The identity of sin²Ф + cos²Ф = 1 is verified below
How to evaluate Trigonometry Ratio ?Trigonometry ratio can be evaluated by following the laid down rules with the the use of table and calculator
Given that tan Ф = 1/√5
Where Tan Ф = Opposite / adjacent
We can calculate the hypotenuse by using Pythagoras theorem
Hyp² = 1² + (√5)²
Hyp² = 1 + 5
Hyp = √6
sin Ф = opp / hyp
sin Ф = 1/√6
sin²Ф = 1/6
cos Ф = adj / hyp
cosФ = √5 / √6
cos²Ф = 5/6
sin²Ф + cos²Ф = 1/6 + 5/6
sin²Ф + cos²Ф = 6/6
sin²Ф + cos²Ф = 1
Therefore, the identity of sin²Ф + cos²Ф = 1 is verified.
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An airplane is heading north at an airspeed of 640 km/hr, but there is a wind blowing from the southwest at 90 km/hr. How many degrees off course will the plane end up flying, and what is the plane's speed relative to the ground? Round your answers to 2 decimal places.
An airplane is heading north at an airspeed of 640 km/hr, but there is a wind blowing from the southwest at 90 km/hr. degrees off course will be 6.25°
How many degrees off course will the plane end up flying, and what is the plane's speed relative to the ground?Generally, the equation for the velocity of the plane with reference to the ground is mathematically given as
Vp= velocity of the plane with reference to wind+ velocity of the wind with reference to ground
Therefore
Vp=Vp'+Vw
[tex]mVp=\sqrt{(640)^2+(90)^2-2*640*90cos45}[/tex]
mVp=579.8km/h
where
[tex]\frac{sintheta}{90}=\frac{sin45}{Vp'}[/tex]
[tex]sin \theta=\frac{90}{579.8}*sin45[/tex]
sin[tex]\theta=0.109[/tex]
[tex]\theta=sin^{-1}(0.109)=6.25[/tex]
In conclusion, degrees off course will be 6.25
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THOMAS OSLER IS BUYING A TOWNHOUSE SELLING FOR $175,000. HIS BANK IS
OFFERING A 30-YEAR FIXED RATE MORTGAGE AT 5.5% WITH A MINIMUM 20% DOWN
PAYMENT. DETERMINE THE AMOUNT OF THE DOWN PAYMENT AND THE MONTHLY PAYMENT
The amount of the down payment and the fixed rate mortgage is $35000 and $9625.
According to the statement
We have given that the
Thomas osler is buying a house selling for $175,000 And we have to find the amount of the down payment and the monthly payment.
So, The down payment is :
percentage of the down payment is 20%
So,
Amount = 20/100*175000
Amount = 35,000$
And the percentage of the fixed rate mortgage with 5.5%
So,
Amount = 5.5/100*175000
Amount = $9625.
This the amount of the down payment with the given percentage and with the fixed rate mortgage.
So, The amount of the down payment and the fixed rate mortgage is $35000 and $9625.
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Please help me with this question <3
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
For two lines to be parallel, there should be angles that follow some specific properties that is usually observed with parallel lines.
We can clearly see that :
[tex] \qquad❖ \: \sf \: \angle7 \cong \angle16[/tex]
( by Alternate interior angle pair )
[tex] \qquad \large \sf {Conclusion} : [/tex]
Lines l and m are parallel to each other.
Which of the following statements is false?
A. The parallel sides of an isosceles trapezoid are congruent.
B. Opposite sides of a parallelogram are congruent.
C. The diagonals of a rhombus are perpendicular and bisect each other.
D. The base angles of an isosceles trapezoid are congruent.
The statement which is false among the answer choices as indicated in the task content is; Choice A; The parallel sides of an isosceles trapezoid are congruent.
Which of the statements indicated in the answer choice is correct?It follows from the answer choice A that The parallel sides of an isosceles trapezoid are congruent.
