Answer:
49.34 i think
Step-by-step explanation:
10. ANEW ACAR. Solve for x and CR. *
X=
Given: ANEW ACAR
NE = 11
EW = 12
I
AR = 4y - 12
CA = 4x + 3
NW = x + y
Applying the definition of congruent triangles, the values of x and y are:
x = 2; y = 6
Length of CR = 8 units.
How to Find the Sides of Congruent Triangles?If two triangles are congruent to each other, based on the CPCTC, all their corresponding angles, and their corresponding sides will be equal to each other.
Given that triangles NEW and CAR are congruent to each other, therefore:
NE = CA
NW = CR
EW = AR
Given the following measures:
NE = 11
EW = 12
AR = 4y - 12
CA = 4x + 3
NW = x + y
Therefore:
NE = CA
11 = 4x + 3
Solve for x:
11 - 3 = 4x
8 = 4x
8/4 = x
x = 2
EW = AR
12 = 4y - 12
12 + 12 = 4y
24 = 4y
24/4 = y
y = 6
NW = CR = x + y
Plug in the values of x and y:
CR = 2 + 6
CR = 8 units.
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on their next training run, pepe averaged a speed of 2/3 of a mile in 5 minutes, while paula averaged 1/4 of a mile in 2 minutes. if pepe and paula each ran at their individual pace for 60 minutes, how many total miles did they cumulatively run?
Answer:
15.5
Step-by-step explanation:
Pepe= 2/3 x 12 = 8
Paula= 1/4 x 30 = 7.5
8+7.5=15.5
Convert the measurement. 28.4 grams ≈ ounces
Answer: 1.0017805194 ounces
Step-by-step explanation:
To change 28.4 g to oz divide the mass in grams by 28.349523125. The 28.4 grams to ounces formula is [oz] = [28.4] / 28.349523125. Thus, we get (all results rounded to 2 decimals):
28.4 g to oz = about 1 oz
28.4 g to ounces = about 1 ounces
28.4 grams to ounces = about 1 oz
Twenty-eight point four grams to ounces equal 1 international avoirdupois ounces, the unit used to measure grocery products (mass) in retail stores in the United States for example.
HOPE THIS HELPS 189328 :)!Find the remainder term Rn in the nth order Taylor polynomial centered at a for the given function. Express the result for a general value of n. f(x)=e^-2x, a=2 Choose the correct answer below. O A. Rn (x) = (-2)^n e^-2c/(n+1)! (x- 2)^n for some c between x and 2. O B. Rn (x) = (-2)^n+1 e^-2c/(n+ 1)! for some c between x and 2. O C. (-2)1+1e 2c Rn(x)=?(n+1)!-(x-2)n + 1 for some c between x and 2. O D. n -2c OD. (x-2)"+1 for some c between x and 2.
The remainder term Rn is equal to (-2) raised to the power of n multiplied by e raised to the power of -2c, divided by n+1 factorial, multiplied by (x-2) raised to the power of n, for some c between x and 2.
The remainder term Rn in the nth order Taylor polynomial centered at a for the given function f(x)=e^-2x, a=2 is equal to (-2) raised to the power of n multiplied by e raised to the power of -2c, divided by n+1 factorial, multiplied by (x-2) raised to the power of n, for some c between x and 2. This remainder term represents the difference between the approximate value of the function given by the Taylor polynomial and the actual value of the function. By taking the limit as n approaches infinity, the remainder term can be used to determine the exact value of the function. The remainder term is an important concept in calculus as it is used to determine the accuracy of a given approximation.
For n=5, R5(x) = (-2)^5 e^-2c/(5+1)! (x-2)^5 for some c between x and 2.
Substituting n=5, x=7 and c=4, we have:
R5(7) = (-2)^5 e^-8/(6)! (7-2)^5
R5(7) = (-2)^5 e^-8/(720) (25)
R5(7) = (-2)^5 e^-8/18000
R5(7) = (-2)^5 e^-8/18000
R5(7) = -1.388888888 x 10^-9
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Find the area of a circle with radius 6 ft.
Answer:
The area of the circle is 113.04 square feet.
Step-by-step explanation:
Given;Radius (r) = 6 ftπ = 3.14Formula;A = πr²A = 3.14 × (6)² ft
A = 3.14 × 36 ft
A = 113.04 square feet.
