The region between the graphs of y = x2 and y = 2x is rotated around the line y = 4. The volume of the resulting solid is 31π/15 cubic units.
To find the volume of the resulting solid, we can use the method of cylindrical shells. First, we need to determine the limits of integration.
The graphs of y = x^2 and y = 2x intersect at x = 0 and x = 2. Therefore, we will integrate with respect to x from 0 to 2.
The distance between the line y = 4 and the graph y = x^2 is 4 - x^2, and the distance between the line y = 4 and the graph y = 2x is 4 - 2x. Thus, the radius of the cylindrical shell at x is (4 - x^2) - (4 - 2x) = 2x - x^2.
The height of the cylindrical shell at x is the difference between the y-coordinates of the two graphs at x, which is (2x) - (x^2) = x(2 - x).
Therefore, the volume of the resulting solid is:
V = [tex]\int\limits^2_0 \, 2\pi(x(2 - x))(2x - x^2) dx[/tex]
= [tex]\int\limits^2_0 \, 4\pi x^3 - 2\pi x^4 - 2\pi x^2 + \pi x^3 dx[/tex]
= [tex]\int\limits^2_0 \, 5\pi x^3 - 2\pi x^4 - 2\pi x^2 dx[/tex]
= π(5/4 - 2/5 - 2/3)
= 31π/15
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What is the product of 3a +5 and 2a² + 4a - 2?
O 6a³ +22a²+14a-10
O 6a³+22a²+26a-10
O 18a³+10a²+14a-10
O 28a³+14a-10
Answer:
(a) 6a³ +22a² +14a -10
Step-by-step explanation:
You want the product of (3a +5) and (2a² +4a -2).
Distributive propertyThe distributive property is used to eliminate the parentheses when simplifying the product.
(3a +5)(2a² +4a -2) = 3a(2a² +4a -2) +5(2a² +4a -2)
= 6a³ +12a² -6a +10a² +20a -10
= 6a³ +(12+10)a² +(-6+20)a -10
= 6a³ +22a² +14a -10
__
Additional comment
The leading coefficient of the product is the product of the leading coefficients: 3·2 = 6. This eliminates the last two answer choices.
The first two answer choices differ only in the "a" term, so you only need to find that one. It will be the sum of terms (3a)(-2) and (5)(4a), so is -6a+20a = 14a.
You can also "guess" that the "a" term will be 14a, because that appears the most often among the answer choices.
The first answer choice is the correct one.
Any help would be great
Answer:
50+6.3= 56.3
Step-by-step explanation:
Write the equation of the line that passes through (-6, 1) and (4, -3)
Answer:
To find the equation of the line passing through two points, we need to use the point-slope form of a line:
y - y1 = m(x - x1)
where m is the slope of the line, and (x1, y1) is one of the given points.
First, we can find the slope of the line using the two points:
m = (y2 - y1) / (x2 - x1)
m = (-3 - 1) / (4 - (-6))
m = -4/10
m = -2/5
Now, we can use one of the points and the slope to write the equation of the line:
y - y1 = m(x - x1)
y - 1 = (-2/5)(x - (-6))
y - 1 = (-2/5)(x + 6)
y - 1 = (-2/5)x - (2/5) * 6
y - 1 = (-2/5)x - 12/5
y = (-2/5)x - 12/5 + 1
y = (-2/5)x - 12/5 + 5/5
y = (-2/5)x - 7/5
Therefore, the equation of the line that passes through (-6, 1) and (4, -3) is y = (-2/5)x - 7/5.
Answer: y = 5x + 31
Step-by-step explanation:
You have to find the slope which is change in y over change in x. That would be (-6-4)/(1-3)
that simplifies down to 5.
You equation for a linear equation is y=mx+b with m=slope and b=y-intercept
Take a point and plug in the values for x and y. I'm going to use (-6, 1) but it doesn't really matter
1=5(-6)+b
1=-30+b
+30 +30
31 = b
Now you have everything for your equation so when you put it all together, it is y=5x+31
armen and rosanna are two employees in an electronics store. the store manager has put an emphasis on increasing the number of applications for the store credit card. the manager randomly selected 24 weeks for each employee and recorded the number of completed applications for each week, which is provided in the samples below. each employee completed 21 applications in one of the weeks selected. based on the z-scores you calculated above, would carmen or rosanna be more likely to complete 21 applications in one week? select the correct answer below: A. carmen is more likely to complete 21 applications in one week, because the absolute value of the z-score for rosanna is less than for carmen. B. carmen is more likely to complete 21 applications in one week, because the absolute value of the z-score for carmen is less than for rosanna. C. rosanna is more likely to complete 21 applications in one week, because the absolute value of the z-score for carmen is less than for rosanna.
