ...
opposite = (7 in)sin(38°) ≈ 4.31 in
adjacent = (7 in)cos(38°) = 5.52 in
The leg lengths of the triangle are 4.31 inches and 5.52 inches.
12. Higher Order Thinking If a clock face
reads 1:00, how many hours must pass for
the hands to form a straight angle?
own
Answers
Thus, Number of hours must pass for the hands to form a straight angle is 5 hours.
Explain about the hand of clock:To read the time on an analogue clock, look in the direction its hands are pointing. The current hour is indicated by the smallest hand, the current minute by the largest hand, and the current second by the thinnest hand.
There are 24 hours in a day, and the hour hand completes two full rotations of the clock because the numbers on the dial only go up to 12. once, from noon to midnight, and once, from midnight to noon.
Given data:
Current time on clock = 1:00
Hour hand at 1 .
Minute hand at 12.
Number of hours must pass for the hands to form a straight angle:
When
minute hand is at 12 and hour hand will be at 6.
Number of hours = 6: 00 - 1: 00
Number of hours = 5 hours
Thus, Number of hours must pass for the hands to form a straight angle is 5 hours.
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if you dilate triangle ABC by a scale factor of 3 and (0,0) is the center, what will be the length of AB?
Answers
the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center would be 3 times the original length of AB.
How to solve the question?
To find the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center, we can use the following formula:
AB' = AB x 3
where AB' is the length of AB after the dilation, and AB is the original length of AB.
However, we need to first determine the length of AB in the original right triangle ABC. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's assume that AB is the hypotenuse of the right triangle ABC, and that AC and BC are the other two sides. Then we have:
AB²= AC²+ BC²
Without more information about the lengths of AC and BC, we cannot determine the value of AB. However, once we have determined the length of AB, we can use the formula above to find the new length of AB after dilation.
Assuming we know the length of AB in the original right triangle ABC, we can now use the formula for dilation to find the new length of AB:
AB' = AB x 3
For example, if AB is 5 units long in the original triangle, then after dilation, AB' would be:
AB' = 5 x 3 = 15 units
Therefore, the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center would be 3 times the original length of AB.
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The owner of a sports complex wants to carpet a hallway connecting to buildings. The carpet costs $1.50 per square foot. How much does it cost to carpet the hallway?
Answers
Answer:
$526.75 i believe
Step-by-step explanation:
please help me with parts 3 and 4
i will mark as brainliest
Answers
The Cartesian equations of line (D) are:
(1) through point A (2, -3) and of direction vector v(-1, 2), 2x + y + 1 = 0
(2) through the points A(3,2) and B(5,- 1), 3x - 2y + 13 = 0
(3) through point A(-5, 2) and parallel to line (L); 3x - 2y + 11 = 0
(4) through C(4, 1) and parallel to line (d), 2x - y - 7 = 0
What is the Cartesian equation of the points?(1) Passing through point A (2, -3) and of direction vector v(-1, 2):
The Cartesian equation of a line passing through point A(x1, y1) and having a direction vector v = (a, b) can be given by:
(x - x1) / a = (y - y1) / b
Substituting the values from the given information:
(x - 2) / (-1) = (y - (-3)) / 2
Simplifying, we get:
-x + 2 = (y + 3) / 2
Multiplying both sides by -2 to eliminate the fraction:
2x - 4 = -y - 3
Rearranging and simplifying, we get the Cartesian equation of the line:
2x + y + 1 = 0
(2) Passing through the points A(3,2) and B(5,-1):
The Cartesian equation of a line passing through two points A(x1, y1) and B(x2, y2) can be given by:
(x - x1) / (x2 - x1) = (y - y1) / (y2 - y1)
Substituting the values from the given points A(3,2) and B(5,-1):
(x - 3) / (5 - 3) = (y - 2) / (-1 - 2)
Simplifying, we get:
(x - 3) / 2 = (y - 2) / (-3)
Multiplying both sides by 2 and -3 to eliminate the fractions:
-3x + 9 = 2y - 4
Rearranging and simplifying, we get the Cartesian equation of the line:
3x - 2y + 13 = 0
(3) Passing through point A(-5, 2) and parallel to line (L); 2x - 3y + 5 = 0:
Since the line is parallel to line (L), it will have the same direction vector as line (L), which is (2, -3).