Thus, The area of the circle is 113.04 square feet.
Answer:
28.274 square feet
Step-by-step explanation:
Multiply the radius by pi
Clarks Inc., a shoe retailer, sells boots in different styles. In early November the company starts selling “SunBoots” to customers for $70 per pair. When a customer purchases a pair of SunBoots, Clarks also gives the customer a 30% discount coupon for any additional future purchases made in the next 30 days. Customers can’t obtain the discount coupon otherwise. Clarks anticipates that approximately 20% of customers will utilize the coupon, and that on average those customers will purchase additional goods that normally sell for $100. Required: 1. How many performance obligations are in a contract to buy a pair of SunBoots? 2. Assume Clarks cannot estimate the standalone selling price of a pair of SunBoots sold without a coupon. Prepare a journal entry to record revenue for the sale of 1,000 pairs of SunBoots.
The journal entry is given in answer part b
What is Discount coupon?A discount coupon is defined as a certificate that entitles the customer to avail of a reduction in the invoice price if the terms mentioned on it are met. It is a technique applied by the sellers to expand sales.
Given that, Clarks Inc., a shoe retailer, sells boots in different styles. In early November, the company starts selling “SunBoots” to customers for $70 per pair. When a customer purchases a pair of SunBoots, Clarks also gives the customer a 30% discount coupon for any additional future purchases made in the next 30 days. Customers can’t obtain the discount coupon otherwise. Clarks anticipates that approximately 20% of customers will utilize the coupon, and that on average those customers will purchase additional goods that normally sell for $100.
a) Two performance obligations are associated with contract. The first one is related to the sun boots, and the other one is associated with discount coupon provided. The discount coupon would be considered a separate obligation because it gives a certain right to the customers that they would not have received if the sale was not made to them.
b) Journal entry =
Account title and explanation Debit ($) Credit ($)
Cash (1000×$70) $70000
Sales Revenue ($70,000-$6000) $64000
Deferred Revenue $6000
Now,
Computation of the deferred revenue:
Deferred revenue = pairs sold × Average purchase price × Discount × Redemption of coupon
= 1000 × 100 × 30% × 20%
= $6000
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If (x+2) is a factor of
Answer: a=3 and b=1
Step-by-step explanation:
Let p(x)=x 2 +ax+2b
If x+2 is a factor of p(x)
Then by factor theorem
p(−2)=0
⇒(−2)
2
+a(−2)+2b=0
⇒4−2a+2b=0
⇒a−b=2 ---(i)
Also given that a+b=4 ---(ii)
Adiing (i) and (ii) we get
2a=6
⇒a=3
putting a=3 in (i) we get
a−b=2
⇒3−b=2
⇒b=1
Therefore we get a=3 and b=1
Alex biked a 48-mile trail in 4 hours. For the first half of the time, he biked twice as fast as he did the second half of the time. What speed did Alex bike the first half?
Alex's speed in the first half of the time is 12 miles per hour
How t find the speed of bikingInformation from the problem include:
For the first half of the time, Alex biked a 48-mile trail in 4 hours.
he biked twice as fast as he did the second half of the time.
Average speed is calculated using the ratio of distance to time spent
this is represented by the formula
= distance / time
The speed for the first half is solved as below
= 48 mile / 4 hours
= 12 mile / hours
= 12 mph
speed for the first half is 12 mph
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Find the Derivative of 1/x+2
Answer: 2x+1
x
Step-by-step explanation:
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15
minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time.
Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84
shoppers per hour make a purchase and each of these shoppers spends an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers,
on average, are waiting in the checkout line to make a purchase at the Good Deals Store
Note that at any time during business hours, about 7 shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store. This is solved using Little Law.
What is littles Law?Little's Law asserts that the long-term average number of individuals in a stable system, L, is equal to the long-term average effective arrival rate,λ, multiplied by the average duration a customer spends in the system, W.
To compute,
First, let's make sure that all the variables use the same time unit.
Thus, if there are 84 shoppers who are making a purchase per house, then there will be: 84/60 minutes
= 1.4
This means that λ (rate) = 1.4 and the average time (W) is 5 minutes
Using Little's Law which states that the queuing formula is:
L = λW
Note that:
λ = 1.4 (computed)
W = 5 mintues (given)
Hence, the Average number of people in queue per time is:
L = 1.4 x 5
L = 7 shoppers.