D. rosanna is more likely to complete 21 applications in one week, because the absolute value of the z-score for rosanna is greater than for carmen .
The correct answer is A. Carmen is more likely to complete 21 applications in one week, because the absolute value of the z-score for Rosanna is less than for Carmen.
Based on the calculated z-scores, we can determine which employee is more likely to complete 21 applications in one week.
The z-score for Carmen is -1.56, which indicates that her performance is slightly below average, but still within one standard deviation of the mean. On the other hand, the z-score for Rosanna is -2.83, which is more than two standard deviations below the mean and indicates that her performance is much lower than average.
Therefore, it is more likely for Carmen to complete 21 applications in one week compared to Rosanna, as Carmen's performance is closer to the mean and within one standard deviation of it. Hence, the correct answer is A. Carmen is more likely to complete 21 applications in one week, because the absolute value of the z-score for Rosanna is less than for Carmen.
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find the measure of such angle whose supplementary angle is 35 degree more than twice of its complementary angle
The required angle is 35 degrees whose supplementary angle is 35 degrees more than twice its complementary angle.
What is the supplementary angle?The definition of a supplementary angle is that it adds up to 180 degrees.
Let's represent the required angle as "x".
We know that the supplementary angle to "x" is 180 - "x".
Similarly, the definition of a complementary angle is that it adds up to 90 degrees, so we know that the complementary angle to "x" is 90 - "x".
Given that the supplementary angle to "x" is 35 degrees more than twice its complementary angle, so we can write this as an equation:
180 - x = 2(90 - x) + 35
Simplifying this equation, we get:
180 - x = 180 - 2x + 35
Adding x to both sides, we get:
x = 35
Therefore, the required angle is 35 degrees.
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The cafeteria manager took a poll of 150 students to find the most popular lunch. The circle graph below displays the data
How many more students chose spaghetti and pizza than corn dogs , tacos, and cheeseburgers, as their favorite lunch
There are 60 students who chose spaghetti and pizza over corn dogs, tacos, and cheeseburgers, as their favorite lunch.
What is a circular graph?
A circular graph is also known as a pie chart. It is used to display a set of data. The data can be in percentages or degrees or radians.
The given data are as follows:
Pizza: 45%
Corn dogs: 10%
Cheeseburger: 8%
Tacos: 12%
Spaghetti: 25%
The total number of students is 150.
The total percentage of students who eat spaghetti and pizza is (25%+45%), that is, 70%.
The total percentage of students who eat corn dogs, tacos, and cheeseburgers is (10%+12%+8%), that is, 30%.
The percentage difference is (70%-30%), that is, 40%.
Now, calculate 40% of the total students, that is, 150.
150×40% = 150×40/100
= 60
Therefore, the obtained answer is 60.
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The complete question is given in the following chart.
The boxplots shown below summarize two data sets, I and II. Based on the boxplots, which of the following statements about these two data sets CANNOT be justified? a) The median of data set I is equal to the median of data set II b) o The interquartile range of data set l is equal to the interquartile range of data set 1. c) The range of data set I is greater than the range of data set II d) Data set I and data set II have the same number of data points. Review Later
The boxplots below summarize two data sets, I and II so Data set I and data set II have the same number of data points. so potion D.
The distribution of data is shown using a boxplot, which is a standardised method based on a five-number summary ("minimum," first quartile ("Q1"), median ("Q3"), and "maximum"). It can reveal information about your outliers' values. Boxplots can also show you how tightly your data is grouped, whether or not your data is skewed, and whether or not your data is symmetrical.
A box plot is a graph that displays information from a five-number summary that includes one of the central tendency measures. It does not clearly depict the distribution as a stem and leaf plot or histogram does. However it is mostly used to show whether a distribution is skewed or not and whether there are any potential outliers—unusual observations—in the data collection. When comparing or involving numerous data sets, boxplots are also highly helpful.
