The number of apples x will be 10 and the number of bananas y will be 4.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that Ethan and his children went into a grocery store and he bought $6 worth of apples and bananas. Each apple costs $0.75 and each banana costs $0.30. He bought a total of 14 apples and bananas altogether.
The number of apples and bananas will be calculated as,
0.75x +0.30y = 6
x + y = 14 or x = 14 - y
0.75(14-y) = 0.30y = 6
10.5-0.75y + 0.30y = 6
y = 10
x = 14 - y
x = 14 - 10 = 4
Therefore, the number of apples x will be 10 and the number of bananas y will be 4.
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NEED HELP NOW Convert 110 kilograms to pounds. (1 kilogram = 2.204 pounds)
The required answer in the given problem, given weight of 5 kilograms and convert it to 11.02 pounds. By multiplying the given weight by the conversion factor of 2.204 pounds/kilogram, obtain the weight in pounds.
Conversion from kilograms to pounds:
To convert a weight value from kilograms to pounds, we use the conversion factor of 1 kilogram equals 2.204 pounds. This means that for every 1 kilogram, there are 2.204 pounds. To convert a weight from kilograms to pounds, we multiply the given value in kilograms by the conversion factor.
To convert 110 kilograms to pounds, we can use the conversion factor that states 1 kilogram is equal to 2.204 pounds. Here's the step-by-step process:
Step 1: Write down the given weight in kilograms, which is 110 kilograms.
Step 2: Multiply the given weight by the conversion factor:
110 kilograms x 2.204 pounds/kilogram
Step 3: Perform the multiplication:
110 x 2.204 = 242.44 pounds
Therefore, 110 kilograms is equal to 242.44 pounds.
In the given problem, given weight of 5 kilograms and convert it to 11.02 pounds. By multiplying the given weight by the conversion factor of 2.204 pounds/kilogram, obtain the weight in pounds.
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Use the formula A=P(1+r/n)ⁿt to solve the compound interest problem
Find how long it takes for $ 900 to double if it is invested at 6 % interest compounded monthly.
The money will double in value in approximately years.
Answer:
11.6 years
Step-by-step explanation:
1800 = 900 x (1 + 0.06/12)^ 12t
1800 = 900 x (1 + 0.005)^12t
1800 = 900 x (1.005)^12t
1800 = 900 x 1.061678^t
1800/900 = (900 x 1.061678^t)/ 900
2 = 1.061678^t
log(1.061678) 2 = t
t = 11.5813
t = 11.6
Determine if the events are independent or dependent.
P(A) = 0.65 P(B) = 0.25, P(A and B)0.40
P(A) = 0.22 P(B) = 0.45, P(A and B) .0.099
P(A) = 0.42 P(B) = 0.62, P(A| B) =
0.2604
You roll a pair of dice at the same time.
You choose two marbles from a bag, without replacement.
The events P(A) = 0.22 P(B) = 0.45, P(A and B) = 0.099 and rolling a pair of dice at the same time are independent events. Others are dependent events.
What is Independent and Dependent Events?Independent events are events for which the occurrence of this event does not depend on any other events.
Dependent events are those events for which it's occurrence depends on other events.
Two events A and B are independent, if
P(A | B) = P(A)
P(A and B) = P(A) × P(B)
P(A) = 0.65, P(B) = 0.25 and P(A and B) = 0.40
P(A) × P(B) = 0.65 × 0.25 = 0.1625 ≠ P(A and B)
∴ A and B are dependent events.
P(A) = 0.22 P(B) = 0.45 and P(A and B) = 0.099
P(A) × P(B) = 0.22 × 0.45 = 0.099 = P(A and B)
∴ A and B are independent events.
P(A) = 0.42 P(B) = 0.62 and P(A| B) = 0.2604
P(A| B) = 0.2604 ≠ P(A)
∴ A and B are dependent events.
If you roll a pair of dice at the same time, probability for any number to occur on dice 1 does not affect the probability of any number to occur on dice 2.
∴ A and B are independent events.