Thus, it is correct to state that at any time during work hours, about 7 shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store.
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Full Question:
If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N=rT. This relationship is known as Little's law.
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time.
Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?
i literally don’t know
∠U is congruent ∠S because all right angles are congruent.
Define congruent.If it is possible to superimpose one geometric figure on the other so that their entire surface coincides, that geometric figure is said to be congruent, or to be in the relation of congruence. When two sides and their included angle in one triangle are equal to two sides and their included angle in another, two triangles are said to be congruent. This concept of congruence appears to be based on the idea of a "rigid body," which may be moved without affecting the internal relationships between its components.
Given
∠U ≅ ∠S
All right angles are congruent.
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what does
QR=
PR=
M
We need to use Right Angled Triangle Properties.
PR≈15 and QR≈12 is the length of the sides.
What is a Right Angled Triangle?
According to the definition of a right triangle, a triangle is referred to as a right-angled triangle or simply a right triangle if one of its angles is a right angle (90o). Triangle PQR is a right triangle in the illustration, and its components are its base, height, and hypotenuse. Here, QR is the base, PQ is the height, and PR is the hypotenuse. The essential side of a right triangle is its hypotenuse, which is also its largest side and sits across from the triangle's right angle.
In a right triangle we have: (Hypotenuse)2 = (Base)2 + (Altitude)2
Area of a right triangle = (1/2 × base × height) square units.
Calculation:PR is the Hypotenuse of the Triangle.
PQ=9
Angle ∡PQR=37° and Angle ∡RPQ=53°
So Sin(theta)=opposite side/Hypotenuse
⇒Sin(37°)=9/Hypotenuse
⇒Hypotenuse=9/0.6018
⇒Hypotenuse=14.955
PR⇒14.955
We know that (Hypotenuse)2 = (Base)2 + (Altitude)2
⇒(14.955)2=(9)2+(QR)2
QR=11.9438
PR≈15 and QR≈12
We need to use Right Angled Triangle Properties.
PR≈15 and QR≈12 is the length of the sides.
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Which of the following represents members of the domain of the graphed function?
See attached picture
Responses
{-4, 2, 3}
{-4, 0, 1}
{1, 2, 3, 4}
{1, 2, 3, 5}
The {-4, 0, 1} represents members of the domain of the graphed function.
What is the domain of the function?
A function is a mathematical object that accepts input, appears to apply a rule to it, and returns the result.
A function can be thought of as a machine that requires in a number, performs some operation(s), and then outputs the result.
The domain of a function is the collection of all its inputs. Its codomain is the set of possible outputs.
The range refers to the outputs which are actually used.
Domain: {-4, 0, 1}.
Simply list the domain as -4 < x < 2, which would imply ALL values between -4 and 2 inclusive.
Yes, this is a function. No x-values repeat, and it passes the Diagonal Line Test for functions.
Hence, the {-4, 0, 1} represents members of the domain of the graphed function.
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If x varies directly as y, find x when y = 8 a) x = 6 when y = 32 b) x = 14 when y = -2
a) The value of x is 42.6 when k =16/3 is directly proportional to y.
b) The value of x is -56 when k is -7 is directly proportional to y.
What kind of variation is one where x and y are directly proportional?
If x = ky for some constant k can be used to indicate the relationship between the variables y and x, then we may say that y varies directly with y or that x is directly proportional to y.x varies directly as y
x α y
x = ky
a) x=6 ; y = 32
x = ky
6 = k(32)
k = 16/3
if y = 8
then,
x = (16/3) * 8
= 128/3
x = 42.6
Hence, the value of x is 42.6 when k =16/3 is directly proportional to y.
b) x= 14 and y = -2
x = ky
14 = k(-2)
k = -7
if y = 8
then,
x = ky
x= (-7) 8
x = -56
Hence, the value of x is -56 when k is -7 is directly proportional to y.
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The square of a number is equal to 72 more than the number. Find all such numbers.
Answer:
You just need to think about this a little. There is a number, that you don't know, x, and if you square it ([tex]x^2[/tex]), that value will be equal to 72 more than the unknown number itself, so you put that into an equation.
Step-by-step explanation:
[tex]x^2=x+72\\x^2-x-72=0\\[/tex]
Apply quadratic formula (you can look it up)
[tex]x_1=\frac{-(-1)+\sqrt{(-1)^2+(-4*1*-72} )}{2(1)} \\\\x_2=\frac{-(-1)-\sqrt{(-1)^2+(-4*1*-72} )}{2(1)} \\\\[/tex]
There are two answers to the quadratic formula.
