The proof that f(2x)=3f(x)+1/f(x)+3 is shown below.
Mathematical functionTo show that f(2x) = 3f(x) + 1/f(x) + 3, we substitute 2x for x in the expression for f(x):
f(x) = x - 1 / x + 1
f(2x) = 2x - 1 / 2x + 1
Next, we substitute x into the expression for f(x) and add the result to itself twice:
f(x) = x - 1 / x + 1
f(x) + f(x) + f(x) = (x - 1 / x + 1) + (x - 1 / x + 1) + (x - 1 / x + 1)
= 3x - 3 / 3x + 3
We can simplify the expression by dividing both the numerator and denominator by 3:
f(x) + f(x) + f(x) = x - 1 / x + 1
= (1/3) (3x - 3) / (1/3) (3x + 3)
= (x - 1) / (x + 1)
= f(x)
Now, we substitute f(x) and f(2x) back into the original equation:
f(2x) = 2x - 1 / 2x + 1
3f(x) + 1/f(x) + 3 = 3(x - 1 / x + 1) + 1/(x - 1 / x + 1) + 3
= (3x - 3 / x + 1) + (1 / (x - 1 / x + 1)) + 3
= (3x - 3 + (x + 1) / (x - 1)) / (x + 1)
= (2x + 2) / (x + 1)
= 2(x + 1) / (x + 1)
= 2
Thus, we have shown that f(2x) = 3f(x) + 1/f(x) + 3.
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A garden is in the shape of a square with a perimeter of 64 feet. The garden is surrounded by two
fences. One fence is around the perimeter of the garden, whereas the second fence is 2 feet from the
first fence on the outside. If the material used to build the two fences is $1.26 per foot, what was the
total cost of the fences?
The perimeter of a square is the sum of its sides and they are all equal, so to obtain the length of each of them we divide the perimeter of the first fence between 4:
[tex]\text{P1}= \dfrac{\text{64 feet}}{\text{4 sides}}[/tex]
[tex]\text{P1}= 16 \ \text{feet}[/tex]
Then, the length of each side of the second fence will increase 2 feet at each end, as shown in the figure. We have then that the perimeter of the second fence is:
[tex]\text{P2 = 20 feet} \times \text{4 sides}[/tex]
[tex]\text{P2 = 80 feet}[/tex]
The sum of the perimeters of both fences is:
[tex]\text{PT = P1 + P2}[/tex]
[tex]\text{PT = 64 feet + 80 feet}[/tex]
[tex]\text{PT = 144 feet}[/tex]
Total cost = $1.26 x 144 feet
Total cost = $181.44
The total cost of the fences was $181.44
Write the equation of the line that passes through (7,- 4) and (-1, -2) in slope-intercept form.
O y = x-1
O y = -x-1
O y=-x-2
O y=-4x-6
What is the solution of this system of equations?
y = -6x + 5
4x - y = 5
The solution of the system of equations given y = -6x + 5 and 4x - y = 5, is (1, -1).
To solve this system of equations, we can use the substitution method. We can rearrange the first equation to solve for y in terms of x:
y = -6x + 5
Next, we can substitute this expression for y in the second equation:
4x - (-6x + 5) = 5
Simplifying this equation, we get:
4x + 6x - 5 = 5
Combining like terms, we get:
10x = 10
Dividing both sides by 10, we get:
x = 1
Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Using the first equation, we get:
y = -6(1) + 5 = -1
Therefore, the solution of the system of equations is (1, -1). This means that the two equations intersect at the point (1, -1) on the coordinate plane, and this is the only point that satisfies both equations simultaneously.
In summary, to solve the system of equations y = -6x + 5 and 4x - y = 5, we used the substitution method to find that the solution is (1, -1).
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Use the table, which shows the number of sofas sold at a furniture store over 5 months. The number of sofas sold in June is 597.
Furniture Store Sofa Sales
January 255
February 234
March 263
April 229
May 248
How does the median change with the new piece of data?
We know that with the new piece of data for the month of June the median is increased by 3.5 which is now 251.5.
What is the median?The median is the value that divides a data sample, a population, or a probability distribution's upper and lower halves in statistics and probability theory.
