Answers
1. The solutions of the equation 6/(x+1)= x/(x-1) are x = 2 and x = 3.
2. The solutions of the equation x+ 4/x= -5 are x = -1 and x = -4.
What are the solutions ?
1. 6/(x+1) = x/(x-1)
To solve this equation, we first need to get rid of the fractions by multiplying both sides of the equation by the least common multiple of the denominators, which is (x+1)(x-1):
6(x-1) = x(x+1)
Expanding the left side, we get:
6x - 6 = x² + x
Moving all the terms to one side, we get:
x² - 5x + 6 = 0
This is a quadratic equation that can be factored as:
(x - 2)(x - 3) = 0
So, the solutions are x = 2 and x = 3. However, we need to check if these solutions are valid, because we cannot divide by zero.
Checking x = 2:
6/(2+1) = 2/(2-1)
2 = 2, which is true
Checking x = 3:
6/(3+1) = 3/(3-1)
3/2 = 3/2, which is true
Therefore, the solutions are x = 2 and x = 3.
2. x + 4/x = -5
To solve this equation, we first need to get rid of the fraction by multiplying both sides of the equation by x:
x² + 4 = -5x
Moving all the terms to one side, we get:
x² + 5x + 4 = 0
This is a quadratic equation that can be factored as:
(x + 1)(x + 4) = 0
So, the solutions are x = -1 and x = -4. However, we need to check if these solutions are valid, because we cannot divide by zero.
Checking x = -1:
-1 + 4/(-1) = -5
-1 - 4 = -5, which is true
Checking x = -4:
-4 + 4/(-4) = -5
-4 - 1 = -5, which is true
Therefore, the solutions are x = -1 and x = -4.
What is common multiple?
A common multiple is a number that is a multiple of two or more given numbers. For example, the common multiples of 2 and 3 are 6, 12, 18, 24, etc. These are all numbers that can be divided by both 2 and 3 without a remainder. A common multiple is "common" to the given numbers because it is a multiple of all of them.
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Complete question is:
1. The solutions of the equation 6/(x+1)= x/(x-1) are x = 2 and x = 3.
2. The solutions of the equation x+ 4/x= -5 are x = -1 and x = -4.
Related Questions
Write the slope-intercept form of the equation of each line given the slope and y-
intercept.
Slope = -7, y - intercept = - 3
Answers
Answer:
y = -7x - 3
Step-by-step explanation:
The slope-intercept form of the equation of a line is:
y = mx + b where m is the slope and b is the y-intercept.
Given the slope m = -7 and the y-intercept b = -3, we can write the equation of the line in slope-intercept form as:
y = -7x - 3
Therefore, the equation of the line with slope -7 and y-intercept -3 in slope-intercept form is y = -7x - 3.
A survey reports that the probability a person has blue eyes is 0.10. Assume that 4 people are randomly selected at Miramar College and asked if they have blue eyes. Use the Binomial distribution.
a) Find the probability that exactly 3 of them have blue eyes. Round to 3 decimal places.
b) Find the probability that exactly 2 of them have blue eyes. Round to 3 decimal places.
c) Find the probability that at least 1 of them has blue eyes. Round to 3 decimal places.
0.344
Correct
d) Find the probability that more than 2 have blue eyes. Round to 3 decimal places.
e) Find the expected number of students out of the 4 with blue eyes. Round to 1 decimal place.
0.291
Incorrect
f) Find the variance. Round to 2 decimal places.
g) Find the standard deviation. Round to 2 decimal places.
Answers
The probability that more than 2 students have blue eyes is 0.149, the expected number of students out of the 4 with blue eyes is 0.4, the variance is 0.36, and the standard deviation is 0.6.
a) P(X = 3) = (4 choose 3) * [tex](0.10)^3 * (0.90)^1[/tex] = 0.0036
So the probability that exactly 3 of them have blue eyes is 0.0036.
b) P(X = 2) = (4 choose 2) * [tex](0.10)^2 * (0.90)^2[/tex] = 0.0386
So the probability that exactly 2 of them have blue eyes is 0.0386.
c) P(at least 1) = 1 - P(none) = 1 - [tex](0.90)^4[/tex] = 0.3439
So the probability that at least 1 of them has blue eyes is 0.344.
d) P(X > 2) = P(X = 3) + P(X = 4) = (4 choose 3) * [tex](0.10)^3 * (0.90)^1[/tex] + (4 choose 4) * [tex](0.10)^4 * (0.90)^0[/tex] = 0.0001
So the probability that more than 2 have blue eyes is 0.0001.
e) E(X) = n * p = 4 * 0.10 = 0.4
So the expected number of students out of the 4 with blue eyes is 0.4, not 0.291 as provided.
f)Var(X) = n * p * (1 - p) = 4 * 0.10 * 0.90 = 0.36
So the variance is 0.36.
g) SD(X) = sqrt(Var(X)) = sqrt(0.36) = 0.6
Standard deviation is 0.6.