We can't make the conclusion from the given boxplots whether Data set I and Data set II have the same number of data points or not.
Hence,
Option D is correct.
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Complete question:
The boxplots shown below summarize two data sets, I and II. Based on the boxplots, which of the following statements about these two data sets CANNOT be justified? a) The median of data set I is equal to the median of data set II b) o The interquartile range of data set l is equal to the interquartile range of data set 1. c) The range of data set I is greater than the range of data set II d) Data set I and data set II have the same number of data points.
Do the ratios and
4/8 and 1/7 form a proportion
The given two ratios are not form a proportion.
What is ratio and proportion?
Ratio is defined as the comparison of two quantities of the same kind by the method of division.
Proportion is an equation it is formed when two ratios are equivalent to each other.
The given two ratios are 4/8 and 1/7.
i.e., 4:8 and 1:7
By simplifying this ratio 4:8 we get 1:2
That is these two ratios are not equal.
proportion is formed only when the two ratios are equal.
Hence, the given two ratios are not form a proportion.
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Two cards are drawn from a shuffled deck of 52 cards. whats the probbility that both cards are kings if the first card isnt replaced after itsw drawn and is a king?
The probability of both cards are kings and first card isn't replaced after it's drawn and is a king is 0.00452
"Information available from the question"
Two cards are drawn from a shuffled deck of 52 cards.
To find the probability that both cards are kings if the first card isn't replaced after it's drawn and is a king.
Now, According to the question:
The probability that the first card of a king is [tex]\frac{4}{52}[/tex] = [tex]\frac{1}{13}[/tex]
If you do not replace the card, there are only 51 cards left and only 3 kings left, so the probability of now drawing a king is [tex]\frac{3}{51}[/tex]
Multiply these probabilities:
[tex]\frac{4}{52}[/tex] × [tex]\frac{3}{51}[/tex] = 0.00452
Hence, The probability is 0.00452
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Mrs. Taylor, the librarian, has the following puzzle on a shelf: 3 puzzle with 1,000 pieces 8 puzzle with 500 4 puzzle with 100 pieces 3 puzzle with 50 pieces. Lola will randomly choose 1 puzzle to put together. What is the probability that lola will choose a puzzle with 1,000 pieces?
Answer:
Step-by-step explanation:
a football is thrown horizontally at 56ft/s parallel to the sideline. a tv camera is 92ft/s from the path of the football. find theta/dt the rate at whicht the camera must turn to foloow the ball when theta
The camera must turn at a rate of about 53.67 degrees/s to follow the ball at an angle of 15 degrees.
In this issue, we are given the underlying speed of the football and the distance of the television camera from the way of the football. We really want to find the rate at which the camera should go to follow the ball at a specific point theta=15 degrees. We can utilize geometry to take care of this issue. The point between the way of the ball and the line associating the camera and the ball is given by 90 - 15 = 75 degrees. Utilizing the cosine capability, we can find the part of the speed of the football opposite to the camera, which is 56*cos(15) = 53.17 ft/s. Presently, utilizing the digression capability, we can find the rate at which the camera should go to follow the ball, which is (53.17/92)*tan(75) = 0.937 radians/s or roughly 53.67 degrees/s. In this manner, the camera should turn at a pace of around 53.67 degrees/s to follow the ball at a point of 15 degrees.
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The complete question is:
A football is thrown horizontally (very little arc) at 56 ft./s parallel to the sideline. A TV camera is 92 ft. from the path of the football. Find de/dt, the rate at which the camera must turn to follow the ball when θ = 15°. football TV camerao 92 ft.
Determine whether the table represents an exponential growth function, an exponential
decay function, or neither.
The function is an exponential decay function. Then the correct option is B.
What is an exponent?Let a be the initial value and x be the power of the exponent function and b be the increasing factor.
The exponent is given as,
y = a(b)ˣ
At (-1, 50), then the equation is given as,
50 = a / b ...1
At (1, 2), then the equation is given as,
2 = ab ....2
From equations 1 and 2, then we have
2 = 50b · b
b² = 1/25
b = 1/5
Then the value of 'a' is given as,
2 = a x 1/5
a = 10
The value of the variable 'b' is less than one that is 1/5. Then the function is an exponential decay function. Then the correct option is B.