It could be referred to as "the middle" value for a data set.
A data set's median value is the point where 50% of the data points have values that are lower or equal to it, and 50% of the data points have values that are higher or equal to it.
Arrange in ascending order:
229 234 248 255 263
When, odd terms are there then the middle term is the median:
Median: 248
New Median:
229 234 248 255 263 597
When there are even terms we will add the middle two terms and divide it by 2 as follows:
248+255/2
Median: 251.5
Median change: 251.5 - 248 = 3.5
Therefore, we know that with the new piece of data for the month of June the median is increased by 3.5 which is now 251.5.
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Side of a triangle is utility pole. Henry wouls like to know how long the guy wire, c, should be. The telephone pole is perpendicular to the ground. Henry knows that a = 20 ft and b = 15 ft. The length of the guy wire should be
The length of the guy wire should be option D √(25) ft. that is 25 ft.
What is perpendicular mean?Perpendicular means that two lines or objects intersect each other at a 90-degree angle, forming a right angle. This means that if you were to draw a line at the point of intersection, it would split the angles on either side of it into two equal parts. In simpler terms, if you imagine a horizontal line and a vertical line intersecting each other, they would be perpendicular to each other. The concept of perpendicularity is important in geometry and is used in many different applications, such as construction, engineering, and design.
We can use the Pythagorean theorem to find the length of the guy wire c. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Therefore:
c² = a² + b²
Substituting the given values:
c² = (20 ft)² + (15 ft)²
c² = 400 ft² + 225 ft²
c² = 625 ft²
Taking the square root of both sides:
c = √(625) ft
c = 25 ft
Therefore, the length of the guy wire should be 25 feet.
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At how many square feet will both companies be at the same amount
If the yard is 2000 feet², both companies will charge the same price.
What is amount?In mathematics, the word "amount" is a broad one that refers to the size or quantity of something, typically given as a numerical value.
It can be used in a number of circumstances where there is a concern with money, measurements, or the quantity of an item.
We must assume both organisations' expenses to be equal and then solve for the yard size to determine the point at which their costs are equal.
Let's assume that x is the yard size at which expenses are equal.
7.5x + 24.5 = 16x + 23.5
When we simplify this equation, we obtain:
0.5x = 1
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Figure A is a scale image of figure b as shown the scale that maps figure a onto figure b is 1:1 3/4
The value of x is 21.75.
We need to convert from mixed number to decimal number. To do this you can follow the steps shown below:
We must divide the numerator of the fraction by the denominator of the fraction.
Now we must add the quotient obtained to the Whole number part.
So, we get:
7 1/4 = 7.25
Then, the scale image that maps "Figure a" onto "Figure b" is: 1 : 7.25.
Finally, in order to find the value of "x", you need to multiply the given length of the corresponding side of "Figure a" by 7.25,
Therefore, you get that the result is:
x = 3(7.25)
x = 21.75
Hence the value of x is 21.75.
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The complete question is ;
Figure a is a scale image of figure b the scale that maps figure a onto figure b is 1:7 1/4 enter the value of x
Both figures are triangles
Figure a is smaller than figure b
Figure a to the right has a 3
Figure b to the right has a x
Is this quadrilateral a parallelogram?
I believe this quadrilateral is a parallelogram.
What is a parallelogram?There are a few ways to choose within the occasion that a quadrilateral may be a parallelogram: Opposite sides are parallel: Within the occasion that inverse sides of a quadrilateral are parallel, at that point it may be a parallelogram.
Inverse sides are said to be consistent: On the off chance that inverse sides of a quadrilateral are consistent, at that point it may be a parallelogram. From the image attached , you can see the both side red are marked equal hence it is a parallelogram
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In one lottery, a player wins the jackpot by matching all five distinct numbers drawn in any order from the white balls (1 through 41) and matching the number on the gold ball (1 through 35). If one ticket is purchased, what is the probability of winning the jackpot?
The probability of winning the jackpot with one ticket is approximately 0.000000003724.
What is the probability?There are a total of C(41, 5) ways to choose five numbers from 41 white balls, and there are 35 possible choices for the gold ball.