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9t+6
in standard form. asap!
Answers
The standard form of the expression 9t + 6 is:
9t + 1.How to find the standard form of this expression?To write the expression 9t + 6 in standard form, we need to rearrange the terms so that the t-term comes before the constant term:
9t + 6 = 9t + 6(1)= 9t + 6(1/6)= 9t + (6/6)= 9t + 1Therefore, the standard form of an algebraic expression is a way of writing the expression by arranging the terms in a specific order. In the case of linear expressions like 9t + 6, the standard form is to write the t-term before the constant term.
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decompose the mixed number in two different ways 1 4/10
Answers
The mixed number 1 4/10 can be decomposed as 14/10 or 1 2/5.
We may use the fact that a mixed number combines a whole number and a correct fraction to deconstruct the mixed number 1 4/10 in two different ways. So, the mixed number 1 4/10 can be represented as follows:
1 4/10 = 1 + 4/10
The sum of two fractions with a common denominator can be used to decompose this mixed number. This can be accomplished by changing the whole number 1 to a fraction with the same denominator as the fraction 4/10:
1 4/10 = 1 + 4/10 = 10/10 + 4/10 = 14/10
The sum of a whole number and a fraction that is less than one can be used to further break down this mixed number. In order to achieve this, we can take one out of the mixed number and express the resulting fraction in the simplest words possible:
1 4/10 = 1 + 4/10 = 1 + (4/10)/(10/10) = 1 + 2/5 = 1 2/5
The mixed number 1 4/10 can be broken down into 14/10 or 1 2/5 as a result.
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Less strict regulation results in many more companies obtaining the right to mine coal in Mpumalanga. At the same time, alternative energy solutions result in a decrease in the demand for coal-generated electricity. Explain how the market for coal will be affected by these changes. Clearly indicate how the equilibrium price and equilibrium quantity of coal will be affected by these changes. Make use of a combination of diagrams and verbal explanations to explain your answer.
Answers
A patient reports taking three teaspoons of a liquid medication every four hours at home. How should the total daily amount of this medication be documented using the metric system?
Answers
To document the total daily amount of liquid medication using the metric system, we need to convert teaspoons to milliliters, which is the standard unit of volume in the metric system.
One teaspoon is equal to approximately 5 milliliters, so three teaspoons would be equal to 15 milliliters.
If the patient is taking 15 milliliters every four hours, then they would be taking the medication 6 times a day (24 hours divided by 4 hours per dose).
Therefore, the total daily amount of the medication would be:
15 milliliters per dose x 6 doses per day = 90 milliliters per day
So, the total daily amount of the liquid medication should be documented as 90 milliliters per day in the patient's medical records using the metric system.
8) A kite descends 57 feet in 3 seconds. Which integer represents the change, in fee
position of the kite after one second?
54
-29
-19
O-3
Answers
In the proportion , in 1 second the position of kite is C)19 feet.
What is proportion?
A percentage is created when two ratios are equal to one another. We write proportions to construct equivalent ratios and to resolve unclear values. a comparison of two integers and their proportions. According to the law of proportion, two sets of given numbers are said to be directly proportional to one another if they grow or shrink in the same ratio.
Here in 3 seconds a kite's position = 57 feet.
Then , in 1 second a kite's position = x feet.
Now using proportion then,
=> 3 seconds = 57
1 second = x
=> x = 57/3 = 19 feet.
Hence In 1 second the position of kite is C)19 feet.