However, it follows from the study of isosceles trapezoids that the base angles of such trapezoids are congruent and their measures are equal, consequently, the isosceles sides have equal length measures.
On this note, the other two sides are parallel but cannot be concluded as congruent.
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Enter the letter for the function graphed
below.
1
a) y = √x - 5
b) y = √x-5
c)y = 5-√√x
d) y = √x + 5
e) y = √x + 5
The letter for the graphed function is y = √(x) - 5
How to determine the letter for the graphed function?The parent function is given as:
y = √x
From the graph, we can see that the function is shifted downwards by 5 unts
This is represented as:
y = √(x) - h
Where h = 5
Substitute the known values in the above equation
y = √(x) - 5
This means that the letter for the graphed function is y = √(x) - 5
Hence, the letter for the graphed function is y = √(x) - 5
So, the complete functions are
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Consider the functions below. f(x)=8x^2+x+3, g(x)=4x-1, h(x)=3x+6. Which of the following statements is true?
A.
As x approaches infinity, the value of g(x) eventually exceeds the values of both f(x) and h(x).
B.
Over the interval [3, 5], the average rate of change of g and h is more than the average rate of change of f.
C.
Over the interval [0, 2], the average rate of change of f and h is less than the average rate of change of g.
D.
As x approaches infinity, the values of g(x) and h(x) eventually exceed the value of f(x).
The functions f(x)=8x^2+x+3, g(x)=4x-1, h(x)=3x+6 C.Over the interval [0, 2], the average rate of change of f and h is less than the average rate of change of g is true.
What is increasing function?
⇒ The function is said to be increasing if the y value increases as the x value increase over a given range
What is average rate of change?
⇒An average rate of change function is a process that calculates the amount of change in one item divided by the corresponding amount of change in another
As x approaches infinity the value of f(x) eventually exceeds the value of both g(x) and h(x)
And it is true for the interval [0,2]
The faster the growth rate higher the average rate of change
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Answer:
C.
Over the interval [0, 2], the average rate of change of f and h is less than the average rate of change of g.
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as
requested.
y5+ x3 = y2 + 12x, slope at (0, 1)
0-2
02
04
The slope of the curve described by the equation at the given point (0,1) as in the task content is; 4.
What is the slope of the curve, the line tangent to the curve at the given point; (0, 1)?According to the task content, it follows that the slope of the curve can be determined by means of implicit differentiation as follows;
y⁵+ x³ = y² + 12x
5y⁴(dy/dx) -2y(dy/dx) = 12 - 3x²
(dy/dx) = (12 -3x²)/(5y⁴-2y)
Hence, since the slope corresponds at the point given; (0, 1); we have;
(dy/dx) = (12 -3(0)²)/(5(1)⁴-2(1))
dy/dx = 12/3 = 4.
Hence, slope, m = 4.
Consequent to the mathematical computation above, it can then be concluded that the slope of the curve, the line tangent to the curve at the given point is; 4.
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Find the variance of 24,30,17,22,22
Answer:
22
Step-by-step explanation:
Determine the five-number summary for this data set, taking into account any outliers. 10 13 15 12 12 4 12 17 12 13 15 18 10 11 20 19 Tiles 11.5 13 16 20 10 12.5 17
The value of the median, lower and upper quartile
Median = 21.5lower quartile = 12upper quartile = 29What is the five-number summary for this data set?
The minimum value = 10
The maximum value = 38
The median is the value that is found in the center of the data when it is ordered from lowest to highest. In the event that there is no value that precisely corresponds to the center, the value will be determined by taking the average of the values that are located on each side of the middle.
10 11 12 15 19 24 27 29 33 38
Median = 21.5
The intermediate value of the data that is located to the left of the median is known as the lower quartile.
10 11 12 15 19
lower quartile = 12
The intermediate value of the data that is located to the right of the median is known as the upper quartile.