Therefore, the total number of ways to win the jackpot is:
C(41, 5) * 35
And the total number of possible outcomes (i.e. all the combinations of 5 white balls and 1 gold ball) is:
C(41, 5) * 35 * C(5, 5)
Since there is only one winning combination, the probability of winning the jackpot is:
1 / (C(41, 5) * 35)
Plugging in the values, we get:
1 / (749,398,832 * 35) = 1 / 26,869,866,120
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A project has estimated annual net cash flows of $55,800 it is estimated to cost $262,260 determine the cash pay bag. Round the answer to one decimal place
The cash payback period for the project is 4.7 years.
What is cash payback period?The cash payback period is a financial metric that calculates the amount of time it takes for a project to generate enough cash flows to recover the initial investment (cost). It is calculated by dividing the initial investment by the estimated annual net cash flows.
What is decimal?Decimal refers to a system of numbers based on the number 10, where each digit represents a different power of 10. It is a way of expressing numbers using a decimal point to separate the whole number from the fractional part.
Given:
Estimated annual net cash flows = $55,800
Initial investment (cost) = $262,260
Cash Payback Period = Initial Investment / Estimated Annual Net Cash Flows
Putting the values:
Cash Payback Period = $262,260 / $55,800
Calculating the cash payback period:
Cash Payback Period = 4.7 years (rounded to one decimal place)
Therefore, the cash payback period for the project is 4.7 years.
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Write an equation in slope intercept form for the line perpendicular to y= -2x +9 and containing the point (4,-3)
Step-by-step explanation:
lets call the line (d)
amd the given eq of the line is (d')
so (d) is perpendicular to (d')
slope(d)×slope(d')=-1
-2(1/2)=-1
so (d): y=1/2x + b
point A (4, -3) belongs to (d) so,
Ya=1/2Xa+b
-3=1/2(4)+b
b=-5
(d):y=1/2x-5
A fair spinner is number from one to four the result of 28 spins are shown below, for which number does the theoretical probability match the experimental probability
The number 2 is the one for which the theoretical probability matches the experimental probability.
How to determine which number does the theoretical probability match the experimental probabilityTo determine which number has a theoretical probability that matches the experimental probability, we need to calculate both the theoretical and experimental probabilities for each of the four numbers.
The theoretical probability of spinning any one of the four numbers is 1/4, or 0.25. This is because there are four equally likely outcomes, and each outcome has a probability of 1/4.
To calculate the experimental probability, we need to count the number of times each number was spun and divide by the total number of spins. Here are the results:
- Number 1 was spun 9 times out of 28 spins, so its experimental probability is 9/28, or approximately 0.321.
- Number 2 was spun 7 times out of 28 spins, so its experimental probability is 7/28, or approximately 0.250.
- Number 3 was spun 6 times out of 28 spins, so its experimental probability is 6/28, or approximately 0.214.
- Number 4 was spun 6 times out of 28 spins, so its experimental probability is 6/28, or approximately 0.214.
To find the number that has a theoretical probability that matches the experimental probability, we can compare the theoretical probability to each of the experimental probabilities. The number that has an experimental probability that is closest to the theoretical probability of 0.25 is number 2, which has an experimental probability of approximately 0.250.
Therefore, the number 2 is the one for which the theoretical probability matches the experimental probability.
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solve for x=? what is the answer
x = 5.5
we can solve this by using tanø formula i.e p/b
Manny's grocery store has a rectangular logo for their business that measures 1.9 meters long with an area that is exactly the maximum area allowed by the building owner.
Create an equation that could be used to determine L, the unknown side length of the logo.
4. Tanya Jackson bought a house and during the first year, was able to rent it for only 7 months at $620 a month. Though the house was vacant for 5 months, Tanya had to pay average expenses of $175 a month for the whole year. She also paid $3,600 in mortgage interest. a. For the first year, what was Tanya's gross income? b. What was her net income or net loss?