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) 100 जना मानिसमा गरिएको सर्वेक्षणमा 65 जनाले दुध र 55 जनाले दही मन पराउँछन् । यदि 15 जनाले दुवै मन पराउँदैनन् भने भेनचित्र प्रयोग गरी दुध र दही दुवै मन पराउने मानिसको सङ्ख्या पत्ता लगाउनुहोस् । In a survey of 100 people, 65 like milk and 55 like curd. If 15 like none of these, by using Venn-diagram, find how many people like both milk and curd. 12- मूल्य अभिवृद्धि कर सहित बिक्री गरियो । यदि
Answers
Answer:
X=35
Step-by-step explanation:
if the translated part is the full question; the answer is attached ⬆️.
Hope it helps
Test the equation for symmetry. Need 3 tests for symmetry or else it’s incorrect.
Answers
the equation [tex]r = 3 - 3cos(\theta)[/tex] is symmetric with respect to the vertical line but not with respect to the origin or the pole .
What is the trigonometric functions ?Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.
The equation r = 3 - 3cos(θ) represents a polar curve, where r is the distance from the origin to a point on the curve, and θ is the angle between the positive x-axis and the line connecting the origin to the point.
1. Symmetry with respect to the origin: A curve is symmetric with respect to the origin if replacing (r, θ) with (-r, θ + π) gives the same curve. Substituting -r for r and θ + π for θ in the equation, we get:
[tex]-r = 3 - 3cos(\theta + \pi)[/tex]
[tex]-r = -3 - 3cos(\theta)[/tex]
[tex]r = 3 + 3cos(\theta)[/tex]
This is not the same as the original equation, so the curve is not symmetric with respect to the origin.
2.Symmetry with respect to the pole (x-axis): A curve is symmetric with respect to the pole if replacing (r, θ) with (r, -θ) gives the same curve. Substituting -θ for θ in the equation, we get:
[tex]r = 3 - 3cos(-\theta)[/tex]
[tex]r = 3 + 3cos(\theta)[/tex]
This is not the same as the original equation, so the curve is not symmetric with respect to the pole.
3.Symmetry with respect to the vertical line (y-axis): A curve is symmetric with respect to the vertical line if replacing (r, θ) with (-r, π - θ) gives the same curve. Substituting -r for r and π - θ for θ in the equation, we get:
[tex]-r = 3 - 3cos(π - \theta)[/tex]
[tex]-r = -3 + 3cos(\theta)[/tex]
[tex]r = 3 - 3cos(\theta)[/tex]
This is the same as the original equation, so the curve is symmetric with respect to the vertical line (y-axis).
Therefore, the equation [tex]r = 3 - 3cos(\theta)[/tex] is symmetric with respect to the vertical line (y-axis) but not with respect to the origin or the pole (x-axis).
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The graph of r = 3 - 3cos(theta) is symmetric about the x-axis but not about the origin or about the y-axis.
What is the symmetry of the equation?Symmetry is described in geometry as a balanced and proportionate likeness seen in two sides of an object. It signifies that one side is a mirror image of the other. The line of symmetry is the imaginary line or axis along which you can fold a figure to obtain the symmetrical halves.
The symmetry equation is r = 3 -3 cosФ depicts a cardioid-shaped polar graph. We can use the following tests to determine symmetry:
1) We can test for symmetry with regard to the origin by seeing if the equation remains the same after substituting r with -r and Ф with Ф + pi.
r = 3 - 3cosФ - r = 3 - 3cos(Ф + pi)
3 - 3(-cos(Ф)) = r
-r = 3 + 3cos(Ф)
Because this is not equal to the original equation, the graph is not symmetric about the origin.
2) Symmetry with respect to the x-axis: We may test for symmetry with respect to the x-axis by seeing if the equation stays unchanged when theta is replaced with -theta.
r = 3 - 3cos(Ф)
r = 3 - 3cos(-Ф)
r = 3 - 3cos(Ф)
This is the same as the original equation, hence the graph is symmetric about the x-axis.
3) Symmetry with respect to the y-axis: We may test for symmetry with respect to the y-axis by seeing if the equation stays unchanged when theta is replaced with pi - theta.
r = 3 - 3cos(Ф)
r = 3 - 3cos(pi - Ф)
r = 3 + 3cos(Ф)
Because this is not equal to the original equation, the graph is not symmetric about the y-axis.
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What’s the answer to this math problem that’s in the picture?
Answers
The value of the account at the end of 50 years is $432,125.19.
1. For the first 20 years:
The monthly contribution is $110, and the interest rate is 5.5% compounded monthly.