The rest should be easy.
The two-way frequency table contains data about students' preferred exercise.
Enjoys swimming Enjoys cycling Row totals
Likes running 28 62 90
Does not like running 46 64 110
Column totals 74 126 200
What is the joint relative frequency of students who do not like to run but enjoy cycling?
64%
55%
32%
23%
The joint relative frequency for those students who don't like to run, but however enjoy cycling, is C. 32%.
How to find the joint frequency ?The joint frequency for students who like cycling, but do not like running, can be found by the formula :
= Number of students who enjoy cycling but don't enjoy running / Number of students in total
Number of students who enjoy cycling but don't enjoy running = 64
Number of students in total = 200
The joint frequency is :
= 64 / 200 x 100%
= 0.32 x 100 %
= 32 %
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Can someone help me on this question?
Answer:
see below for the chart
64 pages in 1 hour
Step-by-step explanation:
You want to fill in the chart showing the analysis of the problem statement saying HP can read 16 pages in 1/4 hour. The problem statement is requesting the number of pages he can read in 1 hour.
ChartSee below for a filled-in chart.
SolutionAnalysis of the bottom line tells you that the right column is 4 times the middle column.
1 = 4 × 1/4
?? = 4 × 16 = 64
Harry Potter can read 64 pages in one hour.
NO LINKS!!
The expoential function given by f(x) = e^x called the (a. natural logarithmic, natural exponential, c. 1 to 1 exponential, d. 1 to 1 algebraic, e. transcendental algebraic) function and the base e is called the (a. algebraic, b. 1 to 1, c. natural, d. rational, e. transcendental)
Answer:
natural exponential
natural
quizlet
Answer:
b. natural exponential
c. natural
Step-by-step explanation:
Given function:
[tex]f(x)=e^x[/tex]
The given exponential function is called the:
natural exponential functionThe base e is called the:
natural baseThe number "e" occurs naturally in math and the physical sciences.
It is the base rate of growth shared by all continually growing processes, and so is called the natural base.
It is an irrational number and named after the 18th century Swiss mathematician, Leonhard Euler, and so is often referred to as "Euler's number".
Sample Response: First, I wrote each ratio as a fraction. Then I found a common denominator of 20. The fraction 4/10 is 8/20. Comparing numerators, 8 is less than 12, so the ratio 4:10 is smaller.
Which did you include in your answer?
Check all that apply.
Write the ratios as fractions.
Find a common denominator.
Compare the numerators to see that 4:10 is the smaller ratio.
In the aforementioned scenario, the choice you must include in your response is to find a common denominator.
In the simplest terms, what is fraction?One can refer to a fraction as being in its lowest term or in its simple form when the numerator and denominator are known to have no common factors besides 1. In general, any number of equal pieces is represented by a fraction, which is a portion of the entire. A fraction, such as one-half, eight-fifths, or three-quarters, indicates the number of parts of a particular size when it is used in daily speech.
The common denominator will be between 10 and 20, and there is a common numerator as well. The common denominator will be 20 because both can divide it and 10 is less than 20.
Finding a common denominator is the solution you should choose in the scenario mentioned above.
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Does anyone know the answer for this?
The Venn diagram has been drawn and the product of LCM and GCF are the same as 24 x 36.
What is a Venn diagram?A Venn diagram is a pictorial representation of data represented in a circle.
The intersection parts of the circle represent the commonality of both sets.
As per the given, 24 and 36
The factors of 24 are as 2³ × 3
The factors of 36 are as 2² × 3²
The common factors 2² × 3
The LCM of 24 and 36 is 72 and the GCF is 12.
LCM x GCF = 24 x 12
72 x 12 = 24 x 12
864 = 864
Hence "After drawing a Venn diagram, it was discovered that the LCM and GCF product equals 24 x 36.".