A- Tanya's gross income for the first year was $7,940, and b- her net income was $5,840 after subtracting her total expenses.
a. To calculate Tanya's gross income for the first year, we need to multiply the rent per month by the number of months the house was rented:
Gross income from rent = $620/month x 7 months = $4,340
Tanya also paid $175/month in average expenses for the whole year, so her total expenses were:
Total expenses = $175/month x 12 months = $2,100
Adding the mortgage interest paid, Tanya's gross income for the first year was:
Gross income = $4,340 + $3,600 = $7,940
b. To calculate Tanya's net income or net loss, we need to subtract her total expenses from her gross income:
Net income = Gross income - Total expenses
Net income = $7,940 - $2,100
Net income = $5,840
Therefore, Tanya's net income for the first year was $5,840.
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Find the mean of the data set:
87,79,92,71,88,70,91,82,85
Round your answer to the nearest tenth.
Provide your answer below:
The mean of the data set is 82.77
How to calculate the mean of the data set?If we want to find the mean, we would have to put it at the back of our minds that the mean is the same as the average. As such, we would have to take the sum of all the values that have been listed in the data set and divide by the total number of the data.
The first step is to arrange the data set orderly from smallest to largest;
70, 71,79, 82,88,85,87,91,92
We have nine values in the data set thus the mean is;
70+71+79+82+88+85+87+91+92/9= 745/9= 82.77
Hence the mean of the data set is 82.77
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Which of the points below correctly plots the point (2,7π/4)?
Points are plotted on a polar graph.
Answer: E is correct
Step-by-step explanation:
Remember that the coordinates (2,7π4) tell us the radius r=2 and the angle θ=7π4. So the point should be on the circle labeled 2 and form an angle of 7π4 with the positive x-axis. Point E is the correct point.
Cual es el nucleo fundamental de la actividad matemática?
The fundamental core of mathematical activity is problem-solving.
What is mathematical activity about ?Mathematical problems can be found in many different fields, and the process of solving these problems involves analyzing the problem, identifying relevant information, formulating a strategy or plan, executing the plan, and checking the solution to ensure it is correct and complete.
This process of problem-solving is central to all areas of mathematics, and is a key skill that can be applied to many different contexts both inside and outside of mathematics.
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Help !!!!!!!!!!!!!!!!!!!!!!!!!!!!
With this strategy, the ball will travel approximately 28.98 meters horizontally before it hits the ground.
How to solveTo solve this problem, we will need to find the time of flight and the horizontal velocity. We can use the following kinematic equations:
Vertical motion:
Maximum height: h = (v^2 * sin^2(angle)) / (2 * g)
Time of flight: t = 2 * v * sin(angle) / g
Horizontal motion:
Range: R = v * t * cos(angle)
Given:
Initial velocity (v) = 20 m/s
Angle = 45 degrees
Acceleration due to gravity (g) = 9.81 m/s²
Calculations:
Convert angle to radians: 45 degrees = 45 * (π/180) = 0.785 radians
Calculate time of flight: t = 2 * 20 * sin(0.785) / 9.81 = 2.04 seconds
Calculate horizontal range: R = 20 * 2.04 * cos(0.785) = 28.98 meters
So, with this strategy, the ball will travel approximately 28.98 meters horizontally before it hits the ground.
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A ball is thrown with an initial velocity of 20 meters per second at an angle of 45 degrees with respect to the horizontal. How far does the ball travel horizontally (range) before it hits the ground? Assume no air resistance.
Can someone please help me I don’t understand
The one to one functions in the options includes
Graph 1Graph 4Graph 6What is one to one functions?A one-to-one function, also known as an injective function, is a type of function in mathematics where every element in the domain maps to a unique element in the range.
So we can say that, each input value corresponds to exactly one output value, and no two distinct input values have the same output value.
Graphically, one-to-one functions pass the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once.
Applying the horizontal line test shows that the one-to-one function are: Graph 1, 4 and 6
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When rounding to the nearest hundredth, which of the
numbers below will round to 18.96? Select all that apply.
A) 18.974
B) 18.963
C) 18.906
D)18.958
E) 1.896
Answer:
D
Step-by-step explanation:
When a number is 5 or more you round up, but when it's 4 or lower you don't round at all. So, when you round the 8, in answer D, you get 6 making the answer 18.96.