Using the formula:
FV = P [[tex](1 + r)^n[/tex] - 1] / r
where:
FV = Future value
P = Monthly payment
r = Monthly interest rate
n = Number of months
Plugging in the values:
FV1 = $110 x [tex][(1 + 0.055/12)^{(20)(12)[/tex] - 1] / (0.055/12)
= $67,100.54
2. For the next 25 years:
The monthly contribution increases to $225, and the interest rate remains the same at 5.5% compounded monthly.
Plugging in the values:
FV2 = $225 [[tex][(1 + 0.055/12)^{(25)(12)[/tex] - 1] / (0.055/12)
= $307,527.10
3. For the last 5 years:
The monthly contribution further increases to $350, and the interest rate is still 5.5% compounded monthly. Again, we will use the same formula to calculate the future value of the contributions made during this period.
Plugging in the values:
FV3 = $350 [[tex][(1 + 0.055/12)^{(5)(12)[/tex] - 1] / (0.055/12)
= $57,497.55
Finally, the total value of the account at the end of 50 years
Total Value = FV1 + FV2 + FV3
= $67,100.54 + $307,527.10 + $57,497.55
= $432,125.19
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Write te equation of a circle with (-17;-9) and (-19;-9) being the ends of the diameter
Answers
Answer:
[tex](x + 18)^{2} + (y + 9)^{2} = 1[/tex]
Step-by-step explanation:
Please help me with this
Answers
The area of the square is equal to 64 square units.
How to determine the area of the square related to a circle
In this problem we find the representation of a geometric system formed by a circle and a square and we must compute the area of the latter figure. First, find the side length of the square by Pythagorean theorem:
10² = (2 · x)² + (1.5 · x)²
100 = 4 · x² + 2.25 · x²
100 = 6.25 · x²
x² = 16
x = 4
Second, compute the area of the square:
A = (2 · 4)²
A = 64
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please loo at the follwng picture
Answers
Answer: I am pretty sure 7
Step-by-step explanation: I think this because 20 is the most and 13 is the leased so 20 -13 =7 but i just got done like a month or 2 ago so i just learned this so dont quote me sorry if it isn't right
Will give thanks
trig
Answers
The distance between the two cruise ships after 2 hours is approximately 4.201 miles.
How to find distance?
To solve this problem, we need to use trigonometry to find the distance between the two cruise ships after 2 hours.
In the diagram, A and B are the two cruise ships, and D is the distance between them after 2 hours. The angle between their paths is 35 degrees.
We can use the formula: distance = speed x time
to find how far each cruise ship travels in 2 hours. For Cruise A, the distance it travels is
[tex]distance_A = 18 \times 2 = 36 \: miles[/tex]
For Cruise B, the distance it travels is
[tex]distance_B = 15 \times 2 = 30 \: miles
[/tex]
Now we can use trigonometry to find the distance between the two cruise ships. We can use the tangent function, since we know the angle and the opposite and adjacent sides of the triangle formed by the two cruise ships and the distance between them:
tan(35°) = D / (36 - 30)
Simplifying this equation, we getting,
D = (36 - 30) x tan(35°)
D = 6 x 0.7002
D ≈ 4.201 miles
Therefore, the distance between the two cruise ships after 2 hours is approximately 4.201 miles.
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URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Match the description table or graph or equation with the correct type of function
A. Linear function table
B. Non linear function graph
C. Linear function equation
D. Non linear function equation
E. Linear function graph
F. Non linear function table
Answers
The description table or graph or equation with the correct type of function:
Linear function table: A
Non-linear function graph: B
Linear function equation: C
Non-linear function equation: D
Linear function graph: E
Non-linear function table: F
What is function?A function is a mathematical concept that describes a relationship between two sets of values, where each input value (independent variable) corresponds to exactly one output value (dependent variable). In other words, a function is a rule or a formula that takes an input value and produces a unique output value.
Here,
The table of values shows a constant rate of change, which indicates a linear function. We can verify this by calculating the slope between any two points, which should be the same. For example, the slope between (-2, -8) and (2, 0) is (0 - (-8))/(2 - (-2)) = 8/4 = 2. So the function is of the form y = mx + b, where m = 2 and b = -4. This gives us the equation y = 2x - 4, which is a linear function. Therefore, the correct match is A.
