Since the longest side cannot be longer than the total of the other two sides, the third side's potential lengths range from 8 to 44.
A triangle's third side must be longer than the difference between the other two sides' lengths and shorter than the total length of the other two sides. The two side lengths in this situation are 18 and 26. The third side must be greater than 8 because the difference between them is 8. The third side must be less than 44 because the total length of the first two sides equals 44. Therefore, the third side length might be any side length between 8 and 44.Since the longest side cannot be longer than the total of the other two sides, the third side's potential lengths range from 8 to 44.
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The complete question is
a triangle has one side length of 26 and another side length of 18. select all of the possible lengths of the third side potential lengths range from 8 to 44?
A normally distributed variable is known to have a mean of 34 and a standard deviation of 2. Between what two values does 95% of the data lie?
O 28 and 40
O 32 and 36
O 30 and 38
O 25 and 35
Answer:
C) 30 and 38
Step-by-step explanation:
In a normal distribution, 95% of the data points fall within two standard deviations of the mean.
Given:
mean μ = 34standard deviation σ = 2To find the two values between which 95% of the data lies, subtract and add two standard deviations to the mean:
⇒ μ - 2σ = 34 - 2(2) = 34 - 4 = 30
⇒ μ + 2σ = 34 + 2(2) = 34 + 4 = 38
Therefore, 95% of the data lies between the values:
30 and 38.Many people have misconceptions about how profitable small, consistent investments can be. In a survey of 1010 randomly selected U.S. adults (Associated Press, October 29, 1999), only 374 responded that they thought that an investment of $25 per week over 40 years with a 7% annual return would result in a sum of over $100,000 (the correct amount is $286,640). Is there sufficient evidence to conclude that less than 40% of U.S. adults are aware that such an investment would result in a sum of over $100,000? Test the relevant hypotheses using α =.05.
No, there is not. We do not have sufficient evidence to conclude that less than 40% of U.S. adults are aware that such an investment would result in a sum of over $100,000.
Two-Proportion Z-Test ResultsTo test the hypothesis that less than 40% of U.S. adults are aware that such an investment would result in a sum of over $100,000, we can use a two-proportion z-test. The null hypothesis is that the proportion of U.S. adults who are aware of the investment's outcome is equal to or greater than 40%, and the alternative hypothesis is that the proportion is less than 40%.
The test statistic is calculated as:
z = (p1 - p2) / √(p(1-p)(1/n1 + 1/n2))
where p1 is the proportion of U.S. adults in the sample who are aware of the investment's outcome (374/1010), p2 is the proportion assumed in the null hypothesis (0.4), and n1 and n2 are the number of observations in the sample and population, respectively.
Using these values, the test statistic is:
z = (0.369 - 0.4) / √(0.4(1-0.4)(1/1010)) = -1.08
Using a two-sided test with a significance level of 0.05, we can look up the critical value in a standard normal table or using software and find that the critical value is ±1.96. Since -1.08 is between -1.96 and 1.96, we fail to reject the null hypothesis.
Therefore, we do not have sufficient evidence to conclude that less than 40% of U.S. adults are aware that such an investment would result in a sum of over $100,000.
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Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?
Answer:
To find out how much interest you will earn, you need to multiply the bond principal amount ($1,000) by the interest rate (5%) and the number of years the bond matures (5).
The interest rate is expressed as a percentage, so you need to convert it to a decimal by dividing it by 100.
Interest = Principal x Interest Rate x Time
Interest = $1,000 x 0.05 x 5
Interest = $1,000 x 0.05 x 5 = $250
So, you will earn $250 in interest over the 5-year period.
The acceleration function (in m/s2) and the initial velocity v(0) are given for a particle moving along a line.
The acceleration function (in m/s2) and the initial velocity v(0) are given for a particle moving along a line.
a(t) = 2t + 6, v(0) = −16, 0 ≤ t ≤ 5
(a) Find the velocity at time t.
v(t)=______ m/s
(b) Find the distance traveled during the given time interval.
By applying the definite integral, it can be concluded that the velocity at time t is v(t) = t² + 6t - 16, and the distance traveled during the given time interval is 36.67 meters.
Definite integral is the integral of a function whose values of the independent variables have a certain interval.