24 27 29 33 38
upper quartile = 29
In conclusion, the 5 number summary is 10, 12, 21.5, 29, 38 → A
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Answer: minimum 10 first quartile 11.5 median 12.5 third quartile 16 maximum 20
Step-by-step explanation:
Suppose that you repeated questions 5 and 6 using two line segments of your choice. The line segments could be any length and in any
orientation as long as the midpoints were marked correctly and coincided with each other. Would you reach the same conclusion that you
reached in question 7? How does your conclusion relate to the diagonals of a parallelogram?
A line segment can be defined as the part of a line in a geometric figure such as a parallelogram, that is bounded by two (2) distinct points and it typically has a fixed length.
In Geometry, a line segment can be measured by using the following measuring instruments:
A scale (ruler).A divider.What is a parallelogram?A parallelogram refers to a geometrical figure (shape) and it can be defined as a type of quadrilateral and two-dimensional geometrical figure that has two (2) equal and parallel opposite sides.
Based on the previous experiment conducted in question 5, 6 and 7, we can logically conclude that the opposite sides of quadrilateral ABCD have the same (equal) slopes, which implies that the opposite sides are parallel. Hence, quadrilateral ABCD is simply a parallelogram by definition.
In this context, yes I would you reach the same conclusion that I reached in question 7 because the line segments that I drew represent the diagonals of a parallelogram.
Therefore, if the point of intersection of the diagonals divide each diagonal in half, then, the quadrilateral belonging to these diagonals forms a parallelogram.
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A baseball travels d meters t seconds after being dropped from the top of the building
Considering the given function, it is found that:
When t = 0, d = 0 meters.When t = 0.5, d = 1.25 meters.When t = 1, d = 5 meters.When t = 1.5, d = 11.25 meters.When t = 2, d = 20 meters.Since the changes for each 5 second interval are not the same, the ball is not traveling at a constant speed.
What is the function for the distance traveled by the ball?The function is:
d = 5t².
Hence:
When t = 0, d = 5 x 0² = 0 meters.When t = 0.5, d = 5 x 0.5² = 1.25 meters.When t = 1, d = 5 x 1 = 5 meters.When t = 1.5, d = 5 x 1.5² = 11.25 meters.When t = 2, d = 5 x 2² = 20 meters.Since the changes for each 5 second interval are not the same, the ball is not traveling at a constant speed.
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A drawer contains loose socks. There are 2 blue and 4 black socks in the drawers. What is the probability that you choose a sock without looking, then choose a second sock (keeping the first in your hand) without looking, and end up with a pair of black socks?
The probability is p = 2/5, the correct option is the third one.
How to find the probability?
There is a total of 6 socks, such that 4 are black and 2 are blue.
Then the probability of first getting a black sock is the quotient between the number of black socks and the number of blue socks, which gives:
P = 4/6
Now there are 5 socks in total, such that 3 are black and 2 blue.
Then the probability of getting another black one is:
Q = 3/5
The joint probability (getting the two black socks) is given by the product of the individual probabilities:
p = (4/6)*(3/5) = 2/5
The correct option is the third one.
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Find the length indicated
Find SR (image)
Answer:
SR = 5
Step-by-step explanation:
We know that the two lines are equal to each other and thus TS + SR = 13
So, we can simply set the two equations of the first line equal to 13:
x + 2 + 2x - 7 = 13
3x - 5 = 13
3x = 18
x = 6
Now we plug in 6 for x in the SR equation:
2 * 6 = 12 - 7 = 5
What expression is equivalent to (9x^2 + 2x -7)(x - 4)
Answer: C
Step-by-step explanation:
We can use the expanding rule to get that one if the expressions is
(9x^2 + 2x - 7)x + (9x^2 + 2x - 7)(-4)
Use the drop-down menus to complete each equation so the statement about its solution is true.