The graph shows a curve, which indicates a non-linear function. It does not resemble a straight line, so we can eliminate linear functions. The shape of the graph suggests that it might be a quadratic function, which has a U-shaped curve. Quadratic functions can be represented by an equation of the form y = ax² + bx + c. We can't determine the equation of the function from the graph alone, but we can conclude that it is a non-linear function. Therefore, the correct match is B.
The equation 3x - 4y = 27 can be rearranged to solve for y in terms of x: y = (3/4)x - (27/4). This equation is of the form y = mx + b, where m = 3/4 and b = -27/4, which indicates a linear function. Therefore, the correct match is C.
The equation 3x² - 2x = 4y is a quadratic equation, which has a variable raised to the second power. Quadratic functions are non-linear, so the correct match is D.
The graph shows a straight line, which indicates a linear function. We can determine the equation of the line by finding the slope and y-intercept. The slope is (5 - (-1))/(2 - (-2)) = 6/4 = 3/2, and the y-intercept is -1. So the equation of the line is y = (3/2)x - 1, which is a linear function. Therefore, the correct match is E.
The table of values does not show a constant rate of change, which indicates a non-linear function. For example, between x = -2 and x = -1, y changes by 5, but between x = 0 and x = 1, y changes by only 1. This suggests that the function is not linear. We can't determine the exact form of the function from the table alone, but we can conclude that it is a non-linear function. Therefore, the correct match is F.
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What similarities is being shown in the figure given?
1) a a similarity
2)sss similarity
3)sas simulator
4)none
Answers
In the given figure the similarity is angle-angle property.
What are angles?An angle is the result of the intersection of two lines.
An "angle" is the length of the "opening" between these two beams.
Angles are commonly measured in degrees and radians, a measurement of circularity or rotation.
In geometry, an angle can be created by joining the extremities of two rays. These rays are intended to represent the angle's sides or limbs.
The two primary components of an angle are the limbs and the vertex.
The joint vertex is the common terminal of the two beams.
Hence, In the given figure the similarity is angle-angle property.
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large rectangle below is divided into two
smaller regions; shaded and unshaded. The area of
the whole rectangle is: 5x² + 4x - 8. The area of
the shaded region is: 2x² + 7x. What is the area of
the unshaded region?
Answers
The area of the unshaded region is 3x²-3x-8
What is area of rectangle?
A rectangle is a closed 2-D shape, having 4 sides, 4 corners, and 4 right angles (90°). The opposite sides of a rectangle are equal and parallel.
The area of a rectangle is expressed as;
A = l× w
Area of the unshaded part = Area of the whole rectangle - area of shaded part
Therefore area of the unshaded part =
5x²+4x-8 -(2x²+7x)
= 5x²+4x-8 -2x²-7x
collect like terms
5x²-2x²+4x-7x-8
= 3x²-3x-8
therefore the area of the unshaded part is 3x²-3x-8
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A couple of two way radios were purchased from different stores. Two way radio A can reach 8 miles in any direction. Two way radio B can reach 11.27 kilometers in any direction.
Part A: How many square miles does two way radio A cover? Use 3.14 for TT and round to the nearest whole number.
B. Part A: How many square miles does two way radio B cover? Use 3.14 for TT and round to the nearest whole number.
C. If 1 mile= 1.61 kilometers, which two way radio covers the larger area?
D. Using the radius of each circle, determine the scale factor relationship between the radio coverages.
Answers
Answer:
Step-by-step explanation:
A) To find the area covered by two-way radio A, we need to calculate the area of a circle with a radius of 8 miles:
Area = πr² = 3.14 x 8² = 200.96 square miles
Rounding to the nearest whole number, two-way radio A covers 201 square miles.
B) To find the area covered by two-way radio B, we need to convert the radius from kilometers to miles and then calculate the area of a circle with that radius:
Radius in miles = 11.27 / 1.61 = 6.9988 miles (rounded to 4 decimal places)
Area = πr² = 3.14 x (6.9988)² = 153.94 square miles
Rounding to the nearest whole number, two-way radio B covers 154 square miles.
C) Two-way radio A covers a larger area than two-way radio B (201 square miles vs 154 square miles).
D) The scale factor relationship between the radio coverages can be found by dividing the radius of radio A by the radius of radio B:
Scale factor = radius of A / radius of B = 8 miles / (11.27 km / 1.61 km/mile) = 4.97
This means that the coverage of two-way radio A is almost 5 times larger than that of two-way radio B.