Using the same approach as in (1), we can write the Cartesian equation of the line as:
(x - (-5)) / 2 = (y - 2) / (-3)
Simplifying, we get:
(x + 5) / 2 = (y - 2) / (-3)
Multiplying both sides by 2 and -3 to eliminate the fractions:
-3x - 15 = 2y - 4
Rearranging and simplifying, we get the Cartesian equation of the line:
3x - 2y + 11 = 0
(4) Passing through C(4, 1) and parallel to line (d): y = 2x + 1:
Since the line is parallel to line (d), it will have the same slope as line (d), which is 2.
Using the same approach as in (1), we can write the Cartesian equation of the line as:
(x - 4) / 1 = (y - 1) / 2
Simplifying, we get:
(x - 4) = (y - 1) / 2
Multiplying both sides by 2 to eliminate the fraction:
2x - 8 = y - 1
Rearranging and simplifying, we get the Cartesian equation of the line:
2x - y - 7 = 0
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h shows the cost of hosting a dance, c, as a function of the number of students attending, n.
What is a restriction on the domain?
Answers
The correct option C: the domain can contain only whole number for the function of the number of students attending dance.
Explain about the domain of function;The collection of all possible inputs for a function is its domain. Graphing and algebra are the two techniques used to determine a function's domain. Use the equation to consider inputs that would result in an undefined value, including such fractions and square roots, in order to obtain the domain algebraically. Decide where the function is graphed and note the x-values of the region(s) to locate it by graphing.
For the given question:
x axis - number of students attending dance classes n.
y axis - cost of hosting a dance
Domain - is the input value taken from x axis only.
So, all value lying on x axis ate taken as the domain for the function.
Thus, the domain can contain only whole number for the function of the number of students attending dance classes.
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Need help on this one please.
The following recursive algorithm computes factorials.
def fact(n) :
if n = 0 :
fact(n) = 1
else:
fact(n) = n · fact(n − 1)
Use mathematical induction to prove that fact(n) = n! for all n ∈ N
Answers
By mathematical induction, we've proven that the given recursive algorithm computes factorials correctly, i.e., fact(n) = n! for all n ∈ N.
How to solveTo prove the algorithm using mathematical induction, we'll proceed in two steps: the base case and the inductive step.
Base case: n = 0
By definition, 0! = 1. According to the given algorithm, when n = 0, fact(0) = 1. Therefore, the base case holds, as fact(0) = 0! = 1.
Inductive step:
Assume that the algorithm is correct for some arbitrary non-negative integer k, i.e., fact(k) = k!. We want to show that the algorithm is also correct for k + 1, i.e., fact(k + 1) = (k + 1)!.
According to the algorithm:
fact(k + 1) = (k + 1) * fact(k)
Using the induction hypothesis that fact(k) = k!, we can substitute this into the equation:
fact(k + 1) = (k + 1) * k!
By definition of the factorial function, (k + 1)! = (k + 1) * k!, so:
fact(k + 1) = (k + 1)!
We've shown that if fact(k) = k!, then fact(k + 1) = (k + 1)!. Thus, the inductive step is complete.
By mathematical induction, we've proven that the given recursive algorithm computes factorials correctly, i.e., fact(n) = n! for all n ∈ N.
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Use the quadratic formula to solve for x.
2x²+2x-1=0
Answers
Answer:
The quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In the equation 2x^2 + 2x - 1 = 0, we have:
a = 2
b = 2
c = -1
Substituting these values into the formula, we get:
x = (-2 ± sqrt(2^2 - 4(2)(-1))) / (2)(2)
Simplifying this expression, we get:
x = (-2 ± sqrt(4 + 8)) / 4
x = (-2 ± sqrt(12)) / 4
x = (-2 ± 2sqrt(3)) / 4
x = (-1 ± sqrt(3)) / 2
Therefore, the solutions to the equation 2x^2 + 2x - 1 = 0 are:
x = (-1 + sqrt(3)) / 2
x = (-1 - sqrt(3)) / 2
Explain what went wrong.
Please solve both.
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