(cscx+sexc)/(sinx + cosx) = cotx + tanx
It is true. The trigonometric identity [tex]\frac{cscx+secx}{sinx + cosx}[/tex] = cotx + tanx
How do we verify or prove that the trigonometric identity (cscx+secx)/(sinx + cosx) = cotx + tanx?To verify that (cscx+secx)/(sinx + cosx) = cotx + tanx
csc = [tex]\frac{1}{sin}[/tex] and Sec = [tex]\frac{1}{cos}[/tex]
If we insert the values, it would be
[tex]\frac{ 1/sinx + 1/cosx}{sinx + cosx}[/tex]
= [tex]\frac{cosx + sinx}{sinx * cosx}[/tex] ÷ ([tex]\frac{1}{sinx + cosx}[/tex])
= [tex]\frac{1}{sinx * cosx}[/tex]
[tex]\frac{Sin^2 x + Cos^2x}{Sin x Cos x}[/tex] = [tex]\frac{Sin^2x}{SinxCosx}[/tex] + [tex]\frac{Cos^2x}{SinxCosx}[/tex]
= cotx + Tanx
Mind you
cotx = [tex]\frac{1}{tanx}[/tex] = [tex]\frac{cosx}{sinx}[/tex]
tanx = [tex]\frac{sinx}{cosx}[/tex]
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A sphere has a radius of 4 in choose true or false for each statement
The true statements are (a) A sphere with half the radius has 1/8 of the volume and (c) A sphere with 3 times the radius has 27 times the volume
Finding the true statements from the radius and volumeGiven that the radius of the sphere is represented as
Radius, r = 4 inches
The formula of the volume of the sphere is represented as
V = 4/3πr³
For the transformed volume, we have
V₂ = 4/3πR³
Where
R = kr and k is the scale factor
So, we have
V₂ = 4/3π(kr)³
V₂ = 4/3πr³k³
This gives
V₂ = Vk³
Testing the statements, we have
(a) A sphere with half the radius has 1/8 of the volume
V₂ = V(1/2)³
Evaluate
V₂ = 1/8V
So, the statement is true
(b) A sphere with twice the radius has 6 times the volume
V₂ = V(2)³
Evaluate
V₂ = 8V
So, the statement is false
(c) A sphere with 3 times the radius has 27 times the volume
V₂ = V(3)³
Evaluate
V₂ = 27V
So, the statement is true
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Complete question
A sphere has a radius of 4 inches.
Choose True or False for each statement.
(a) A sphere with half the radius has 1/8 of the volume
(b) A sphere with twice the radius has 6 times the volume
(c) A sphere with 3 times the radius has 27 times the volume
Please help me write a summary of the 3 rules on segments
1) When 2 chords intersect inside a circle, and 4 segments are formed
2) When 2 secants intersect outside a circle, and 4 segments are formed
3) When 1 secant and 1 tangent intersect outside a circle, and 3 segments are formed
1. The rule states that the product of the lengths of the two segments of one chord equals the product of the lengths of the two segments of the other chord (i.e., (a1 x a2) = (b1 x b2), where a1 and a2 are segments of chord A, and b1 and b2 are segments of chord B).
2. The rule states that the product of the length of the external segment of one secant and the length of the entire secant equals the product of the length of the external segment of the other secant and the length of the entire secant (i.e., (e1 (e1 + i1)) = (e2 (e2 + i2)), where e1 and e2 are the external segments and i1 and i2 are the internal segments).