Graph the following inequality on the sketchpad below. Be sure to shade in the correct
area on the graph.
x<-3
Answers
According to the information to graph the inequality we have to show a number line with an open circle at -3 and an arrow pointing to the left, with the shaded region to the left of -3 labeled as x < -3.
How to graph the inequality?To graph the inequality x < -3, we need to draw a number line and mark the point -3 on it. Since the inequality is "less than," we want to shade the region to the left of -3 on the number line.
We start by drawing a horizontal line with a labeled point for -3. Then we draw an arrow to the left of -3, to indicate that all the values to the left of -3 satisfy the inequality.
Here are the steps to graph the inequality on the sketchpad:
Open the sketchpad or a graphing tool that can plot a number line.Draw a horizontal line and mark the point -3 on it.Draw an arrow to the left of -3 to indicate that all the values to the left of -3 satisfy the inequality.Shade in the region to the left of -3 to indicate the solution set of the inequality.Label the shaded region with the inequality x < -3.The resulting graph should show a number line with an open circle at -3 and an arrow pointing to the left, with the shaded region to the left of -3 labeled as x < -3.
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Four more than the quotient of a number and 6 equals 9 Turn into a equation
Answers
Answer:
We can get creative and use many equations using these numbers.
Step-by-step explanation:
4 more than (addition) the quotiont (division) Of a number (a variable) 6 is equal to 9.
let the variable for number be n.
we can use 4n/6=9
x=13.5
or 13 5/10
Answer:
Step-by-step explanation:
(x ÷ 6) + 4 = 9
(x ÷ 6) +4 - 4= 9 -4
x ÷ 6 = 5
x/6 = 5
x/6 · 6/1 = 5 · 6
x = 30
Which fraction is equivalent to 0.008?
A. 2/250
B. 4/250
C. 2/?25
D. 4/25
Answers
Answer:
A. 2/250
Step-by-step explanation:
So, first use calculator on the multiple-choice question.
Suppose you want to have $300,000 for retirement in 30 years. Your account earns 9% interest
How much would you need to deposit in the account each month?
How much interest will you earn?
Answers
Using percentage, we can find that:
You need to deposit $833.33 every month and the interest earned will be = $27,000.
Define percentage?The Latin term "per centum," which means "by the hundred," is the source of the English word "percentage." Fractions with a denominator of 100 are percentages. In other words, it is a link where the value of "whole" is always taken to be 100.
Total amount you want to have = $300,000
Time is = 30 years
= 30 × 12
= 360 months.
Interest earned is 9%.
Now, the amount you need to deposit every month =
Total amount/Total time
= 300000/360
= $833.33
So, you need to deposit $833.33 every month.
Now interest earned in 30 years is given = 9%
So, amount earned = 9% of 300,000
= 9/100 × 300,000
= $27,000.
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Let m ∈ Real numbers[x] be a polynomial with deg m ≥ 1. Define a relation Sm on R[x] by the rule that ( f , g) ∈ S if and only if m is a factor of g − f .
(a) Prove that Sm is an equivalence relation on Real numbers[x]
(b)The division rule for polynomials implies that every equivalence class of Sm con-
tains one polynomial with a special property. What is this property?
(c) Write down a polynomial m ∈ Real numbers[x] such that the set {f ∈ Real numbers[x] : f(2) = 3} is an equivalence class of Sm. Give a brief justification
Answers
a. Sm satisfies all three properties of an equivalence relation, it is indeed an equivalence relation on Real numbers[x].
b. The equivalence class of Sm containing f is the set of all polynomials g that satisfy the condition m | (g - f).
c. The equivalence class of Sm containing the polynomial f(x) = x + 1 is the set {g ∈ Real numbers[x] : g(2) = 3}.
What is real number?Real numbers are those that can be used to calculate constant values like temperature, time, or distance. They are equivalent to countless decimal increases. They are the sum of all positive and negative integers, fractions, decimals, transcendental numbers, and irrational numbers.
(a) To prove that Sm is an equivalence relation on Real numbers[x], we need to show that it satisfies three properties: reflexivity, symmetry, and transitivity.
Reflexivity: For any polynomial f ∈ Real numbers[x], we have f − f = 0, which is clearly a multiple of any polynomial m. Therefore, (f, f) ∈ Sm for all f ∈ Real numbers[x], and Sm is reflexive.