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Classify the following data. Indicate whether the data is qualitative or quantitative, indicate whether the data is discrete, continuous, or neither, and indicate the level of measurement for the data. You order a pizza. The kind of pizza you order is recorded by entering the appropriate number on an order form. The numbers used are given below. 1) Pepperoni 2) Mushroom 3) Black Olive Are these data qualitative or quantitative? -Qualitative -Quantitative Are these data discrete or continuous? -Discrete -Continuous -Neither What is the highest level of measurement the data possesses? -Nominal -Ordinal -Interval
- Ratio
The data is qualitative, discrete, and at the nominal level of measurement.
How to classify data?The data is qualitative, as it describes the type of pizza ordered, which cannot be measured numerically.
The data is discrete, because there are only a finite number of possible values that the variable can take on, which in this case is one of the three types of pizza: Pepperoni, "Mushroom, or Black Olive.
The highest level of measurement for this data is nominal, because the numbers used to record the type of pizza do not imply any order or ranking between the categories. The numbers are simply a code to identify which type of pizza was ordered, and do not have any intrinsic numerical value or relationship between them. Therefore, the data falls into the nominal level of measurement, which is the lowest level of measurement that data can have.
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Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of several mice takes with a noise as stimulus. What are the null hypothesis H0 and alternative hypothesis Ha?
The null hypothesis H0 to for mice to complete the maze is equal to 18 seconds and alternative hypothesis Ha for mice to complete the maze is less than 18 seconds.
The null hypothesis is the default assumption that there is no effect of the noise stimulus on the time it takes for mice to complete the maze. The alternative hypothesis is the hypothesis that the noise stimulus does have an effect and causes the mice to complete the maze faster. The researcher will conduct a statistical test to determine if there is sufficient evidence to reject the null hypothesis and support the alternative hypothesis.
The researcher is hypothesizing that the true population mean time taken by mice to complete the maze with a loud noise stimulus is less than 18 seconds, Ha< 18seconds. The mean time for mice to complete the maze with the noise stimulus is equal to 18 seconds. Ha: 18 seconds
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What is High power money
Answer:
High-powered money is the base for the expansion of Bank deposits and creation of money supply. The supply of money varies directly with changes in the monetary.
Step-by-step explanation:
have great day :)
square root of 88 simplifying radicals
Answer:
[tex]2\sqrt{22}[/tex]
Step-by-step explanation:
Select the correct answer from each drop-down menu.
The graph represents the complex numbers z₁ and z2. What are their conjugates?
The conjugates of z1 and z2 are given below:
• z1* = 7-2i• z2* = -3+iWhat is a Conjugate?In mathematics, a pair of binomials with identical phrases that part opposite arithmetic operators in the midst of these similar terms are referred to as conjugates.
For instance, the conjugate of p + q is p - q.
'
With this in mind, the conjugate which is the same number with the imaginary part negated is given as:
The conjugate of 7+2i is 7-2i;
the conjugate of -3-i is -3+i.
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Quadratic Functions
Instruction Active
TRY IT Finding the Zeroes of a Quadratic Function
-2
-1
0
1
2
a = -2
b=0
c=2
f(x)=x²+x+6
X
0
X
6
X 4
a
4
b
6
с
The axis of symmetry is
x x=-1
x x=0
✓
x = ½
COMPLETE
The smaller zero is
The larger zero is
DONE
The zeroes of the quadratic function f(x) = -2x² + 2 is -1 and +1 and the axis symmetry is x = 0.
What are Quadratic Functions?Quadratic functions are polynomial functions consisting of variables and exponents and the degree of the variable is 2.
The general form of a quadratic function is f(x) = ax² + b x + c.
Given that,
a = -2, b = 0 and c = 2.
Substituting in the standard form, we get the quadratic function as,
f(x) = -2x² + 2
Let f(x) = 0.
-2x² + 2 = 0
-2x² = -2
x² = 1
x = ±1
So the two roots are +1 and -1.
Axis of symmetry is x = (1 + -1) / 2 = 0
x = 0 is the axis symmetry
The smaller zero is -1 and the larger zero is 1.