No Solutions 2x+5+2x+3x= x +
One Solution 2x+5+2x+3x= x +
Infinitely Many Solutions 2x+5+2x+3x= x +
Since there is only one value of x, hence the equation given has only one solution
Equations and expressionsEquations are expressions separated using mathematical operations. Given the equation below;
2x+5+2x+3x = x
Collect the like terms
2x+2x+3x -x = -2
7x -x = -2
Simplify
6x = -2
x = -2/6
x = -1/3
Since there is only one value of x, hence the equation given has only one solution
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The three circles in the diagram have the same centre and have radii 3cm, 4cm and 5cm.
What percentage of the area of the largest circle is shaded?
Answer:
Fufusyyigywngd, hdj4snwhsjtc
Bahr Ltd flu not ld6wlw
Remy obtains a 30-year mortgage in the amount of $625,000 for a co-op. She secures a 7/1 ARM at an initial interest rate of 3%. Her initial monthly payment is $2,635.03. After 7 years, the interest rate on her loan changes to 4.925%. Calculate her new monthly payment in year 8 of the loan. (Round your answer to the nearest cent.)
The new monthly payment is $3,181.53
What is ARM?
ARM stands adjustable rate mortgage, which means that the interest rate applicable to the mortgage would change during the life of the mortgage, in this case, 7/1 ARM means that interest rate of 3% is applicable for the first 7 years and the rate would change in year 8 to another interest rate, which is 4.925% as hinted at in the question.
To determine the monthly payment in year 8 , we need to compute the outstanding balance at the end of year 7, which is the opening balance in year 8 using the present value formula of an ordinary annuity since monthly payments would occur at the end of each month
PV(balance at the end of year 7)=PMT*(1-(1+r)^-N/r
PMT=initial monthly payment=$2,635.03
r=initial monthly interest rate=3%/12=0.0025
N=number of monthly payments in the remaining 23 years(i.e 30-7)=23*12
N=number of monthly payments in the remaining 23 years(i.e 30-7)=276
PV=$2,635.03*(1-(1+0.0025)^-276/0.0025
PV=$2,635.03*(1-0.502008144755209)/0.0025
PV=$2,635.03*0.497991855244791/0.0025
PV=$ 524,889.39
With the balance at the end of year 7, we can now compute monthly payment in year 8 using the same formula as above where the unknown is the PMT, PV=$ 524,889.39 and r= 4.925%.
$524,889.39=PMT*(1-(1+4.925%/12)^-276/4.925%/12)
$524,889.39=PMT*(1-(1.00410416666666667
)^-276/0.00410416666666667
$524,889.39=PMT*(1-0.32289378683267300
)/0.00410416666666667
$524,889.39=PMT*164.98019407122700000
PMT=$524,889.39/164.98019407122700000
PMT=$3,181.53
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In 2001 the total cost of manufacturing an article was sh.1250 and this was divided between the cost of materials was doubled,labour cost increased by 30% and transport costs increased by 20%.calculate the cost of manufacturing the article bin 2004
The cost of manufacturing the article bin 2004 will be 1875.
What is Percentage Increased ?Percentage increase is the ratio of increase to the initial value in percentage.
Given that In 2001 the total cost of manufacturing an article was sh.1250 and this was divided between the cost of materials was doubled, labor cost increased by 30% and transport costs increased by 20%
Given that in 2001, the cost of manufacturing = 1250
If this cost was divided between the Material, labor, and transportation equally, then, cost for each sector will 1250/3 = 416.67
cost of materials was doubled, that is 416.67 x 2 = 833.33labor cost increased by 30% , that is 130/100 x 416.67 = 541.67transport costs increased by 20% that is, 120/100 x 416.67 = 500The cost of manufacturing = 833.33 + 541.67 + 500
The cost of manufacturing = 1875
Therefore, the cost of manufacturing the article bin 2004 will be 1875.