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a duck flew at 18 miles per hour for 3 hours than at 15 miles per hour for 2 ours how far did the duck fly in all
The distance the duck fly in all is 99 miles
How to determine how far the duck fly in allFrom the question, we have the following parameters that can be used in our computation:
Distance 1: 18 miles per hour for 3 hoursDistance 2: 15 miles per hour for 2 hoursThe distance covered in all can be calculated as
Distance = The sum of the product of speed and time
Substitute the known values in the above equation, so, we have the following representation
Distance = 18 * 3 + 15 * 2
Evaluate the products
This gives
Distance = 54 + 45
Evaluate the sum
Distance = 99 miles
Hence, the distance is 99 miles
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Staci pays $32.70 for 5 cell phone cases. Each case costs the same amount. How much does each case cost?
Part A
Which expression represents the problem?
$32.70 × 5
$32.70 ÷ 5
$32.70 + 5
$32.70 – 5
Part B
Evaluate the expression from Part A.
$ ( ??? )
Answer:
$32.70 ÷ 5
Each case is $6.54.
Step-by-step explanation:
$3270 for 5 cases, meaning you would split $32.70 into 5.
$32.70 ÷ 5 phone cases = $6.54
This also means each case is $6.54. To prove this, multiply by 5.
Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)
f ''(x) = 2x + 4ex
After integration, the required function f is (2x³ - sin (x) + Cx + D).
What is the integration of 'xⁿ' and 'sin (x)'?
[tex]\int {x^{n} } \, dx = \frac{x^{n+1} }{n+1} + C\\\\\\\int {sinx} \, dx = -cosx + C[/tex]
Given, f''(x) = 12x + sin x
Therefore,
[tex]\int {f''(x)} \, dx \\\\=\int {f'(x)} \, dx \\\\\\= \int{12x + sin x} \, dx + C\\\\= 6x^{2} - cosx + C\\[/tex]
Again, f'(x) = 6x - cos (x) + C
Therefore,
[tex]\int {f'(x)} \, dx\\ \\=\int {f(x)} \, dx \\\\= \int {6x^{2} - cosx + C } \, dx \\\\= 2x^{3} - sinx + Cx + D[/tex]
Therefore, the required function is (2x³ - sin (x) + Cx + D).
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Write an equation for the line through the given point with the given slope in slope-intercept form. (10, –9); m = –2
Please explain.
Considering the definition of a line, the equation of the line that passes through the point (10, -9) and has a slope of -2 is y= -2x +11.
Linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin and represents the coordinate of the point where the line crosses the y axis.Line in this caseIn this case, you know:
The line has a slope of -2.The line passes through the point (10, -9).Substituting the value of the slope m and the value of the point y=mx+b, the value of the ordinate to the origin b is obtained as follow:
-9= -2×10 + b
-9= -20 + b
-9 + 20= b
11= b
Finally, the equation of the line is y= -2x +11.
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Question 12 options:
Nick determines the remainder of
, using the remainder theorem.
How does she proceed to the correct answer?
Fill in the blanks from the work bank below: (do not enter spaces)
x = 1 x = -1 x = 0 -23 -19 -15 -29
Nick evaluates the numerator of the rational expression when
. He concludes that the remainder of the division is
The input value for the expression is 1 and the remainder is -15.
What is the remainder theorem?The Remainder Theorem begins with an unnamed polynomial p(x), where "p(x)" simply means "some polynomial p with variable x". The Theorem then discusses dividing that polynomial by some linear factor x a, where an is simply a number.
Given polynomial is 5x⁴⁵ - 3x¹⁷ +2x⁴ - 19 and is divided by x + 1. Find the value of the polynomial at 1 and then calculate the remainder by synthetic solution.
F(x) = 5x⁴⁵ - 3x¹⁷ +2x⁴ - 19
F(1) = 5 - 3 + 2 - 19
F(1) = -19 + 4
F(1) = -15
The remainder will be calculated by synthetic calculation,
1 __5 ___-3___2_____-19
|
| ______5___ 2______4
5 2 4 -15
The remainder of the given expression is -15.
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5x^2-13+6=0
i need helppp
Answer:
answer is 2.5
Step-by-step explanation:
A circle with center w is shown in the figure below.
Radius = TW
Diameter = TX
Chord = UV
The length of TX is 6 units.
What is a circle?A circle is a two-dimensional figure with a radius and circumference of 2 x pi x r.
The area of a circle is given as πr².
We have,
From the figure:
Diameter = TX
Radius = TW or WX or WY
Chord = UV
WY = 3 units
TX = TW + WX = 3 + 3 = 6 units.
Thus,
TX is 6 units.
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You buy 9 granola bars for a camping trip. Each bar costs $0.95. What
is the total cost of 9 granola bars before tax?
Answer:
Step-by-step explanation: 9 multiplied by .95 = $8.55 before tax
Here we show that function defined on an interval value property cannot have (a; b) and satisfying the intermediate removable or (a) jump discontinuity. Suppose has & jump discontinuity at Xo € (a,b) and lim f (x) lim f (x) xx0 {ix0 Choose 0 such that lim f (x) < 0 < lim f (x) and 0 + f(xo) xI*o Xx0 In Exercise & we showed there is interval [xo 0,.Xo) such that f(x) < 0 if Xe [xo 6,xo): Likewise, there an interval (xo, Xo + 6] such that f(x) > 0 if xe(xo, Xo + 6]. Conclude that does not satisly the intermediate value property on [xo 6,xXo + 6]. (6) Suppose has a removable discontinuity at Xo € (a,b) and a = lim f(x) < f(xo) Show that there is an interval [xo = 6,Xo) such that f(x)< a+[f(xo) - &] if x e[xo 6,Xo]: Conclude that f does not satisfy the intermediate value property
f cannot have a jump discontinuity at [tex]$x_0 \in(a, b)$[/tex] and [tex]$$ \lim _{x \uparrow x_0} f(x) < \lim _{x \mid x_0} f(x) .$$[/tex]
f cannot have a removable discontinuity at [tex]$$x_0 \in(a, b) $$[/tex] and [tex]\alpha=\lim _{x \rightarrow x_0} f(x) < f\left(x_0\right)[/tex]
Let f be a function defined on (a, b) satisfies intermediate value property.
Claim: f ca not have removable on jump discontinuity.
Suppose f has a jump discontinuity at [tex]$x_0 \in(a, b)$[/tex]
We take [tex]$\theta$[/tex] such that
[tex]$$\lim _{x \rightarrow x_0} f(x) < \theta < \lim _{x \downarrow x_0} f(x) \text { and } \theta \neq f\left(x_0\right)$$[/tex]
Now there exist [tex]$\delta > 0$[/tex] such that [tex]$f(x) < \theta$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0\right)$[/tex] and [tex]$f(x) > \theta$[/tex] for all [tex]$x \in\left(x_0, x_0+\delta\right]$[/tex]
Now [tex]$f\left(x_0-\delta\right)[/tex][tex]< \theta < f\left(x_0+\delta\right)$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0+\delta\right] \backslash\left\{x_2\right\}$[/tex] and [tex]$f\left(x_0\right) \neq \theta$[/tex].
Therefore the point [tex]$\theta$[/tex] has no preimage under f
that is, there does not exists [tex]$y \in\left[x_0-\delta, x_0+\delta\right][/tex] for which
[tex]$$f(y)=\theta[/tex] because [tex]\left\{\begin{array}{l}y=x_0 \Rightarrow f(y) \neq \theta \\y > x_0 \Rightarrow f(y) > \theta \\y < x_0 \Rightarrow f(y) < \theta\end{array}\right.$$[/tex]
Therefore f does not satisfies intermediate value property on [tex]$\left[x_0-\delta, x_0+\delta\right]$[/tex],
Hence f does not satisfies IVP on (a, b) which is not possible because we assume f satisfies IVP on (a, b),
Therefore f can not have a jump discontinuity.
Suppose f has a removable point of discontinuity at [tex]$x_0 \in(a, b)$[/tex],
Let [tex]$\alpha=\lim _{\alpha \rightarrow x_0} f(x)$[/tex],
Let [tex]\alpha < f\left(x_0\right)$[/tex] so [tex]$f\left(x_0\right)-\alpha > 0$[/tex].
Now [tex]$\lim _{x \rightarrow x_0} f(x)=\alpha$[/tex] then [tex]\exists$ \delta > 0$[/tex] such that
[tex]$$\begin{aligned}& |f(x)-\alpha| < \frac{f\left(x_0\right)-\alpha}{2} \text { for all } x \in\left\{x_0-\delta, x_0-\alpha\right]-\left\{x_0\right\} \\& \Rightarrow \quad f(x) < \alpha+\frac{f\left(x_0\right)-\alpha}{2} \text { for all } x \in\left[x_0-\delta, x_0+\delta\right]-\left\{x_0\right\}\end{aligned}$$[/tex]
So [tex]$f(x) < \frac{f\left(x_0\right)+\alpha}{2}$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0\right]-\left\{x_0\right\}$[/tex]
Now [tex]$f\left(x_0\right) > \alpha$[/tex].
And [tex]$f(x) < \frac{f\left(x_0\right)+\alpha}{2} < f\left(x_0\right)$[/tex] for all [tex]$x \in\left[\left(x_0 \delta, x_0\right)\right.$[/tex]
Let [tex]$\mu=\frac{f\left(x_0\right)+\alpha}{2}$[/tex].
Then there does not exist [tex]$e \in\left[x_0-\delta, c\right]$[/tex] such that [tex]$f(c)=\mu$[/tex]
Because for [tex]$e=x_0 \quad f(e) > \mu$[/tex]
for [tex]$c < x_0 \quad f(c) < \mu$[/tex].
Therefore f does not satisfy IVP on [tex]$\left[x_0-\delta_1 x_0\right]$[/tex] which contradict our hypothesis,
therefore [tex]$\alpha \geqslant f\left(x_0\right)$[/tex]
Let [tex]$\alpha > f\left(x_0\right)$[/tex]. so [tex]$\alpha-f\left(x_0\right) > 0$[/tex]
[tex]$\lim _{x \rightarrow x_0} f(x)=\alpha$[/tex]
Then [tex]\exists $ \varepsilon > 0$[/tex] such that
[tex]$|f(x)-\alpha| < \frac{\alpha-f\left(x_0\right)}{2}$[/tex] for all [tex]$\left.x \in\left[x_0-\varepsilon_0 x_0+\varepsilon\right]\right\}\left\{x_i\right\}$[/tex]
[tex]$\Rightarrow f(x) > \alpha-\frac{\alpha-f\left(x_0\right)}{2}$[/tex] for all [tex]$x \in\left[x_0-\varepsilon_1, x_0+\varepsilon\right] \backslash\left\{x_0\right\}$[/tex]
[tex]$\Rightarrow f(x) > \frac{\alpha+f\left(x_0\right)}{2}$[/tex] for all [tex]$x \in\left[x_0-\varepsilon_1, x_0\right)$[/tex]
Now [tex]$f\left(x_0\right) < \alpha$[/tex]
The [tex]$f(x) > \frac{f\left(x_0\right)+\alpha}{2} > f\left(x_0\right)$[/tex].
So [tex]$f\left(x_0\right) < \frac{f\left(x_0\right)+\alpha}{2} < f(x)$[/tex] for all [tex]$x \in\left[x_0 \varepsilon, \varepsilon_0\right)$[/tex]
Let [tex]$\eta=\frac{f\left(x_e\right)+\alpha}{2}$[/tex]
Then there does not exist [tex]$d \in\left[x_0-\varepsilon, x_0\right]$[/tex] such that [tex]$f(d)=\xi$[/tex].
Because if [tex]$d=x_0, f(d)=f\left(x_0\right) < \eta$[/tex] if [tex]$d E\left[x_0-\varepsilon, x_0\right)$[/tex]
Then [tex]$f(d) > \eta$[/tex]
Therefore f does not satisfies IVP on [tex]$\left[x_0-\varepsilon, x_0\right]$[/tex] which contradict olio hypothesis.
Therefore [tex]$\alpha \leq f\left(x_0\right)$[/tex] (b) From (a) and (b) it follows [tex]$\alpha=f\left(x_0\right)=\lim _{x \rightarrow x_0} f(x)$[/tex]. Therefore f can not have a removable discontinuous
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a persons weight varies directly with gravity if a person weights 180 pounds on earth they will weigh only 30 pounds on the moon if harsh weights 54 pounds on earth how much would he weight on the moon
The weight of harsh on the moon if he weighs 54 pounds on earth is 9 pounds
How to calculate direct variation?Weight varies directly with gravity
Let
Weight of a person = w
Gravity = g
So,
w = k × g
Where,
k = constant of proportionality
If w = 180 pounds and g = 30 pounds
w = k × g
180 = k × 30
180 = 30k
divide both sides by 30
k = 180/30
k = 6
If w = 54 pounds g = ?
w = k × g
54 = 6 × g
54 = 6g
divide both sides by 6
g = 54/6
g = 9 pounds
Therefore, harsh weighs 9 pounds on the moon.
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