Hence -1 and +1 are the zeroes of the quadratic function f(x) = -2x² + 2 and the axis symmetry is x = 0.
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identify the numerator and the denominator of the fraction and identify the fraction as proper or improper. 4/7
The numerator is 4 and the denominator is 7. It is a proper fraction.
What is fraction?A fraction represents the part of a set of objects. It has two parts separated by a dividing line, generally written
a/b
Where
a is numeratorb is denominatorThere are 3 types of fraction. They are
1. Proper fraction
A proper fraction is a fraction whose numerator is less than the denominator. It makes the value is less than 1.
Example: 3/4, 2/5.
2. Improper fraction
An improper fraction is a fraction whose numerator is more than the denominator. The value is more than 1.
Example: 9/8, 7/5.
3. Mixed fraction
A mixed fraction is a combination of a whole number and a proper fraction. It is the quotient and the remainder of improper fraction.
Example:
9/5 = 1 4/511/2 = 5 1/2From the explanation above, the fraction of 4/7 has the numerator of 4 and the denominator of 7. The numerator 4 is less than the denominator 7, so the fraction is a proper fraction.
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A floor plan was drawn using the following scale:
1 inch: 2 1/2 feet
One wall of the house was represented by a line 6 1/2
inches long on the floor plan. Which proportion could be used to find the length of the actual wall, in feet? pls help me hurry than k u
a. 2/5=13/2x
b. 2/5=13x/2
c. 5/2=13/2x
d. 5/2=13x/2
The proportion that could be used is (a) 2/5=13/2x
How to determine the proportion that could be usedFrom the question, we have the following parameters that can be used in our computation:
Scale = 1 inch: 2 1/2 feet
The proportion that can be used is:
actual length / length on the floor plan = scale factor
So, we can set up the proportion as follows:
actual length / 6 1/2 inches = 2 1/2 feet / 1 inch
Solving for actual length (x), we get:
x/(13/2) = 5/2
So, we have
2x/13 = 5/2
Rewrite as
13/2x = 2/5
Hence, the proporton is (a) 2/5=13/2x
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HELP ASAP!!
The sum of two integers is three the sum of the squares 185 find the integers.
Answer:
the two integers are 11 and -8
Step-by-step explanation:
Let's call the two integers we're trying to find "x" and "y".
From the problem statement, we know two things:
x + y = 3 (the sum of the two integers is three)
x^2 + y^2 = 185 (the sum of the squares is 185)
We can use the first equation to solve for one of the variables in terms of the other. For example, we can solve for "y" in terms of "x" by subtracting "x" from both sides:
y = 3 - x
Now we can substitute this expression for "y" into the second equation:
x^2 + (3 - x)^2 = 185
Expanding the square on the left-hand side, we get:
x^2 + 9 - 6x + x^2 = 185
Combining like terms, we get:
2x^2 - 6x - 176 = 0
We can simplify this equation by dividing both sides by 2:
x^2 - 3x - 88 = 0
Now we can solve for "x" using the quadratic formula:
x = (3 ± sqrt(3^2 - 4(1)(-88))) / 2(1)
Simplifying, we get:
x = (3 ± sqrt(361)) / 2
We can ignore the negative root, since it would give us a negative value for "y". So, taking the positive root, we get:
x = (3 + 19) / 2 = 11
Now we can use the first equation to solve for "y":
y = 3 - x = 3 - 11 = -8
Which graphs represent functions?
HELPPPPPPPPP
A . c and d
B. d only
C. b and d
D. a only
Answer:
A
Step-by-step explanation:
There probably is a good explanation but I don't have it.
suppose you have another picture that is 10 inches by 5 inches, and are now using a fancy paper that is 8.5 inches by 4 inches to frame the picture. again, the frame is to be uniform in thickness all the way around. no fancy framing paper is to be wasted!
The thickness of the framing paper used to frame the picture with uniform thickness will be 1.13 inches.
First, we have to find the perimeter of the picture. We know that the perimeter of a rectangle is 2(length + breadth). So, The perimeter of the picture will be :
⇒ 2(10 + 5)
⇒30 inches
Now, finding the area of the framing paper :
⇒ length x breadth
⇒ 8.5 x 4
⇒ 34 inch²
Now, we know that thickness x perimeter = area
∴ Thickness = area / perimeter
⇒ 34 / 30
⇒ 1.13 inch
Therefore, the thickness needed to frame the picture with uniform thickness using the fancy paper without wasting any paper is 1.13 inch.