This kind of integral is frequently used for calculating the acceleration and velocity of a moving object.
a(t) = dv/dt
v(t) = ds/dt
To find the velocity at time t, we look at the given acceleration function:
a(t) = dv/dt
2t + 6 = dv/dt
dv = (2t + 6) dt
v(t) = ∫(2t + 6) dt
= t² + 6t + C
Knowing that v(0) = -16, we put this into the previous equation so we get C = -16
Now we have the velocity at the time t as follows:
v(t) = t² + 6t - 16
To find the distance traveled, we look at the velocity function:
v(t) = ds/dt
t² + 6t - 16 = ds / dt
ds = (t² + 6t - 16) dt
s = ∫(t² + 6t - 16) dt
= 1/3t³ + 3t² - 16t + C
Then we calculate the distance traveled during 0 ≤ t ≤ 5:
s = s(5) - s(0)
= (1/3 * 5³) + (3 * 5²) - (16 * 5) + C - 0 - 0 + 0 - C
= 125/3 + 75 - 80
= 125/3 - 5
= 125/3 - 15/3
= 110/3
≈ 36.67 meters.
Thus, the velocity at time t is v(t) = t² + 6t - 16, and the distance traveled during the given time interval is 36.67 meters.
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Calculate the limit:
lim[(x,y) -> (0,0)] [3(x^2)y / (x^2 + y^2)]
The limit does not exist because the function is not defined at (x,y) = (0,0)You can see this by plugging in (x,y) = (0,0) into the function, you get 3(0^2) * 0 / (0^2 + 0^2) = 0 / 0 which is an indeterminate form.
Why is limit defined?In Mathematics, a limit is defined as a value that a function approaches the output for the given input values. Calculus and mathematical depend on limits, which are also used to determine integrals, derivative, and continuity.
What are types of limits?There are each limits (left- and proper limits), infinite limits, and limitations at infinity in addition to regular, two-sided limits.
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480,000 in scientific notation.
The number 480,000 in scientific notation is 4.8 * 10⁵
How to determine the scientific notation of the number
From the question, we have the following parameters that can be used in our computation:
Number = 480,000
Multiply the number by 1
So, we have the following representation
Number = 480000 * 1
Express 1 as 10⁵/10⁵
Som we have
Number = 480000 * 10⁵/10⁵
This gives
Number = 4.8 * 10⁵
Hence, the expression is 4.8 * 10⁵
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what number is equal to 4 hundreds + 3 tens + 7 ones + 6 hundredths + 8 thousandths
Answer:
Four hundred thirdy seven and six hundred and eight thousandths.
437.068
Step-by-step explanation:
Hope it helps! =D
The number formed by these terms is 437.068.
To solve this problem, we first need to understand the place value system and what each term represents.
As per the place value system used in decimals:
- Hundreds: It is the third digit to the left of the decimal point and each unit represents 100.
- Tens: It is the second digit to the left of the decimal point and each unit represents 10.
- Ones: It is right next to the decimal point, all the digits to the left of it are the ones. Each unit represents 1.
- Hundredths: It is the second digit to the right of the decimal point and each unit represents 0.01.
- Thousandths: It is the third digit to the right of the decimal point and each unit represents 0.001.
In the given context, we know that there are 4 hundreds, 3 tens, 7 ones, 6 hundredths, and 8 thousandths.
To start, we calculate each term separately:
- 4 hundreds = 4 x 100 = 400
- 3 tens = 3 x 10 = 30
- 7 ones = 7 x 1 = 7
- 6 hundredths = 6 x 0.01 = 0.06
- 8 thousandths = 8 x 0.001 = 0.008
After evaluating each term separately, we sum them up together for the final answer:
- Sum = 400 + 30 + 7 + 0.06 + 0.008 = 437.068
Hence, the number formed by these terms is 437.068.
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(50 POINTS AND BRAINLYEST)
The side length of the square is 14 units long, what is the diameter of the circular base of the cylinder from the picture in number 1?
Answer: the diameter of the circle is about 19.8
Step-by-step explanation:
Since the length of the square is 14 units, we can use the Pythagorean theorem to find the diagonal:
14^2 + 14^2 = c^2
196 + 196 =c^2
392 = c^2
c = about 19.8
Answer:
19.8
Step-by-step explanation:
The diameter of the base of the cylinder is equal to the diagonal of the inscribed square and is equal to the hypotenuse of an equilateral triangle into which the square is divided by diagonals ... By the Pythagorean theorem, the diameter (hypotenuse) is equal to the square root of the sum of the squares of the legs and is equal to 19.8
In how many ways can 3 novels, 2 mathematics books, and 1 chemistry book be arranged on a bookshelf if (a) the books can be arranged in any order? (b) the mathematics books must be together and the novels must be together? (c) the novels must be together, but the other books can be arranged in any order?