3. The rule states that the square of the length of the tangent segment equals the product of the length of the external segment of the secant and the length of the entire secant (i.e., t^2 = e * (e + i), where t is the length of the tangent segment, e is the external segment, and i is the internal segment).
what are the 3 rules on segments all about?The three rules of segments are:
A segment is a part of a line that consists of two endpoints and all the points between them.Two segments are congruent if they have the same length.A segment bisector is a line, segment, or ray that divides a segment into two equal parts, creating two congruent segments.Find more exercises on rules of segments;
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Given a circle with center C and m∠SCR = 42°, determine each of the following.
mSR⌢ =_______ degrees
mSCQ⌢ =______ degrees
mSQR⌢ =______ degrees
mQV⌢ =-_____ degrees
m∡QSV =______ degrees
m∡QVR = ___ degrees
(45 pointswill give brainiest for effort)
The measures of the angles are
MSR = 42
MSCQ = 138 degree
mSQR = 318 degrees
mQV = 42 dgerres
How to solve for the anglesMSCQ + MSCR = 180
MSR = MSCR = 42
MSCQ + 42 = 180
MSCQ = 138 degree
MSQR = 360 - 42
= 318 degrees
MQV = MSR vertically opposite
= 42 degree
MQSR = 1/2 x 42
= 21 degrees
MQV = 1/2 x 180 = 90 degrees
The measures of the angles have been solved for
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7
Assignment Active
Describing Exponential Functions
Which statements describe the function f(x) = 3()*? Check all that apply.
Each successive output is the previous output divided by 3.
As the domain values increase, the range values decrease.
The graph of the function is linear, decreasing from left to right.
Each successive output is the previous output multiplied by 3.
The range of the function is all real numbers greater than 0.
The domain of the function is all real numbers greater than 0.
The correct statements regarding the exponential function 3(1/3)^x is given as follows:
Each successive output is the previous output divided by 3.As the domain values increase, the range values decrease.The range of the function is all real numbers greater than 0.How to define an exponential function?An exponential function has the definition presented as follows:
y = ab^x.
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.The function for this problem is given as follows:
f(x) = 3(1/3)^x.
The parameter b is of 1/3 < 1, hence:
Each successive output is the previous output divided by 3.The function is decreasing.The range is all real values greater than 0, as a = 3 is positive and the function has an horizontal asymptote at y = 0.
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Asynchronous activity.
The given information describes a pair of congruent triangles. The corresponding vertices, angles, and sides of the triangles are congruent. The congruent angles are ∠REM ≅ ∠MOC, ∠ERM ≅ ∠MCO, and ∠CMO ≅ EMR. The congruent sides are ER ≅ CO, MO ≅ EM, and CM ≅ RM. The congruent triangles are ΔMER ≅ ΔMOC.
What is the analysis of the two triangles given above?1) the corresponding vertices are
R ~ C and
E ~ 0
2) The corresponding angle are
∠E ~∠O and
∠ R ~ ∠C
3)
The corresponding sides are;
ER ~ CO;
MO ~ EM
CM ~ RM
4) the congruent angles are:
∠REM ≅ ∠MOC
∠ERM ≅ ∠MCO
∠CMO ≅ EMR
5) the congruent sides are:
ER ≅ CO;
MO ≅ EM
CM ≅ RM
6) the congruent triangles are;
ΔMER ≅ ΔMOC
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NEED HELP ASAP!!!!!!! Let f(p) be the average number of days a house stays on the market before being sold for price p in $1,000s. Which statement best describes the meaning of f(275)?
A. Houses sell on the market for an average of $275,000 and stay on the market an average of 275 days before being sold
B. Houses sell for an average of $275,000.
C. f(275) indicates houses stay on the market an average of 275 days before being sold.
D. f(275) represents the average number of days houses stay on the market before being sold for $275,000.
f(275) represents the average number of days houses stay on the market before being sold for $275,000.
Which statement best describes the meaning of f(275)?From the question, we have the following parameters that can be used in our computation:
f(p) is the average number of days a house stays on the market before being sold for price p in $1,000s
This means that
In f(275), we have
p = 275
So, the meaning is:
f(275) is the average number of days a house stays on the market before being sold for price $275,000s
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Help me solve and show my work please
The above prompts are related to the concepts of Degrees, Radians, Sines and Cosines. See the solutions below.
What are the solutions to the above prompts?
1)
(a) Cos 150 degrees
To convert 150 degrees to radians, we multiply by π/180:
150 degrees = (150π/180) radians = (5π/6) radians
We can use the unit circle and reference angle of π/6 to find the cosine of 5π/6:
cos(5π/6) = -cos(π/6) = -(√3/2) = -√3/2
Therefore, cos 150 degrees = -√3/2.