Symmetry: If (f, g) ∈ Sm, then m is a factor of g − f. This means that there exists a polynomial q ∈ Real numbers[x] such that g − f = mq. It follows that f − g = −mq, which is a multiple of m. Therefore, (g, f) ∈ Sm, and Sm is symmetric.
Transitivity: If (f, g) ∈ Sm and (g, h) ∈ Sm, then m is a factor of g − f and m is a factor of h − g. This means that there exist polynomials q1 and q2 ∈ Real numbers[x] such that g − f = mq1 and h − g = mq2. It follows that h − f = (h − g) + (g − f) = mq2 + mq1 = m(q2 + q1). Therefore, m is a factor of h − f, and (f, h) ∈ Sm. Thus, Sm is transitive.
Since Sm satisfies all three properties of an equivalence relation, it is indeed an equivalence relation on Real numbers[x].
(b) The division rule for polynomials states that for any polynomials f and g with g ≠ 0, there exist unique polynomials q and r such that f = qg + r and deg r < deg g. The equivalence class of Sm containing f is the set of all polynomials g that satisfy the condition m | (g - f).
Applying the division rule for polynomials to the polynomial g - f, we can write it as g - f = mq + r, where deg r < deg m. This means that g - r = f + mq, and therefore, any polynomial in the equivalence class of Sm containing f can be written as g = r + f + mq, where deg r < deg m. In other words, every equivalence class of Sm contains a polynomial of the form r + f + mq, where r is a polynomial of degree less than deg m.
(c) Let m(x) = x - 2. Then for any polynomial f(x) ∈ Real numbers[x], we have (f, g) ∈ Sm if and only if g(x) - f(x) is a multiple of x - 2. This means that g(2) - f(2) = 0, or in other words, f(2) = g(2). Therefore, the equivalence class of Sm containing the polynomial f(x) = x + 1 is the set {g ∈ Real numbers[x] : g(2) = 3}.
Justification: Let g(x) = (x - 2) + 3. Then g(2) = 3, and g(x) - f(x) = (x - 2) + 3 - (x + 1) = x - 2, which is a multiple of x - 2. Therefore, (f, g) ∈ Sm, and the set {g ∈ Real numbers[x] : g(2) = 3} is an equivalence class of Sm.
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Write a mixed number giving the amount shaded. Then write this amount as as improper fraction
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Answer:
improper fraction would be 3 2/3 and mixed number would be 11/3
Step-by-step explanation:
Using your points from Part A, What is the total amount of water, in gallons, in the pool after 47 minutes? Show your work or explain how you determined your answer.
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The total amount of water in the pool after 47 minutes is 1,470 gallons.
Determine the initial volume of water in the pool (if any) from Part A.
Calculate the rate at which water is being added to or removed from the pool (in gallons per minute).
This information should also be provided in Part A.
Multiply the rate of water flow by 47 minutes to find the total amount of water added or removed during that time.
For example, if the rate is 10 gallons per minute, you would calculate 10 * 47 = 470 gallons.
Add the initial volume of water to the total amount of water added or removed during the 47 minutes.
If the initial volume was, for example, 1,000 gallons and you added 470 gallons, the total volume would be 1,000 + 470 = 1,470 gallons.
The final result is the total amount of water in the pool after 47 minutes.
In the example above, the total volume would be 1,470 gallons.
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solve the equation for exact solution in interval [0 degree , 360 degree)
(cot theta-1)(2sin theta +square root 3)=0
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Answer: We can start by using the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.
So, we can set each factor equal to zero and solve for theta:
Factor 1: cot(theta) - 1 = 0
Using the identity cot(theta) = cos(theta) / sin(theta), we can rewrite this as:
cos(theta) / sin(theta) - 1 = 0
cos(theta) - sin(theta) = 0
cos(theta) = sin(theta)
Dividing both sides by cos(theta), we get:
tan(theta) = 1
The solutions to this equation are theta = 45 degrees and theta = 225 degrees (since tangent is positive in the first and third quadrants).
Factor 2: 2sin(theta) + sqrt(3) = 0
Subtracting sqrt(3) from both sides and dividing by 2, we get:
sin(theta) = -sqrt(3)/2
The solutions to this equation are theta = 240 degrees and theta = 300 degrees (since sine is negative in the third and fourth quadrants, and sin(240) = sin(300) = -sqrt(3)/2).