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A DC10 airplane travels 3000 km with a tailwind in 3 hr. It travels 3000 km with a headwind in 4 hr. Find the speed of the plane and the speed of the wind.
The speed of the plane will be 875km/h and the speed of the wind will be 125km/h.
How to calculate the speed?It should be noted that the speed is calculated as:
= Distance/Time
Based on the information, the DC10 airplane travels 3000 km with a tailwind in 3 hr. It travels 3000 km with a headwind in 4 hr.
Therefore, (v + w) = 3000/3
v + w = 1000 .... I
(v - w) = 3000/4
v - w = 750 .... ii
We'll then add both equations together. Therefore, the speed of the plane will be 875km/h and the speed of the wind will be 125km/h.
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A box contains orange balls and green balls. The number of green balls is seven more than four times the number of orange balls. If there are 102 balls altogether, then how many green balls and how many orange balls are there in the box?
green balls = 83
orange balls = 19
Step-by-step explanation:
green balls = g
orange balls = r
g = 7 + 4r
g + r = 102
7 + 4r + r = 102
5r = 95
r = 19
g + 19 = 102
g = 83
cual es el valor x-3=11
A _[blank 1]_ is a number whose only factors are 1 and itself. If a number has other factors besides 1 and itself, it is called a _[blank 2]_. You can use divisibility rules or _[blank 3]_ to help you determine whether a number is prime or composite.
Match each blank with the option that correctly fills in that blank.
1. the sieve of Eratosthenes.
2.whole number.
3. composite number.
4.counting number.
5. the standard algorithm
6. prime number.
Numbers whose only factors are 1 and itself is known as prime numbers , numbers whose factors besides 1 and itself are composite numbers, we can use the sieve of Eratosthenes to determine whether the number is prime or composite.
Given a paragraph in which there are blanks:
A _____ is a number whose only factors are 1 and itself. If a number has factors besides 1 and itself, it is called a _____. You can use divisibility rules or _______ to help you determine whether a number is prime or composite.
We are required to fill the blank with appropriate options.
We have to fill "prime numbers" in the first blank.
We have to fill "composite numbers" in the second blank.
We have to fill "the sieve of Eratosthenes" in the third blank.
Hence numbers whose only factors are 1 and itself is known as prime numbers , numbers whose factors besides 1 and itself are composite numbers, we can use the sieve of Eratosthenes to determine whether the number is prime or composite.
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Your Turn
Solve the equation by using Properties of Equality.
3. 5x − 10 = 20
4.
+9=21
Answer: for number 3, x= 6
Step-by-step explanation:
Please select the best answer from the choices provided .
Answer: B: (1-5x²)³, all real numbers
Step-by-step explanation:
The notation [tex]f\circ g(x)[/tex] is called function composition, which is where you pass one function in as the value of another function. In other words, [tex]\( f\circ g(x)=f(g(x)) \)[/tex].
Since we have the values for f(x) and g(x), let's plug them in.
[tex]f(g(x))\\f(1-5x^2)\\(1-5x^2)^3[/tex]
Hence, the best answer choice is B, as all real numbers would work and it is the cube of 1 - 5x².