If you choose two marbles from a bag, when the first marble is chosen, then the second marble you choose will not be the first marble in any case. So second event depends on the first event.
∴ A and B are dependent events.
Hence, as P(A and B) = P(A) × P(B) satisfies for the events P(A) = 0.22 P(B) = 0.45, P(A and B) = 0.099, is is independent events. Also rolling a pair of dice at the same time is also independent events. The rest of the events are dependent events as it does not satisfies the condition for independent events.
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Write a quadratic function to model the graph to the right.
f(x)=____
f(x) = x² + 2x + 2
Solution:
Given vertex (1,1)
a = 1
Formula of quadratic equation is :
y = a(x + b)² + c
where, a =1, x=x, b = x1, c = y1
Substituting the values in the equation, we get
y = x² + 2x + 2
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A quadratic function to model the graph to the right is f(x) = x² + 2x + 2
What is a quadratic equation?
A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.
Solution:
Given vertex (1,1)
a = 1
Formula of quadratic equation is :
y = a(x + b)² + c
where, a =1, x=x, b = x1, c = y1
Substituting the values in the equation, we get
y = x² + 2x + 2
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ABDE is an isosceles trapezoid. Select all parts that are marked congruent.
Side AB and Side AE
Angle E and Angle D
Angle E and Angle B
Angle A and Angle B
Angle A and Angle D
Side AE and Side BD
Side AB and Side ED
The concurrent parts in the given ABDE are Side AB ≅ Side ED and Side AE ≅ Side BD.
Concurrent triangles:Two triangles are said to be congruent if all three corresponding sides and all three corresponding angles have the same size. These triangles may be moved, flipped, twisted, and turned to get a similar effect. When relocated, they are parallel to one another. The concurrence sign is '≅'.
Here we have,
ABDE is an isosceles trapezoid.
Draw 2 diagonals in a given trapezoid
Then the 2 diagonal will divide the trapezoid into 4 triangles
Here ΔAOB, ΔBOD, ΔEOD, and AOE
From the formed triangles
ΔAOB and ΔEOD will be two concurrent triangles
Then Side AB ≅ Side ED
ΔBOD and AOE will be two concurrent triangles
Side AE ≅ Side BD
Therefore,
The concurrent parts in the given ABDE are Side AB ≅ Side ED and Side AE ≅ Side BD.
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if you form a triangle from three given side lengths, will you always get one triangle or more than one triangle?
If the three side lengths are all different then you will only get one triangle. However, if two or more of the side lengths are the same then you can get more than one triangle.
What is known as a triangle?A triangle is a polygon with three vertices and three sides. The angles of the triangle are formed by the connection of the three sides end to end at a point. The triangle's three angles add up to 180 degrees in total
How is a triangle made?Three diagonals must meet at three different locations to form a triangle. There are five possible arrangements of three diagonals. We categorize them according on how many different diagonal endpoints there are. The quantity of triangles with 3, 4, and 5 endpoints will be counted directly.
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A password is required to log into a system that only recognizes numbers and letters
of the alphabet (not case sensitive). You want to create a password using 7 numbers
or letters, but cannot reuse any of them.
How many different passwords are possible?
There are total 42,072,307,200 different passwords are possible .
What are combinations ?
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order.
Combinations can be confused with permutations. However, in permutations, the order of the selected items is essential. For example, the arrangements ab and ba are equal in combinations (considered as one arrangement), while in permutations, the arrangements are different.
Given: A password is required to log into a system that only recognizes numbers and letters of the alphabet
Since , there are total 26 letters and 10 numbers are possible that can be used to create a password . Hence , there are 36 character in total to be used.
Also , repetition of character is not allowed. Therefore , the chance of selecting every character decreases by 1 in comparison to its preceding character .
Chance of selecting first character = 36
Chance of selecting second character = 35
Similarly for seventh = 30
Thus , different passwords that are possible using 7 numbers or letters, without reusing any of them = 36*35*34*33*32*31*30 = 42,072,307,200 .
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3. For the graph above, what is the approximate y-value of the point of intersection?
0-1
04
03
02
Answer: 02
Step-by-step explanation: w
Valentino's Pizzeria sells four types of pizza crust. Last week, the owner tracked the number sold of each type, and this is what she found. Type of Crust Number Sold Thin crust 293 Thick crust 216 Stuffed crust 174 Pan style 195 Based on this information, of the next 3000 pizzas she sells, how many should she expect to be thin crust? Round your answer to the nearest whole number. Do not round any intermediate calculations. pizzas X
Answer:
1120
Step-by-step explanation:
Given the information below:
The Relative Frequency of selling a pan style pizza =
Therefore, if Valentino's Pizzeria were to sell 4000 pizzas, the expected number of Pan style pizza sold will be:
Number of Pan Style Pizza Sold = Relative Frequency of pan Style Pizza Purchase X Total Number of Sales
They should expect to sell 1120 pan Style pizzas.
Help me with geometry
Answer:
34
Step-by-step explanation:
This is somewhat hard to explain for me. I attached a picture/diagram to help.
They are congruent angles.
Angle N, inside the MNP triangle is 34. Which means the adjacent angle is 56.
The interior angles of a triangle add up to 180.
So to find angle M in triangle LNM, we can do 180 - 90 - 56.
Therefore the m<LMN = 34
Which diagram shows a pair of angle measures that prove lines a and b are parallel?
Answer:
c
...................
Answer:
the answer is the option C
There are five questions listed below. Each question includes the quantity 22. Match the 22 in each question on the left to which part of the problem it represents on the right -- the base, percent, or amount. Some answer options on the right will be used more than once.
Percent is from the Latin adverbial phrase per centum meaning “by the hundred.”
What is percentage?In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.percentage, a relative value indicating hundredth parts of any quantity. One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entirety and 200 percent specifies twice the given quantity.Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. The formula used to calculate percentage is: (value/total value)×100%.Using concepts of percentage,
3 is 16.63% of 22.
22 is 13.41% of 164.
0.88 is 4% of 22.
8 is 22% of 36.36.
1.1 is 5% of 22.
A percentage is a number or ratio expressed as a fraction of 100.
Let 3 be x% of 22.
=x/100*22=3
=x=3*100/22=13.63
3 is 13.63% of 22.
Let 22 be y % of 164.
=y/100*164=22
=y=22*100/164=13.41
13.41% of 164 is 22.
4% of 22 = 0.88
=4/100*22=0.88
Let 8 is 22% of z.
=22/100*z=8
=8*100/22=36.36
5% of 22 = 1.1
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Question 12 of 25
Anthony surveys a group of students at his school about whether they play a
sport. This table shows the results broken down by gender.
Boys
Girls
Total
Play a sport
95
76
171
Do not play a
sport
45
59
104
Total
overe pot independent because Nair) 0.10
140
135
275
Are being a girl and playing a sport independent events? Why or why not?
Being a female and participating in sports are independent events for the provided data because P(girl) = P(girl participates in sports)
What are independent events?These are the occurrences that are independent of other occurrences.
Occurrences A and B are independent events when the likelihood of one occurring is unaffected by the occurrence of the other.
Probability of independent events formula:
"P(A∩B) = if A and B are independent occurrences P(A).P(B)
What exactly is probability?Finding the chance that an event will occur is the focus of this area of mathematics."
P(A) = n(A)/n(S), where n(A) is the number of positive outcomes and n(S) is the total number of possible outcomes for an event A.
For given question,
We have been given a table that shows the results broken down by gender.
Play sports Do not play sports Total
Boy 95 45 140
Girl 76 59 135
Total 171 104 275
Using the definition of probability:
⇒ P(girl) = Number of girls/ Number of students
⇒ P(girl) = 135/275
⇒ P(girl) = 0.49 ....................(1)
Now we find the probability of girls that play a sport.
Using the definition of probability:
P(girl plays a sport)
= Number of girls who plays a sport/ number of students who plays a sport
= 76/ 171
= 0.44 ......................(2)
From (1) and (2),
P(girl) = P(girl plays a sport)
Thus, being a girl and playing a sport are Independent events.
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Book b is twice as long as book a. book c is 30 pages longer then book b. together the three books have 280 pages.
I just need help with question number 10
It has to be solved algebraically Can someone please help me with this problem
Answer:
book c has 130 pages
Step-by-step explanation:
let book a = x pages
book b = 2x
book c = 2x + 30
according to the question:
x + 2x +2x + 30 = 280
5x + 30 = 280
5x = 280-30
5x = 250
x = 250/5
x = 50
thus book c has 2(50) + 30 = 130 pages
What is the solution to 9|x – 8| < 36?
4 < x < 12
x < –4 or x > 12
x > –12 or x < 8
–4 < x < 8
Answer:
4 < x < 12
Step-by-step explanation:
x - 8 ≥ 0
x ≥ 8
If x ≥ 8
9(x-8) < 36
9x - 72 < 36
9x < 36 + 72
9x < 108
9/9 x < 108/9
x < 12
8 ≤ x < 12
if x < 8
9(-x+8) < 36
-9x + 72 < 36
-9x < 36 - 72
-9x < -36
-9/-9 x > -36/-9
x > 4
4 < x < 8
4<x<12
Expand the log completely
Answer:
1/3log₂(x) +1/3log₂(y) +1/3log₂(z)
Step-by-step explanation:
You want the complete expansion of ...
[tex]\log_2{\sqrt[3]{x\cdot y\cdot z\vphantom{b}}}[/tex]
Relevant rulesThe rules of logs and exponents relevant to this expansion are ...
[tex]\log_b{(xy)}=\log_b{(x)}+\log_b{(y)}\\\\\log_b{(x^a)}=a\cdot\log_b{(x)}\\\\\sqrt[n]{\vphantom{b}a}=a^{\frac{1}{n}}[/tex]
ExpansionApplying these rules to the given expression, we get ...
[tex]\log_2{\sqrt[3]{x\cdot y\cdot z\vphantom{b}}}=\log_2{((xyz)^{\frac{1}{3}})}=\dfrac{1}{3}\log_2{(xyz)}\\\\=\boxed{\dfrac{1}{3}\log_2{(x)}+\dfrac{1}{3}\log_2{(y)}+\dfrac{1}{3}\log_2{(z)}}[/tex]
Equation that passes through 4,-3 and -4,7
Target charges $4.40 for 8 sodas. Walmart charges $7.08 for 12 sodas. Which store has
the best unit price? How much cheaper is it?
Answer:
Target
Step-by-step explanation:
$4.40/8 is 0.55 and $7.08/12 is 0.59
The doubling period of a bacterial population is 15 minutes. At time t=120 minutes, the bacterial population was 90000.
What was the initial population at time t=0 ?
Find the size of the bacterial population after 3 hours.
The doubling period of a bacterial population is 15 minutes. the initial population at time t=0 is 351.56. the size of the bacterial population after 3 hours is:359,997.44.
How to find the initial population?y = total amount of bacteria
a = initial bacterial at time 0
b = growth factor
t=120 minutes =15 minutes 8 times
So,
t=8
90000 = a (2)^8
90000 = a (256)
90000/256 = a
351.56 = a (initial population)
Equation:
y = 351.56 (2)t
After 3 hours (15 minutes 10 times), or at t = 10
y = 351.56 (2)^10
y= 359,997.44
Therefore the initial population at time t=0 is 351.56. the size of the bacterial population after 3 hours is:359,997.44.
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f(x) = 2(x²-7)
Find f(2).
Suppose that you’d like to retire in 40 years and you want to have a future value of $ 500000 in a savings account. Also suppose that your employer makes regular monthly payments into your retirement account.
If you can expect an APR of 7% for your account, how much do you need your employer to deposit each month?
Employer Contribution =
The formulas we have been using assume that the interest rate is constant over the period in question. Over a period of 40 years, though, interest rates can vary widely. To see what difference the interest rate can make, let’s assume a constant APR of 4% for your retirement account. How much do you need your employer to deposit each month under this assumption?
Employer Contribution =
a) If only your employer makes the monthly deposits into your retirement account for the future value of $500,000 in 40 years at 7% APR, your employer contributions should be $190.49.
b) If only your employer makes the monthly deposits into your retirement account for the future value of $500,000 in 40 years at 4% APR, your employer contributions should be $423.03.
What is the future value?The future value represents the amount expected to be the balance in the investment account after a period.
The future value can be used to determine the periodic payments required to meet the target balance, using an online finance calculator.
Periodic Deposits at 7%:N (# of periods) = 480 months (40 years x 12)
I/Y (Interest per year) = 7%
PV (Present Value) = $0
FV (Future Value) = $500,000
Results:
PMT (Periodic deposits) = $190.49
Sum of all periodic payments = $91,435.07
Total Interest = $408,564.93
Periodic Deposits at 4%:N (# of periods) = 480 (40 years x 12)
I/Y (Interest per year) = 4%
PV (Present Value) = $0
FV (Future Value) = $500,000
Results:
PMT (Periodic deposits) = $423.03
Sum of all periodic payments = $203,052.33
Total Interest = $296,947.67
Thus, the periodic deposits to meet the future value will depend on the interest rate.
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5.6.8 let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e). show that this need not be true if f is continuous but not uniformly continuous.
Proof. (1): To prove {f(xn)} is a Cauchy sequence just need to prove ∀ > 0, ∃N, s.t.,∀n, m > N,
have |f(xn) − f(xm)| < . Since f is uniformly continuous on set E, thus ∀ > 0, ∃δ > 0, s.t.,
∀x, y ∈ E, if |x − y| < δ,then |f(x) − f(y)| < .as {xn} is a Cauchy sequence, then ∃N, s.t.,
∀n, m > N, |xn − xm| < δ, thus |f(xn) − f(xm)| < which proves {f(xn)} is a Cauchy sequence.
(2): for example f(x) = 1
x
, x ∈ (0, 2) which is continuous but not uniformly continuous. {
1
n
} is
a Cauchy sequence, however, {f(xn)} does not converge which proves that it is not a Cauchy
sequence.
Showing that this need not be true if f is continuous but not uniformly continuous.
Given :
let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e).
( 1 )
If f is uniformly continuous on E, then given ε > 0 there is δ > 0 such that if x, y are in E and |x−y| < δ,
then |f(x) − f(y)| < ε. Let (xn) be a Cauchy sequence in E. Then given δ > 0 there is N such that if p, q > N,
then |xp − xq| < δ, and thus |f(xp) − f(xq)| < ε, implying that (f(xn)) is a Cauchy sequence.
( 2 )
Let E = {1, 1/2, 1/3, · · } and f(1/n) = 1 is n is odd, f(1/n) = −1 if
n is even. Then f is continuous but not uniformly continuous. The sequence (xn) = (1/n) in E is Cauchy but the
sequence (f(xn)) = (1, −1, 1, −1, · · ·) is not Cauchy.
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A waterproof speaker sells for $179, which is marked up by 30% of the selling price. What is the cost of the speaker? A. $152.30 B. $132.50 C. $123.50 D. $125.30
The original cost of the speaker will be equal to $125.3. The correct option is D.
What is the percentage?The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.
Given that a waterproof speaker sells for $179, which is marked up by 30% of the selling price.
The cost price of the speaker will be calculated as,
Cost price = 179 x 0.7
Cost price = $125.3
Therefore, the original cost of the speaker will be equal to $125.3. The correct option is D.
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A new computer game costs $160,000 to research and develop. Once completed, individual games can be produced for just $.20 each. If the first 275 are given away as samples, how many games must be sold for the average price to be $2?
Answer: The correct answer is (d) 89,200 games.
To find the number of games that must be sold, we need to set up an equation to represent the total revenue from the game sales and solve for the number of games sold.
The total revenue from the game sales is equal to the number of games sold multiplied by the price per game, minus the cost of research and development.
Let X be the number of games sold
Total revenue = X * $2 - $160,000
We also know that the total cost of producing the games is equal to the number of games produced multiplied by the cost per game.
Total cost = X * $0.20
We can set the total revenue equal to the total cost and solve for X:
X * $2 - $160,000 = X * $0.20
X = 89,195
Since we gave away the first 275 games for free, we need to add 275 to the number of games sold to find the total number of games that must be sold.
Total number of games sold = X + 275 = 89,195 + 275 = 89,470
Therefore, the correct answer is (d) 89,200 games.
Two machines produce sound pressure levels of approximately 80 dB(A) each. What is their approximate combined sound pressure level?
83.01dB is approximate combined sound pressure level.
What is sound pressure level?
The result of the pressure changes in the air caused by sound waves is sound pressure, often known as sound pressure level. The hearing threshold is the lowest sound pressure that a human can hear, and the pain threshold is the maximum sound pressure that a person can tolerate.
The difference between sound pressure at the pain threshold and the hearing threshold is a million times higher.
We have sound pressure level denoted as (Lp) of each machine
i.e 80 dB,(given).
Therefore, we can calulate sound pressure level from both machines combined by given formula,
SPL = 10 * Log(10^(SPL(1) /10) + 10^(SPL(2) /10) +10^(SPL(3) /10) + ............)
where,
SPL= sound pressure level(dB)
=> SPL = 10 * Log(10^(80 /10) + 10^(80 /10))
=> SPL = 10 * Log(10^8 + 10^8)
=> SPL = 10 * 8.301
=> SPL = 83.01dB
Hence, 83.01dB is approximate combined sound pressure level.
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A store is instructed by corporate headquarters to put a markup of 11% on all items. An item costing 18$ is displayed by the store manager at a selling price of 2$. As an employee, you notice that this selling price is incorrect. Find the correct selling price. What was the manager's likely error?
if the markup is 11%, well, we simply add that to the cost of the item, in this case 18.
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{11\% of 18}}{\left( \cfrac{11}{100} \right)18}\implies 1.98~\hfill \underset{selling~price}{\stackrel{18 ~~ + ~~1.98}{\text{\LARGE 19.98}}}[/tex]
B OA) A Which statements are necessary to prove the two triangles are congruent by SSS? O B) D =CD, AB = C, mzBDA = 90° AD, BD = CD, mzBDA = 90° O a OD) AD AD, BD = CD, AB = AC BD = CD, AB= AC
The statements that are necessary to prove the two triangles are congruent by SSS are AD≅ AD, BD ≅ CD, and AB ≅ AC.
What is congruence?
In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.
Here,
To prove that two triangles are congruent using the SSS Congruence criterion, we must establish that all three sides of one triangle are equal to all the corresponding sides of the other.
Thus,
Side AD in ∆ADB is congruent to side AD in ∆ADC.
Side BD in ∆ADB is congruent to side CD in ∆ADC.
Side AB in ∆ADB is congruent to side AC in ∆ADC.
Hence, based on the SSS congruence criterion, it is concluded that both triangles are congruent. i.e AD≅ AD, BD ≅ CD, AB ≅ AC.
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Answer:
D
Step-by-step explanation:
To prove that two triangles are congruent using the SSS Congruence criterion, we must establish that all three sides of one triangle are equal to all the corresponding sides of the other.Thus,Side AD in ∆ADB is congruent to side AD in ∆ADC.Side BD in ∆ADB is congruent to side CD in ∆ADC.Side AB in ∆ADB is congruent to side AC in ∆ADC.Therefore, based on the SSS congruence criterion, it is concluded that both triangles are congruent.The statements necessary to prove that they are congruent by SSS is: D
Need an answer pls will mark Brainliest
Answer:
4 m^3 - 38 m^2 + 57 m - 72 = (4 m^2 - 6 m + 9)(m - 8) + 0
Step-by-step explanation:
Compute (4 m^3 - 38 m^2 + 57 m - 72) ÷ (m - 8) using polynomial long division:
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
To eliminate the leading term of the numerator, 4 m^3, multiply m - 8 by 4 m^2 to get 4 m^3 - 32 m^2. Write 4 m^2 on top of the division bracket and subtract 4 m^3 - 32 m^2 from 4 m^3 - 38 m^2 + 57 m - 72 to get -6 m^2 + 57 m - 72:
| | | 4 m^2 | | | |
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
| -(4 m^3 | - | 32 m^2) | | | |
| | | -6 m^2 | + | 57 m | - | 72
To eliminate the leading term of the remainder of the previous step, -6 m^2, multiply m - 8 by -6 m to get -6 m^2 + 48 m. Write -6 m on top of the division bracket and subtract -6 m^2 + 48 m from -6 m^2 + 57 m - 72 to get 9 m - 72:
| | | 4 m^2 | - | 6 m | |
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
| -(4 m^3 | - | 32 m^2) | | | |
| | | -6 m^2 | + | 57 m | - | 72
| | | -(-6 m^2 | + | 48 m) | |
| | | | | 9 m | - | 72
To eliminate the leading term of the remainder of the previous step, 9 m, multiply m - 8 by 9 to get 9 m - 72. Write 9 on top of the division bracket and subtract 9 m - 72 from 9 m - 72 to get 0:
| | | 4 m^2 | - | 6 m | + | 9
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
| -(4 m^3 | - | 32 m^2) | | | |
| | | -6 m^2 | + | 57 m | - | 72
| | | -(-6 m^2 | + | 48 m) | |
| | | | | 9 m | - | 72
| | | | | -(9 m | - | 72)
| | | | | | | 0
The quotient of (4 m^3 - 38 m^2 + 57 m - 72)/(m - 8) is the sum of the terms on top of the division bracket. Since the final subtraction step resulted in zero, m - 8 exactly divides 4 m^3 - 38 m^2 + 57 m - 72, and there is no remainder.
| | | 4 m^2 | - | 6 m | + | 9 | (quotient)
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72 |
| -(4 m^3 | - | 32 m^2) | | | | |
| | | -6 m^2 | + | 57 m | - | 72 |
| | | -(-6 m^2 | + | 48 m) | | |
| | | | | 9 m | - | 72 |
| | | | | -(9 m | - | 72) |
| | | | | | | 0 | (remainder) invisible comma
(4 m^3 - 38 m^2 + 57 m - 72)/(m - 8) = (4 m^2 - 6 m + 9) + 0
Write the result in quotient and remainder form:
Answer: 4 m^3 - 38 m^2 + 57 m - 72 = (4 m^2 - 6 m + 9)(m - 8) + 0
Determine whether the expressions are equivalent.
0.4(k+3) is to 0.4 k+1.2
Equivalent. 0.4(k+3)=0.4 k+1.2
Not equivalent. 0.4(k+3)=0.4 k+0.8
Equivalent. 0.4(k+3)=0.4 k+0.4
Not equivalent. 0.4(k+3)=0.4 k+0.4
Both the expressions are equivalent to each other.
What do you mean by algebraic expression?
The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added before a variable and then multiplied by it.
Given expression:
LHS
0.4(k+3)
= 0.4k + 0.4 × 3
= 0.4k + 1.2
RHS
0.4 k+1.2
LHS = RHS
Therefore, both the expressions are equivalent to each other.
To learn more about the algebraic expression from the given link.
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Does it make sense to discuss the similitude between 2 rombii, 2 squares, 2 rectangles, 2 parralelegrams, and 2 conguent polygons
Yes, it makes sense to discuss the similarity between these shapes.
How to determine if it make sense to discuss the similitude between 2 rombii, 2 squares?Two shapes are considered similar if they have the same shape, but possibly different sizes. This means that the angles of the shapes are the same and the sides of the shapes are in the same ratio.
Two rhombi are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two squares are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two rectangles are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two parallelograms are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two congruent polygons are similar by definition, since congruent shapes are identical in size and shape.
It is important to note that similarity is different from congruence. Congruent shapes are identical in size and shape, whereas similar shapes have the same shape but possibly different sizes.
Learn more about conguent polygons on:
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