(b) Sin 240 degrees
To convert 240 degrees to radians, we multiply by π/180:
240 degrees = (240π/180) radians = (4π/3) radians
We can use the unit circle and reference angle of π/3 to find the sine of 4π/3:
sin(4π/3) = -sin(π/3) = -(√3/2) = -√3/2
Therefore, sin 240 degrees = -√3/2.
2)
To use the law of cosines, we need either a side and the two angles opposite that side, or two sides and the angle between them. In this case, we have two angles and the side opposite one of them, so we can use the law of sines to find the other side and then use the law of cosines to find the remaining side or angle.
Using the law of sines, we have:
sin A / AB = sin B / AC
where AC is the side opposite angle B. Substituting the given values, we get:
sin 52 / 26.7 = sin 70 / AC
Solving for AC, we get:
AC = sin 70 / sin 52 * 26.7 = 30.4
Now we can use the law of cosines to find the remaining side, x:
x^2 = AB^2 + AC^2 - 2ABACcos A
Substituting the given values, we get:
x^2 = 26.7^2 + 30.4^2 - 226.730.4*cos 52
Solving for x, we get:
x ≈ 22.1
Therefore, the missing side is approximately 22.1.
To find the area of the triangle, we can use the formula:
Area = (1/2) * a * b * sin C
where a and b are the lengths of the two sides, and C is the included angle. Substituting the given values, we get:
Area = (1/2) * 7 * 9 * sin 30
Using the fact that sin 30 = 1/2, we can simplify this to:
Area = (1/2) * 7 * 9 * (1/2) = 15.75
Therefore, the area of the triangle is 15.75 square units.
From the law of sines, we have:
sin A / a = sin B / b
Substituting the given values, we get:
sin 45° / (7√2) = sin B / 7
Solving for sin B, we get:
sin B = (7/7√2) * sin 45° = √2/2
Therefore, B = 45° or 135°. We can use the fact that the angles of a triangle sum to 180° to find C:
C = 180° - A - B = 90° or 0°
If C = 0°, then the triangle is degenerate and the sides a and b lie on top of each other. Therefore, we must have C = 90°.
Now we can use the Pythagorean theorem to find the remaining side:
c^2 = a^2 + b^2 - 2ab*cos C
Substituting the given values, we get:
c^2 = (7√2)^2 + 7^2 - 2(7√2)(7)*cos 90°
Solving for c, we get:
c = √(98 + 49) = √147 = 7√3
Therefore, the sides of the triangle are:
a = 7√2
b = 7
c = 7√3
Using the law of cosines to find x:
In triangle ABC with side lengths AB = 18, AC = 15, and angle A = 108 degrees, we want to find the length of side BC (denoted by x). Using the law of cosines:
x^2 = 18^2 + 15^2 - 2(18)(15)cos(108 degrees)
We can evaluate the cosine of 108 degrees using the identity cos(180 - 108) = -cos(108):
cos(108 degrees) = -cos(180 - 108 degrees) = -cos(72 degrees)
Substituting this into the equation above, we get:
x^2 = 18^2 + 15^2 - 2(18)(15)(-cos(72 degrees))
Simplifying:
x^2 = 729
Taking the square root of both sides:
x = ±27
Since x represents a length, we take the positive root:
x = 27
Therefore, the length of BC is 27 units.
Using the law of cosines to find theta:
In triangle ABC with side lengths AB = 20, BC = 12, and AC = 10, we want to find the measure of angle B (denoted by theta). Using the law of cosines:
cos(B) = (12^2 + 10^2 - 20^2) / (2(12)(10)) = -1/2
Since -1/2 is the cosine of an angle in the second quadrant, we have:
B = 180 degrees - arccos(-1/2)
Using a calculator, we find:
B ≈ 150.5 degrees
Therefore, the measure of angle B is approximately 150.5 degrees (or theta = 150.5 degrees).
Learn more about radians at:
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Please help I’m really bad at this
Answer:
the first one
Step-by-step explanation:
you move N to the left of the equation,
and then K to the right of the equation.
voila.