please help! will give brainliest to whoever answers
maths functions
Answer:
1. f(x) is reflected across the x-axis
2. f(x) is translated 1 unit up
3. f(x) is vertically scaled by a factor of 2
4. f(x) is reflected across the x-axis AND is vertically scaled by a factor of 2
5. f(x) is vertically scaled by a factor of 3 AND is translated 1 unit down
6. f(x) is vertically scaled by a factor of 1/6 AND is translated 1 unit up
Solving question:
(1) [tex]g(x) = -f(x)[/tex]
This graph has been reflected in the x axis. Equation: [tex]\sf g(x) = -\dfrac{2}{x}[/tex]
(2) [tex]g(x) = f(x) + 1[/tex]
Graph has been translated 1 units up vertically. Equation: [tex]\sf g(x) = \dfrac{2}{x} +1[/tex]
(3) [tex]g(x) = 2f(x)[/tex]
This graph has been stretched vertically by a factor of 2. Equation: [tex]\sf g(x) = \dfrac{4}{x}[/tex]
(4) [tex]g(x) = -2f(x)[/tex]
This graph has been reflected in the x axis and stretched vertically by a factor of 2. Equation: [tex]\sf g(x) = -\dfrac{4}{x}[/tex]
(5) [tex]g(x) = 3f(x) - 1[/tex]
This graph has been stretched vertically by a factor of 3 and translated 1 units down. Equation: [tex]\sf g(x) = \dfrac{6}{x} -1[/tex]
(6) [tex]g(x) = \frac{1}{6} f(x) + 1[/tex]
This graph has been stretched vertically by a factor of 1/6 and translated 1 units up. Equation: [tex]\sf g(x) = \dfrac{1}{3x} +1[/tex]
Find the points of intersection of the equation: xy=2 and x+y =4
Answer:[tex]\Large\boxed{(2+\sqrt{2},~2-\sqrt{x} )~~and~~ (2-\sqrt{2},~2+\sqrt{x} )}[/tex]
Step-by-step explanation:
Given the system of equations
[tex]1)~xy=2[/tex]
[tex]2)~x+y=4[/tex]
Divide x on both sides of the 1) equation
[tex]xy=2[/tex]
[tex]xy\div x=2\div x[/tex]
[tex]y=\dfrac{2}{x}[/tex]
Current system
[tex]1)~y=\dfrac{2}{x}[/tex]
[tex]2)~x+y=4[/tex]
Substitute the 1) equation into the 2) equation
[tex]x+(\dfrac{2}{x} )=4[/tex]
Multiply x on both sides
[tex]x\times x+\dfrac{2}{x}\times x=4\times x[/tex]
[tex]x^2+2=4x[/tex]
Subtract 4x on both sides
[tex]x^2+2-4x=4x-4x[/tex]
[tex]x^2-4x+2=0[/tex]
Use the quadratic formula to solve for the x value
[tex]x=\dfrac{-(-4)\pm\sqrt{(-4)^2-4(1)(2)} }{2(1)}[/tex]
[tex]x=2\pm\sqrt{2}[/tex]
Substitute the x value into one of the equations to find the y value
[tex]xy=2[/tex]
[tex](2+\sqrt{2} )y=2[/tex]
[tex]y=2-\sqrt{2}[/tex]
[tex]OR[/tex]
[tex]xy=2[/tex]
[tex](2-\sqrt{2} )y=2[/tex]
[tex]y=2+\sqrt{2}[/tex]
Therefore, the points of intersection are
[tex]\Large\boxed{(2+\sqrt{2},~2-\sqrt{x} )~~and~~ (2-\sqrt{2},~2+\sqrt{x} )}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Answer:
(2 - [tex]\sqrt{2}[/tex] , 2 + [tex]\sqrt{2}[/tex] ) and (2 + [tex]\sqrt{2}[/tex], 2 - [tex]\sqrt{2}[/tex] )
Step-by-step explanation:
xy = 2 → (1)
x + y = 4 ( subtract x from both sides )
y = 4 - x → (2)
substitute y = 4 - x into (1)
x(4 - x) = 2
4x - x² = 2 ( multiply through by - 1 )
x² - 4x = - 2
using the method of completing the square
add ( half the coefficient of the x- term)² to both sides
x² + 2(- 2)x + 4 = - 2 + 4
(x - 2)² = 2 ( take square root of both sides )
x - 2 = ± [tex]\sqrt{2}[/tex] ( add 2 to both sides )
x = 2 ± [tex]\sqrt{2}[/tex] , that is
x = 2 - [tex]\sqrt{2}[/tex] , x = 2 + [tex]\sqrt{2}[/tex]
substitute these values of x into (2) for corresponding values of y
x = 2 - [tex]\sqrt{2}[/tex] , then
y = 4 - (2 - [tex]\sqrt{2}[/tex])
= 4 - 2 + [tex]\sqrt{2}[/tex]
= 2 + [tex]\sqrt{2}[/tex] ⇒ (2 - [tex]\sqrt{2}[/tex] , 2 + [tex]\sqrt{2}[/tex] ) ← 1 point of intersection
x = 2 + [tex]\sqrt{2}[/tex] , then
y = 4 - (2 + [tex]\sqrt{2}[/tex] )
= 4 - 2 - [tex]\sqrt{2}[/tex]
= 2 - [tex]\sqrt{2}[/tex] ⇒ (2 + [tex]\sqrt{2}[/tex] , 2 - [tex]\sqrt{2}[/tex] ) ← 2nd point of intersection
Find two positive numbers whose difference is 9 and whose product is 2950.