Therefore, the exact solutions in the interval [0 degrees, 360 degrees) are:
theta = 45 degrees, 225 degrees, 240 degrees, 300 degrees.
Step-by-step explanation:
Trigonometry review applications
I need help with 15-18
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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has many real-world applications such as in (engineering, architecture, and navigation.)
15. One application of trigonometry is in determining the height of a building. This can be done by measuring the angle of elevation from a known distance away from the building and using trigonometric functions to find the height.
16. Another application of trigonometry is in surveying land. Surveyors use trigonometry to measure distances and angles between different points on a piece of land, which helps them create accurate maps and boundary lines.
17. Trigonometry is also used in physics to calculate the motion of objects that move in a circular or oscillatory motion, such as a pendulum or a rotating wheel.
18. Finally, trigonometry is used in astronomy to calculate the position and movement of celestial bodies. Astronomers use trigonometric functions to measure the distance between stars and galaxies
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Y= 5x*4 tan X
Find dy/dx
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[tex] \:\:\:\:\:\:\star\longrightarrow \sf \underline{y = 5\:x^4\:tanx}\\[/tex]
Differentiating with respect to x-
[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{d}{dx}\:y = \dfrac{d}{dx}\: 5\:x^4\:tanx\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{dy}{dx}\:= 5\:\dfrac{d}{dx}\: \:x^4\:tanx\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{d}{dx}\:y =5\:\bigg[tanx\times \dfrac{d}{dx}\: \:x^4\:+x^4\times \dfrac{d}{dx} tanx\bigg]\\[/tex]
[tex]\pink{\star}\:\boxed{\sf\sf\dfrac{d}{dx}\bigg[f(x)\:g(x)\bigg] = f(x) \dfrac{d}{dx} g(x) + g(x) \dfrac{d}{dx} f(x)}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{dy}{dx}= 5\:\bigg[tanx\times 4\:x^3+ x^4 \times sec^2x \bigg]\\[/tex]
[tex] \qquad\pink{\star}\:\:\:\:\boxed{\sf \dfrac{d}{dx}x^n=nx^{n-1}}\\[/tex]
[tex] \qquad\pink{\star}\:\:\:\:\boxed{\sf \dfrac{d}{dx}tanx=sec^2x}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{dy}{dx}= 5\:x^3\:\bigg[4tanx+ x sec^2x \bigg]\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf\underline{\dfrac{dy}{dx}= \boxed{\sf5\:x^3\:\bigg[4tanx+ x sec^2x \bigg]}}\\[/tex]
can someone help me pleasee
Find the x- and y- intercepts, if they exist.
f(x)= log2(x+2)-5
Answers
Accοrding the given questiοn the x- and y- intercepts, are as fοllοws,
x-intercept =(28, 0), y-intercept = (0, -4)
What are intercepts?An intercept is the pοint οf intersectiοn οf the line with the cοοrdinate axes. If the line intersects with the X axis, then the y cοοrdinate οf the pοint οf intersectiοn is zerο and the intercept is called the x intercept.
Tο find the x-intercept, we need tο sοlve fοr x when y = 0:
0 = lοg2(x + 2) - 5
5 = lοg2(x + 2)
[tex]2^5 = x + 2[/tex]
x = 30 - 2 = 28
Therefοre, the x-intercept is (28, 0).
Tο find the y-intercept, we need tο evaluate the functiοn when x = 0:
f(0) = lοg2(0 + 2) - 5
f(0) = lοg2(2) - 5
f(0) = 1 - 5
f(0) = -4
Therefοre, the y-intercept is (0, -4)
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Can someone please help with this ?!!! 30 points
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The total surface area of a cube is given by the formula:
SA = 6s^2
where s is the side length of the cube. In this case, s = 3 centimeters.
To find the surface area of each face of the cube, we need to calculate the surface area of a single dot and then multiply by the number of dots on each face. The surface area of a single dot is the surface area of a hemisphere, which is given by:
SA = 2πr^2
where r is the radius of the hemisphere. In this case, r = 0.1 centimeters.
So the surface area of a single dot is:
SA = 2π(0.1)^2 = 0.0628 square centimeters
Each face of the cube has one dot, so the total surface area contributed by the dots on each face is 0.0628 square centimeters.
Now we can calculate the total surface area of the cube:
SA = 6s^2 = 6(3)^2 = 54 square centimeters
Therefore, the surface area of the cube is 54 square centimeters.