The probability (a) 6! ways; (b) 3! ways; (c) 3!2! ways. 6! = 6 x 5 x 4 = 120, 3! = 3 x 2 x 1 = 6, 3!2! = 3 x 2 x 1 x 2 x 1 = 12.
(a) In this instance, the three novels, two volumes on mathematics, and one book on chemistry can be arranged on the bookshelf in any sequence. By multiplying the number of things in each category, you can arrive at the answer: 3 novels x 2 math books x 1 chemistry book = 6. These things can be arranged in 6! (6 factorial) different ways, which is equal to 120 when multiplied by 6 x 5 x 4.
(b) In this instance, both the novels and the mathematical books must be in the same location. You can figure this out by multiplying the number of things in each category; for example, 3 novels times 1 collection of math books is 3. These things can be arranged in 3! (3 factorial) different ways, which is equal to 3 x 2 x 3.
(c) The other books may be put in any sequence, but the novels in this instance must be together. By multiplying the amount of items in each category, you can arrive at the answer: 3 novels x 2 math books and 2 x 2 chemistry books = 6. These objects can be arranged in 3!2! (3 factorial x 2 factorial) ways, which is equal to 12 when multiplied by 3 x 2 x 1 x 2 x 1.
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After measuring a set of individuals, a researcher finds that Bob's score is three times greater than Jane's score. These measurements must come from a(n) ___ scale.a. nominal b. interval C. ratio d. ordinal
After measuring a set of individuals, a researcher finds that Bob's score is three times greater than Jane's score. These measurements must come from a ratio scale
A ratio scale is a type of measurement scale that allows for the comparison of two or more values by using division. After measuring a set of individuals, a researcher finds that Bob's score is three times greater than Jane's score.
In this case, the researcher is comparing Bob's score to Jane's score, and the comparison involves division (Bob's score is three times greater than Jane's score).
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Morgan ran 5 miles in 57 days and a half minutes. How long did it take her to run each mile
Answer:
70
Step-by-step explanation:
The time that should take her to run each mile is approx 11.5 minutes.
Given that,
She runs 5 miles in 57 and a half minutes.
Based on the above information,
The time that should be taken by her is
= 57.5 ÷ 5 miles
= 11.5 minutes.
Therefore we can conclude that the time that should take her to run each mile is approx 11.5 minutes......
Before you work on your homework each week, please make sure you've read through the assigned sections in the book.
The assigned reading for this lesson begins with this quote:
Quote:
Human beings only use ten percent of their brains. Ten percent! Can you imagine how much we could accomplish if we used the other sixty percent?
According to the textbook, who said this quote?
Please help 30 points
The quote that "Human beings only use ten percent of their brains. Ten percent! Can you imagine how much we could accomplish if we used the other sixty percent?" was made by Ellen DeGeneres.
What is a quote?The representation of an utterance in oral speech that is preceded by a quotative marker, such as a verb of saying, is known as a paraphrase. When doing a brainstorming exercise, quotes can help spark ideas that then influence the ideas for the piece.
Through the presentation of other people's ideas, they can steer you in new directions. Finally, using short quotes that won't keep a writer from writing for too long can help them as they put a piece together.
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3a-5b when a= -3and b= -2
The numerical value of the expression 3a - 5b when a = -3 and b = -2 is 1.
What is the numerical value of the given expression?Given the expression in the question;
3a - 5b
a = -3b = -2
To determine the value of the expression, replace a with -3 and b with -2 for all occurrence of a and b in the expression.
3a - 5b
Plug in a = -3
3( -3 ) - 5b
Plug in b = -2
3( -3 ) - 5( -2 )
-9 -+ 10
10 - 9
Subtract 9 from 10
1
Therefore, the value of the expression is 1.
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imagine that you have a tennis ball and a baseball at different locations. the center of the tennis ball is at <-3,-1,0>m, and the center of the baseball is at <3,5,0>m. the position vector of the baseball is . the position vector of the tennis ball is . the position vector of the tennis ball relative to the baseball is . which of the following images best corresponds to the described geometry?
So, the two balls are located on the same plane, the x-y plane, with the baseball at <3,5,0>m and the tennis ball at <-3,-1,0>m, with the vector pointing from baseball to the tennis ball is <-6,-6,0>m.
The position vector of the baseball is <3,5,0>m. The position vector of the tennis ball is <-3,-1,0>m.
The position vector of the tennis ball relative to the baseball is the vector pointing from the baseball to the tennis ball and it is calculated by subtracting the position vector of the baseball from the position vector of the tennis ball: <-3,-1,0>m - <3,5,0>m = <-6,-6,0>m
So the two balls are located on the same plane, the x-y plane, with the baseball at <3,5,0>m and the tennis ball at <-3,-1,0>m, with the vector pointing from baseball to the tennis ball is <-6,-6,0>m.
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Two vectors A and B are added together to form a vector C. The relationship between the magnitudes of the vectors is given by A+B=C. Which one of the following statements concerning these vectors is true?
The true statement is " A and B must point in the same direction."
For complete questions refer belowTwo vectors A and B are added together to form a vector C. The relationship between the magnitudes of the vectors is given by A + B = C. Which one of the following statements concerning these vectors is true?
A and B must be displacements.A and B must have equal lengths.A and B must point in opposite directions.A and B must point in the same direction.A and B must be at right angles to each other.According to the question
False: All types of vectors with the same unit may be added using vector analysis, and we add vectors component by component. so, this defilement choice (1).A vector's length is not a factor to take into account before performing the operation, therefore vectors of different lengths can be added.This goes against the assertion since vectors moving in the opposite direction might create a new vector with a negative value.This is true because vectors pointing in the same direction add together to form new vectors.false: This presumption is factually incorrect.To know more about vectors on brainly : brainly.com/question/29740341
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an individual who exhibits a mixture of schizophrenic symptoms most likely has undifferentiated schizophrenia. T/F
False. Undifferentiated schizophrenia is a type of schizophrenia where a person displays general symptoms of the disorder such as hallucinations and delusions, but does not meet the criteria for any of the other subtypes of schizophrenia.
A person exhibiting a mixture of schizophrenic symptoms would likely be diagnosed with schizophrenia, but not as specifically undifferentiated schizophrenia.This disorder is characterized by a mixture of positive and negative symptoms, including delusions, hallucinations, disorganized speech, disorganized behavior, lack of emotion, and social withdrawal. However, a person who exhibits a mixture of schizophrenic symptoms could have any type of schizophrenia, not just undifferentiated schizophrenia. For example, a person could have paranoid schizophrenia, disorganized schizophrenia, or catatonic schizophrenia. To diagnose schizophrenia, a doctor must analyze the patient's symptoms and determine which type of schizophrenia they have.
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Mr. Mogi borrowed $9000 for 10 years to make home
improvements. If he repaid a total of $20,000 at what
interest rate did he borrow the money?
The interest rate that Mr. Mogi borrowed the money, given the amount borrowed and the amount repaid, is 17. 8%
How to find the interest rate ?To find the interest rate that Mr. Mogi borrowed the money at, first find the amount he paid per year:
= Total payment / Number of years
= 20, 000 / 10
= $ 2, 000 per year
You can then use the RATE function on Spreadsheet to find the interest:
NPER = 10 years
PMT = $ 2, 000
PV = 9, 000
FV = $ 20, 000
The interest rate would be 17. 8%
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Suppose one-sixth of a certain company's sales are made in New England. If New England sales amount to $700,000, what are the total sales (in dollars) of the company?
Answer:
$4,200,000
Step-by-step explanation:
let x = total sales
(1/6)x = 700,000
x = 700,000(6/1) = 4,200,000
**dividing by 1/6 is the same as multiplying by 6
The coldest temperature recorded in the city of Texas was -22.3°F. The coldest temperature recorded at Vostok station. Antarctica is four times is called what equation can be used to represent the coldest temperature recorded in Vostok station in Fahrenheit. Enter the correct equation in the box.
The coldest temperature recorded in Vostok station in Fahrenheit is -89.2°F
How to calculate the equation?An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
It is important to note that an equation is the mathematical statement which can be made up of two expressions which are connected by an equal sign.
The coldest temperature recorded in Vostok station in Fahrenheit is:
= -22.3 × 4
= -89.2°F
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Let (lim)f(x) = 7 and (lim)g(x) = -2. Find lim [f(x) - g(x)].
lim [f(x) - g(x)] is9.
What is a limit of a function?Generally, In mathematics, a limit is a value that a function "approaches" as the input of the function "approach" some value.
More formally, given a function f(x) and a value L, we say that "the limit of f(x) as x approaches a is L" if the values of f(x) can be made arbitrarily close to L by taking x sufficiently close to a, but not equal to a.
In other words, the limit of a function describes the behavior of the function as the input gets arbitrarily close to a particular value, but doesn't necessarily equal that value
The limit of the difference between the two functions is the difference of their individual limits. So,
lim [f(x) - g(x)] = lim f(x) - lim g(x)
= 7 - (-2)
= 9.
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Please answer this question for me
The value of x and y in the parallel lines are 14 and 60 respectively.
How to find angles in parallel lines?When parallel lines are cut by a transversal line, angle relationships are formed such as corresponding angles, alternate exterior angles, alternate interior angle, linear angles, vertically opposite angles etc.
Therefore, let's find the values of x and y in the parallel lines as follows:
4x + 12 = 68 (vertically opposite angles)
Vertically opposite angles are congruent.
Hence,
4x + 12 = 68
4x = 68 - 12
4x = 56
divide both sides by 4
x = 56 / 4
x = 14
Therefore, let's find the value of y.
68 + 2y - 8 = 180(same side interior angles)
Same side interior angles are supplementary
68 + 2y - 8 = 180
2y = 180 - 68 + 8
2y = 120
y = 120 / 2
y = 60
Therefore,
x = 14
y = 60 degrees
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Which graph shows the new position of the rectangle after a translation?
3rd image is translations are rigid transformations that do not change the orientation of the original image.
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
A translation is a movement of the graph either horizontally parallel to the -axis or vertically parallel to the -axis.
3rd image is translations are rigid transformations that do not change the orientation of the original image.
2 and 4 are rotations and 5 is a dilation.
Hence, 3rd image is translations are rigid transformations that do not change the orientation of the original image.
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4) Estimate the sum. Round each num
1/2 + 2/1/20
4
4
8
The sum is approximately
Submit
The sum of expression 1/2 + 2/(1/2) is obtained to be option B: 4.
What is sum?
The outcome of adding two or more numbers or phrases is known as the sum in mathematics. The sum is a method of bringing things together as a result. To put it another way, adding two or more numbers together results in a new result or total.
The fractional expression is - 1/2 + 2/(1/2)
Apply the fraction reciprocal rule -
If there is a fraction in denominator then it is reciprocated and multiplied.
= 1/2 + 2/(1/2)
= 1/2 + 2(2/1)
= 1/2 + 4
Now, take the LCM and multiply it with the numerators -
= 1/2 + 4
Find the sum of the numbers -
= (1 + 8)/2
= 9/2
= 4.5
When estimated and rounded the number is equivalent to 4.
Therefore, the fractional expression results in 4.
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Estimate the sum. Round the number.
1/2 + 2/(1/2)
The sum is approximately -
A: 0
B: 4
C: -4
D: 8
If a,b and c are constants is a(x-b)-c=ax-(ab+c). Explain
The statement a(x-b)-c = ax-(ab+c) is true because it follows the rules of algebraic operations and the distributive property.
How to explain a(x-b)-c=ax-(ab+c)?On the left side of the equation, we have a(x-b)-c.
Here, a is a constant and it is being multiplied by (x-b). According to the distributive property, we can rewrite this as:
ax-ab-c.
On the right side of the equation, we have ax-(ab+c)
Here, we have a constant a being multiplied by x and then we have another constant ab and c being subtracted, which we can also rewrite as:
ax - ab -c
As you can see, both sides of the equation are equivalent and the statement is true.
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Insert < or > between the pair of integers to make a true statement.
-12 -5
Help me please. asap!
Answer:
-12 < -5
Step-by-step explanation:
its simple math. -12 is less than -5 because its more negative.
Can someone please explain this to me step by step? I am rather confused as to what to do.
The coordinate of P that divides line AB in the ratio 4:1 is (6.6, 3.8)
What is an equation?An equation is an expression that shows the relationship between numbers and variables.
If a point O(x, y) divides line segment AB with endpoints A(x₁, y₁) and B(x₂, y₂) in the ratio of n:m. The coordinates of O is:
[tex]x = \frac{n}{n+m}(x_2-x_1)+x_1 \\\\y = \frac{n}{n+m}(y_2-y_1)+y_1[/tex]
Given that P(x, y) divides line AB in the ratio 4:1. The coordinates are A(1, 3) and B(8, 4). Therefore:
[tex]x=\frac{4}{4+1}(8-1) +1=6.6\ unit\\\\y=\frac{4}{4+1}(4-3) +3=3.8\ unit[/tex]
The coordinate of P is (6.6, 3.8)
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twelve people each spin a coin. find the number of ways in which exactly five heads may be obtained
the probability exactly five heads may be obtained is 77%.
What is probability?
By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1.
twelve people each spin a coin.
n = number of trials = 12
x = number of successes(head) among n trials = 5
p = probability of success(head) in any one trial = 1/2
probability of failure in any one trial (q = 1 - p) = 1/2
exactly five heads may be obtained 12 C 5 * (1/2)[tex].{5}[/tex] *(1/2)[tex].{5}[/tex] = 792/1024 =0.77
Hence the probability exactly five heads may be obtained is 77%.
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find a polynomial polynomial the sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that may be real or complex that represents the perimeter perimeter the length of the outer edge of a shape. of the rectangle rectangle a quadrilateral containing four right angles. .
The perimeter of a rectangle can be represented by a polynomial expression. The expression will consist of two terms, one representing the width and one representing the length. Each term will contain a variable, which is to the power of one, multiplied by a coefficient. The sum of the two terms produces the perimeter of the rectangle.
For example, if the width of the rectangle is w and the length is l, then the polynomial expression for the perimeter would be: P = w + l. This expression can also be written in expanded form as: P = w + w + l + l.
By adding more terms to this polynomial expression, the perimeter of more complex rectangles can be found. For example, if the rectangle is not a perfect square, then two additional terms can be added to the polynomial expression. One term will represent the extra width and one term will represent the extra length. By adding these two additional terms, the perimeter of the rectangle can be calculated.
In conclusion, the perimeter of a rectangle can be represented as a polynomial expression, which consists of terms that have variables raised to the power of one and are multiplied by coefficients. By adding more terms to the expression, the perimeter of more complex rectangles can be found.
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the purpose of the following exercises is to familiarize you with the system you will be using for the rest of your course. these exercises are not intended to teach or test your knowledge of any specific subject material. therefore, you will not be penalized for using hints or submitting incorrect answers. part a (figure 1) (the link above tells you that this part has an illustration.) how many months of the year have 28 days?
Every month of the year has
a constant volume of cookie dough is formed into a cylinder with a relatively small height and large radius. when the cookie dough is placed into the oven, the height of the dough decreases as the radius increases, but it retains its cylindrical shape. at time t, the height of the dough is 6 mm, the radius of the dough is 20 mm, and the radius of the dough is increasing at a rate of 2 mm per minute. Part A: At time t, at what rate is the area of the circular surface of the cookie dough increasing with respect to time? (5 points)
Part B: At time t, at what rate is the height of the dough decreasing with respect to time?
So, the rate at which the area of the circular surface of the cookie dough is increasing with respect to time is 80π mm^2/min and the rate at which the height is decreasing with respect to time is -2 mm/min.
Part A: The area of the circular surface of the cookie dough is given by A = πr^2, where r is the radius of the dough. The rate at which the area is changing with respect to time is given by dA/dt = 2πr*dr/dt. At time t, the radius is 20 mm and the rate at which it is increasing is 2 mm/min. Therefore, the rate at which the area of the circular surface of the cookie dough is increasing with respect to time is dA/dt = 2π(20 mm)(2 mm/min) = 80π mm^2/min.
Part B: The height of the dough is given by h, and the rate at which the height is decreasing with respect to time is given by dh/dt. Since the dough is retaining its cylindrical shape, the decrease in the height of the dough is exactly the opposite of the increase in the radius. At time t, the rate at which the radius is increasing is 2 mm/min, so the rate at which the height is decreasing with respect to time is dh/dt = -2 mm/min.
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