The two positive numbers whose difference is 9 and whose product is 2950 are 50 and 59
How to determine the positive numbers?As a general rule, it should be noted that positive numbers are numbers that have their value greater than 0
So, we start by representing the two positive numbers with x and y.
So, we have the following equations
x - y = 9
xy = 2950
Make x the subject in the first equation x - y = 9
x = y + 9
Substitute y + 9 for x in the second equation
(y + 9) * y = 2950
Expand the equation
y^2 + 9y - 2950 = 0
Using a graphing tool, we have the solution of the above equation to be
y = 50
Recall that:
x = 9 + y
So, we have:
x = 9 + 50
Evaluate
x = 59
Hence, the two positive numbers whose difference is 9 and whose product is 2950 are 50 and 59
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The vertices of a composite figure are given. Find the area of the figure.
G(-5, -1), H(-5, 1), I(2, 4), J(5, -1), K(1, -3)
Check the picture below.
so the composite is really a trapezoid and two triangles, let's get their area and sum them all up.
[tex]\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h~~=height\\ a,b=\stackrel{parallel~sides}{bases~\hfill }\\[-0.5em] \hrulefill\\ a = 2\\ b = 5\\ h = 7 \end{cases}\implies A=\cfrac{7(2+5)}{2} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{\LARGE Areas}}{\stackrel{yellow~trapezoid}{\cfrac{7(2+5)}{2}}~~ + ~~\stackrel{blue~triangle}{\cfrac{1}{2}(\stackrel{b}{3})(\stackrel{h}{5})}~~ + ~~\stackrel{orange~triangle}{\cfrac{1}{2}(\stackrel{b}{10})(\stackrel{h}{2})}} \\\\\\ 24.5~~ + ~~7.5~~ + ~~10\implies \text{\LARGE 42}[/tex]
Let σ(n) be the sum of all positive divisors of the integer n and let p be any prime number.
Show that σ(n) < 2n holds true for all n of the form n = p²
The statement that "σ(n) < 2n holds true for all n of the form n = p²" has been proved.
Let p be any prime number, and let σ(n) be the sum of all positive divisors of the integer n.
As p is a prime number, and 2 is the smallest prime number, so, p[tex]\geq[/tex]2
So, the positive divisors of the integer n are: 1,p,p².
As σ(n) represents the sum of all positive divisors of the integer n.
σ(n)=1+p+p²
In order to prove that σ(n) < 2n,for all n of the form n = p².
1+p+p²<2p²
p²-p-1>0
It is know that, p[tex]\geq[/tex]2.
So, p²-p-1[tex]\geq[/tex]1
Thus, σ(n) < 2n holds true for all n of the form n = p².
Learn more about prime numbers here:
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5. Write the following inequality in slope-intercept form. −8x + 4y ≥ −52 y ≤ 2x − 13 y ≥ 2x − 13 y ≤ 2x + 13 y ≥ 2x + 13
Answer:35443
Step-by-step explanation: