Answers
We have the following conjecture: "The square of an odd number is always odd".
4.1) We select the odd numbers 1, 3, 5 and check the conjecture:
[tex]\begin{gathered} 1^2=1, \\ 3^2=9, \\ 5^2=25. \end{gathered}[/tex]We notice that the results of the squares are odd numbers, so the conjecture is true for these numbers.
4.2) We can't find an example where the conjecture is false. In point 4.4, we will prove that the conjecture is true.
4.3) We agree with the conjecture. We believe that it is TRUE.
4.4) We prove that the conjecture is true for all odd numbers.
By definition, an odd number can be written as:
[tex]m=2n+1.[/tex]Where:
• n is an integer,
,• 2n is an even number because 2n is divisible by 2.
Now, we compute the square of m:
[tex]m^2=(2n+1)^2=(2n+1)\cdot(2n+1)=4n^2+4n+1=2\cdot(2n^2+2n)+1.[/tex]We see that the result is the sum of an even number 2*(2n²+2n) plus 1, so the result is an odd number for every odd number m = 2n + 1. This result proves the conjecture.
The conjecture is TRUE for all odd numbers.
Related Questions
Write the name of the decimal in words.4.79
Answers
four and seventy-nine hundredths
Explanation:[tex]\begin{gathered} \text{Given: 4.79} \\ \\ We\text{ are to write the numbers in words} \end{gathered}[/tex]4 = units
7 = tens
9 = hundredth
Combining the words using the place value:
4 = four
79 = seventy nine hundredths (hundred has th due to the decimal)
4.79 becomes four and seventy nine hundredths
Find the average weekly earnings.Unit rate: $1.15; Units per week: 475, 348, 516,402 (Note: 4 weeks of work)
Answers
Given:
The weekly earning for 4 weeks is,
475, 348, 516,402.
The average of the earning for 4 weeks is,
[tex]\begin{gathered} A=\frac{475+348+516+402}{4} \\ A=\frac{1741}{4} \\ A=435.25 \end{gathered}[/tex]Unit rate is $1.15,
[tex]\begin{gathered} A=1.15\times435.25 \\ A=500.5375 \end{gathered}[/tex]Answer: The average weekly earnings is $500.5375.
An online ticket broker charges a flat service fee of 6.50 per ticket sold you are interested in the total amount of money you must pay for a given number of tickets
Answers
The broker charges $6.50 per ticket, therefore, the total amount will be directly related to the total number of tickets sold.
The two quantities that are changing in the situation are:
[tex]Total\text{ amount, number of tickets.}[/tex]Now, the total amount depends on the number of tickets sold, therefore, if we define
[tex]\begin{gathered} Total\text{ amount=T,} \\ number\text{ of tickets=n,} \end{gathered}[/tex]we can set the following equation:
[tex]T=6.50n.[/tex]Answer:Part a) Total amount, and the number of tickets.
Part b)
[tex]\begin{gathered} Total\text{ amount =T,} \\ number\text{ of tickets=n.} \end{gathered}[/tex]T is the dependant variable, and n is the independent variable.
Part c)
[tex]T=6.50n.[/tex]I just need the answer for number 2 and a simple explanation
Answers
We have to find the area of the regular hexagon with side length of 18 ft.
We can use the formula:
[tex]A=\frac{3\sqrt[]{3}}{2}s^2[/tex]where s is the side length.
Replacing s = 18 ft, we get:
[tex]\begin{gathered} A=\frac{3\sqrt[]{3}\cdot}{2}(18)^2 \\ A\approx2.5981\cdot324 \\ A\approx841.8 \end{gathered}[/tex]Answer: the area of the regular hexagon is 841.8 square feet.
picture of the lesson.
Answers
Using the Pythagorean theorem:
[tex]\begin{gathered} c=\sqrt[]{90^2+90^2} \\ c=\sqrt[]{16200} \\ c=90\sqrt[]{2} \\ c\approx127ft \end{gathered}[/tex]Use this function to predict the amount it will take in 2009 and in 2016 to equal the value of 1 currency unit in 1918
Answers
1) Since we have a Linear Function that models this situation, then we can plug it into that function.
2)
a)
Let's begin by predicting the value of that currency in 2009, i.e. 19 years since 1990:
[tex]\begin{gathered} V(x)=0.4226x+13.4784 \\ V(19)=0.4226(19)+13.4784 \\ V(19)=21.5078\approx21.51 \end{gathered}[/tex]b) Now, in 2016, i.e. 26 years since 1990:
[tex]\begin{gathered} V(26)=0.4226(26)+13.4784 \\ V(26)=24.466\approx24.47 \end{gathered}[/tex]Thus, in 2009 we would need $21.51, and in 2016, $24.47 to equate the value of 1 currency in 1918.
If f(x) = x^2 + 4 and g(x) = V1 - x, what is the value off(g(-3)) ?
Answers
First, substitute -3 into g ( x ) for x
[tex]Hence,g(-3)=\text{ }\sqrt{1\text{ - ( -3 ) }}\text{ = }\sqrt{1\text{ + 3 }}\text{ = }\sqrt{4\text{ }}\text{ = 2 }[/tex]So g ( - 3 ) = 2 , Next, sujbstitute 2 for x in f (x)
[tex]f(g(-3))=f(2)=(2)^2\text{ + 4 = 4 + 4 = 8 }[/tex]Therefore the value of f (g ( -3 ) ) is 8
6. Winston and Larry both added chemicals to their pools. Winston added 6 milliliters of chemicals for every gallon of water. Larry added 8 milliliters of chemicals for every 2 gallons of water. Which statement below is true? Winston used more chemicals per gallon than Larry because 6:1 is greater than 8:2. B. Larry used more chemicals per gallon than Winston because 8:2 is greater than 6:1. C. Winston used more chemicals per gallon than Larry because 6 milliliters is greater than 2 milliliters. D. Larry used more chemicals per gallon than Winston because 8 milliliters is greater than 6 milliliters.
Answers
Winston added 6ml of chemical for every gallon of water
Larry added 8ml chemical for every 2 gallons of water
This means that 4ml chemical was added to 1 gallon of water
Then we can conclude that
If we compare the amount of chemical added to 1 gallon of water, we will discover that
Winton used more chemical per gallon than Larry because ratio wise, 6:1 is greater than
8:2
Hence, Option A is correct
The highest scorer of the women's basketball championship was Jessica Bradley. She scored 146 more points than Tina Harner, her teammate. Together, Bradleyand Harner scored 1592 points. How many points did each player score during the championship?
Answers
Let Tina Harner's point be represented by 'x'.
Also, let Jessica Bradley's point be
[tex]x+146[/tex]Therefore,
[tex]x+x+146=1592[/tex]Evaluate for x
[tex]\begin{gathered} 2x+146=1592 \\ 2x=1592-146 \\ 2x=1446 \\ x=\frac{1446}{2}=723 \\ \therefore x=723points \end{gathered}[/tex]Then, Jessica Bradley's point is
[tex]x+146=723+146=869[/tex]Final answers
[tex]\begin{gathered} Tina\text{ Harner=723points} \\ Jessica\text{ Bradley=869points} \end{gathered}[/tex]Point G is the point (3, -1). Which point is 5 units from point G
Answers
We have
Point G( 3,-1)
Then
Point A is 4 units to left from Point G
Point D is 4 units up, from Point G
and
Point B is 5 units left,from Point G
Then answer is
OPTION B) Point B
A rectangular page is to contain square inches of print. The margins on each side are to be inches. Find the dimensions of the page such that the least amount of paper is used.
Answers
Let l represent the length of the printed rectangular region of the page.
Given that the area of the printed rectangular region is 36, then
width of the printed portion or region = 36/l
The margin left on both sides is 1.5 inches. Thus,
length of page = l + 1.5(2) = l + 3
width of page = 36/l + 1.5(2) = 36/l + 3
Area = length x width
Area = (l + 3)(36/l + 3)
Area = 36 + 3l + 108/l + 9
Area = 36 + 9 + 3l + 108/l
A(l) = 45 + 3(l + 36/l)
We would minimise A(l)
For Amin, A'(l) = 0
3(1 - 36/l^2) = 0
3 = 0 or 1 - 36/l^2 = 0
1 = 36/l^2
l^2 = 36
l = ±√36
l = ±6
Also,
A''(l) > 0
A''(l) = 3(0 - 36(-2)l^-3) = 72/l^3
Substituting l = 6,
72/6^3 > 0
Thus,
l = 6 gives Amin
The dimensions would be
length = l = 6 + 3 = 9
width = 36/6 + 3 = 6 + 3 = 9
Length = 9 inches
width = 9 inches
Find the degree of the polynomial f(x) = x6 + x2 + 5x.
Answers
Given
[tex]f(x)=x^6+x^2+5x[/tex]Find
degree of the polynomial
Explanation
Degree of the polynomial is the highest power of the variable in the equation
Here the highest power of x is 6
hence the degree of the polynomial is 6
Final Answer
degree = 6
option (b) is correct
How do I go about converting the general form 4x^2 + 6y = -4y^2 + 12 to its standard form?
Answers
Answer:
[tex][/tex]Explanation:
Given:
[tex]4x^2+6y=-4y^2+12[/tex]To find:
convert from general to standard form
The standard form of a circle:
[tex](x\text{ - a\rparen}^2\text{ + \lparen y - b\rparen}^2\text{ = r}^2[/tex]To convert the given form to the standard form, we will write x and y into perfect squares using the complete the square:
[tex]\begin{gathered} first\text{ divide through by 4 \lparen we need the coefficient of x}^2\text{ and y}^2\text{ to be 1\rparen} \\ \frac{4x^2}{4}\text{ + }\frac{6y}{4}\text{ = }\frac{-4y^2}{4}\text{ + }\frac{12}{4} \\ \\ x^2\text{ + }\frac{3}{2}y\text{ = -y}^2\text{ + 3} \\ \\ x^2\text{ + y}^2\text{ +}\frac{3}{2}y\text{ = 3} \end{gathered}[/tex][tex]\begin{gathered} for\text{ x, we only have x}^2.\text{ There is no coefficient of x. We can't complete the square} \\ \\ for\text{ y: }y^2\text{ + }\frac{3}{2}y \\ We\text{ will complete the square:} \\ coefficient\text{ of y = 3/2; half the coefficient of y = 3/4} \\ square\text{ of half the coefficient of y = \lparen}\frac{3}{4})\placeholder{⬚}^2 \\ \\ Add\text{ square of half the coefficient of y to both sides of the equation:} \\ x^2\text{ + y}^2\text{ + }\frac{3}{2}y\text{ + \lparen}\frac{3}{4})\placeholder{⬚}^2\text{ = 3 + \lparen}\frac{3}{4})\placeholder{⬚}^2 \\ \\ x^2\text{ + \lparen y + }\frac{3}{4})\placeholder{⬚}^2\text{ = 3 + }\frac{9}{16} \end{gathered}[/tex][tex]\begin{gathered} x^2\text{ + \lparen y + }\frac{3}{4})\placeholder{⬚}^2\text{ = }\frac{3(16)\text{ + 9}}{16} \\ \\ x^2\text{ + \lparen y + }\frac{3}{4})\placeholder{⬚}^2\text{ = }\frac{57}{16} \end{gathered}[/tex]Consider parallelogram ABCD below. Use the information given in the figure to find x, measure of EDA, and measure of EAD
Answers
In a parallelogram the diagonals bisct each other. Then:
[tex]14=4x-2[/tex]Use the equation above to find the value of x:
[tex]\begin{gathered} 14+2=4x-2 \\ 16=4x \\ \frac{16}{4}=x \\ \\ 4=x \end{gathered}[/tex]To find the meaure of the angles you need to know the next about the angles formed by the diagonals in a parallelogram:
Angles in blue are congruent
Angles in green are congruent
[tex]\begin{gathered} m\angle EDA=45º \\ m\angle EAD=57º \end{gathered}[/tex]If the ratio of tourists to locals is 3:4 and there are 120 tourists at the opening of a new muesum, how many locals are in attendance?
Answers
Let the number of locals be "x".
From the information given, we can write two ratios and equate:
[tex]\frac{3}{4}=\frac{120}{x}[/tex]We can easily solve for "x" by cross multiplication >>>
[tex]\begin{gathered} \frac{3}{4}=\frac{120}{x} \\ 3x=4\times120 \\ 3x=480 \\ x=160 \end{gathered}[/tex]So,
The
What is the extended ratio relating to the side lengths of a 30-60-90 triangle?A. x: x√2: 3xB. x: x√3 : 2xC. x: x: x√3D. x: x: x√2
Answers
The side length of a 30-60-90 triangle follows the ratio:
[tex]x:x\sqrt{3}:2x[/tex]See the illustration below.
Hence, the answer is Option B.
Which digit is in the hundredths place?6.537O A. 5B. 6C. 3D. 7
Answers
We can get the place value of all the digits of the number 6.537 as shown in the diagram below
The hundredth
Find the smallest distinct positive numbers that provide a counterexample to show the statement is false,The sum of any two different odd numbers is divisible by 4.The counterexample isХContinue© 2021 McGraw HN LLC. All Rights Reserved. Terms of UseType here to searchooFIL
Answers
The counter example is 1 + 27
EXPLANATION
The sum of two different odd numbers is divisble by 4.
The above statement is not always true. The counterexample that will disprove this statement is:
Two odd numbers such as ;
1 is an odd number and 27 is an odd number
1 + 27 = 28
28 is not divisible by 4.
Hence, the counter example is 1 + 27
Area and perimeter of polynomialsFind the area and perimeter of the closet with the given dimensions:
Answers
Answer:
Area = (10y²-27y+5) square units
Perimeter = (14y-12) units
Explanation:
The dimensions of the closet are:
• 5y-1
,• 2y-5
Area
[tex]\begin{gathered} Area=Length\times Width \\ =(5y-1)(2y-5) \\ =5y(2y-5)-1(2y-5) \\ =10y^2-25y-2y+5 \\ =10y^2-27y+5 \end{gathered}[/tex]The area of the closet is (10y²-27y+5) square units.
Perimeter
[tex]\begin{gathered} Perimeter=2(Length+Width) \\ =2(5y-1+2y-5) \\ =2(5y+2y-1-5) \\ =2(7y-6) \\ Perimeter=14y-12 \end{gathered}[/tex]The perimeter of the closet is (14y-12) units.
A runner estimated that he ran about 12 miles. Select all of the rates and times that the runner could have run, 6.NS.3 3.3 miles per hour for 3.8 hours 6.1 miles per hour for 1.9 hours 5.8 miles per hour for 2.3 hours 0 2.75 miles per hour for 4.4 hours Course 1. Chapter 3 . Compute with
Answers
We get that
[tex]2.75\cdot4.4=12.1[/tex]so the correct option is D
the radius of a circle is 1 what is the length of an arc that subtends an angle of pi/4 radians.
Answers
In order to calculate the length of this arc, we can use a rule of three, knowing that the complete circle has an angle of 2π and a length of 2πr:
[tex]\begin{gathered} \text{angle }\to length \\ 2\pi\to2\pi r \\ \frac{\pi}{4}\to x \\ \\ x\cdot2\pi=\frac{\pi}{4}\cdot2\pi r \\ x=\frac{\pi}{4}\cdot r \\ x=\frac{\pi}{4} \end{gathered}[/tex]So the length of the arc is π/4.
Santa received 7/10 of the profits on the goods he sold. He gave 5/6 of what he received to Mrs. Claus, and she gave 1/7 of that amount to the elf. What fraction of the total profits did the elf receive?
Answers
Given:
Amount of profit received = 7/10
Amount he gave to Mrs. Claus = 5/6
Amount she gave to the elf = 1/7 of the amount Mrs. Claus received.
Let's find the fraction of total profits the elf received.
Let's first find the fraction of profit Mrs. Claus received.
We have:
[tex]\begin{gathered} \frac{5}{6}*\frac{7}{10} \\ \\ =\frac{5*7}{6*10} \\ \\ =\frac{35}{60} \\ \\ =\frac{7}{12} \end{gathered}[/tex]Now, since the elf received 1/7 of the amount Mrs. Claus received, the total of profits the elf received will be:
[tex]\begin{gathered} \frac{1}{7}*\frac{7}{12} \\ \\ =\frac{1*7}{7*12} \\ \\ =\frac{1}{12} \end{gathered}[/tex]Therefore, the fraction of the total profits the elf received is 1/12.
ANSWER:
[tex]\frac{1}{12}[/tex]If f(x) = -6x + 6, then f(x) =
Answers
Answer:
(6-x)/6
Step-by-step explanation:
We are given the following function:
f(x) = -6x + 6.
This function can also be written as:
y = -6x + 6
Finding the inverse:
We switch y with x, then isolate y. So
[tex]x=-6y+6[/tex][tex]6y=6-x[/tex][tex]y=\frac{6-x}{6}[/tex]The inverse function is:
[tex]f^{-1^{}}(x)=\frac{6-x}{6}[/tex]The answer is (6-x)/6
Nick is 100 miles away from Isaac. They are traveling towards each other. If I Isaac travels 6 mph faster than Nick and they meet after 5 hours, how fast was each traveling?Nick is traveling _____ mphIsaac is traveling ______ mph
Answers
Consider the following picture.
Lets say that Nick is on the left side and Isaac
Mia is at Arby's and is getting her drink. She decides to go crazy andmix Dr. Pepper, Sprite, and Tea together. What is the sample space for the probability of any certain order?
Answers
from the question;
Mia is at Arby's and is getting her drink. She decides to go crazy and
mix Dr. Pepper, Sprite, and Tea together.
from this our sample space will be;
[tex]\lbrace DST,DTS,STD,SDT,TDS,TSD\rbrace[/tex]Where D = Dr. Pepper
S = sprite
T = tea
and this can be combined in different order
The correct opition is A
mike bought four new baseball cards to add to his collection the next day his dog ate half of the collection. there are twenty -nine cards left. How many did he start with?
Answers
Since the dog ate half of the collection, and 29 remained, we deduce that twice the value of remaining cards must be the value of the total collection with the 4 new cards. Simply multiply 29 by 2 and subtract four from the previous result and this is the answer.
29*2= 58
58 - 4 = 54
Answer: He started with 54 cards.
27–34: Describing Distributions. Consider the following distributions.-How many peaks would you expect the distribution to have? Explain.-Make a sketch of the distribution.The annual snowfall amounts in 50 randomly selected American cities
Answers
The number of expected peaks = 1
because the cities are not on the same time zone
the sketch will be as shown at the previous image
it will be similar to that with 50 rectangles represents 50 cities with one peak and different height
Good luck
Which group of side lengths will NOT create a triangle? 7, 6, 10 4, 2, 7 3, 4, 5 3, 3,3
Answers
Triangle inequality
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side.
Let's analyze each group of side lengths:
• 7,6,10
Note that:
7+6 > 10
7+10 > 6
6+10 > 7
Since all the conditions are met, this group can create a triangle
• Now for 4,2,7
Testing each sum:
4+2 > 7
This inequality is false since 6 is not greater than 7
Thus, this group cannot create a triangle
• Now for 3,4,5
Testing each sum:
3 + 4 > 5
3 + 5 > 4
4 + 5 > 3
Since all the conditions are met, this group can create a triangle
• Finally, for 3,3,3
Testing each sum:
3 + 3 > 3
This happens three times. They form in fact an equilateral triangle
Thus, the only group that cannot create a triangle is 4,2,7
can you help me solve thea. domainb. rangec.f(-3)d. the values of x for which f(x) =-2e. the point where the graph of f crosses the x-axisf. the point where the graph of f crosses the y-axisg. values of x for which f(x) < 0h. Is f(-8) positive or negative?
Answers
We are given a function and his graph and we are asked the following:
a. The domain of the function: The domain is the values of "x" that the function takes, in this case, we can represent that mathematically as follows:
[tex]D(f(x))=\lbrace x\parallel-2\le x\le6\rbrace[/tex]b. The range of the function: the range is the values the dependent variable takes, in this case, the values of "y", we can represent this as follows:
[tex]R(f(x))=\lbrace y\parallel-2\le y\le6\rbrace[/tex]c. f(-3) is the value the function takes when x=-3, from the graph we can see that this value is outside of the domain, therefore, it is undefined for this function
d. the values of x for which f(x) = -2, we can see from the graph that the only value of x that yields -2 is x = 6
e. the point where the graph of f crosses the x-axis. We can see from the graph that the function crosses the x-axis at x = 4
f. the point where the graph of f crosses the y-axis. We can see from the graph that the function crosses the y-axis ath y = 4
g. values of x for which f(x) < 0. From the graph we can look for the values of "x" where "y" is a negative number, those values can be represented mathematically as follows:
[tex]4It should be noted that the value at x =4 is not included, since the function yields zero at that point
h. Is f(-8) positive or negative?: The point x=-8 is outside the domain of the equation, therefore is undefined
The figure shows the height, h, of a thrown ballas a function of time. Height is measured in feetand time is measured in seconds. How long willit take for the ball to return to the ground?4030Height in Feet20101 2 3 4 5Time in Seconds
Answers
Answer:
C. 5. 0 seconds
Explanation:
From the graph given, we see that the ball returns to the ground after 5 seconds (by returning to the ground we mean the height = 0 again).
Therefore, the correct answer choice is C.