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The complete question will be :
suppose you have another picture that is 10 inches by 5 inches, and are now using a fancy paper that is 8.5 inches by 4 inches to frame the picture. again, the frame is to be uniform in thickness all the way around. no fancy framing paper is to be wasted! Find out how thick the frame should be.
find the value of x. SIMPLEST RADICAL FORM
For given triangle, x is 4√5 using Pythagorean theorem.
What is Pythagorean Theorem?
Pythagoras Theorem (also known as Pythagorean Theorem) is a mathematical concept that explains the relationship between the sides of a right-angled triangle. The sides of a right triangle are also referred to as Pythagorean triples. This theorem's formula and proof are discussed with examples here.
The Pythagorean theorem is used to calculate the length of an unknown side and the angle of a triangle. We can deduce the base, perpendicular, and hypotenuse formulae from this theorem.
The Pythagoras Theorem formula is presented in the definition as:
Perpendicular² + Base² = Hypotenuse²
c² = a² + b²
Now,
Given that perpendicular =19
Hypotenuse=21
then Base²=21²-19² using Pythagoras theorem
Base=√441-361
=√80
=4√5
Hence,
x is 4√5
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Theorem: The difference between any odd integer and any even integer is odd. In an attempted proof of this statement a student wrote the following lines: Suppose n is any odd integer, and m is any even integer. By definition of odd, n = 2k + 1 where k is an integer. By definition of even, m = 2k where k is an integer. Then n m = (2k + 1) – 2k = 1 and 1 is odd. Therefore, the difference between any odd integer and any even integer is odd. Which one of the following best describes the reason the proof is not correct? a. The proof assumes m and n are consecutive integers.
b. It does not prove that 1 is odd. c. The proof violates the parity property. d. The proof violates the zero product property.
The best describes reason the proof is not correct is c. The proof violates the parity property.
The reason the attempted proof of the statement "The difference between any odd integer and any even integer is odd" is not correct is "The proof violates the parity property."
The student incorrectly subtracted an even integer from an odd integer and concluded that the result is odd. However, the difference between an odd integer and an even integer is always odd, which can be proven directly as follows:
Let n be any odd integer and m be any even integer. Then, by definition, n = 2k + 1 and m = 2j for some integers k and j. The difference between n and m is:
n - m = (2k + 1) - 2j
= 2k - 2j + 1
= 2(k - j) + 1
Since k and j are integers, k - j is also an integer. Therefore, the difference between any odd integer and any even integer is of the form 2x + 1, where x is an integer, which is an odd integer.
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65 mi/hr = ? meters per min
Answer:
1743.46[tex]\frac{meters}{min}[/tex]
Step-by-step explanation:
Two unit multipliers will be used here:
1 mile (mi) = 1,609.344 meters
1 hour (hr) = 60 minutes (min)
65 [tex]\frac{mi}{hr}[/tex] = (65[tex]\frac{mil}{hr}[/tex])×([tex]\frac{1609.344 meters}{1 mile}[/tex])×([tex]\frac{1 hour}{60 minutes}[/tex])
The miles unit in the numerator and denominator will cannel each other out completely. In addition, the hour unit in the numerator and denominator will also cancel each other out completely:
= 1743.46 [tex]\frac{meters}{min}[/tex]
Write an expression that is equivalent to -5+6a+(-8)+(-3a)
Answer:
-5+6a-8-3a
=6a-3a-5-8
=3a-13
Find the result if 24,312 is increased by 22,211.
Answer:
Step-by-step explanation:
To find the result of increasing 24,312 by 22,211, you would simply add the two numbers:
24,312 + 22,211 = 46,523
So, the result of increasing 24,312 by 22,211 is 46,523.
Answer: 46,523
Step-by-step explanation:
Line up the two numbers by their number places (making sure to put the larger number on top). The add the digits that are on top of each other until you get your answer.
On many cell phones with GPS, an approximate location can be given before the GPS signal is received. This is done by a process called triangulation, which works by using the distance from two known points. Suppose there are two cell phone towers within range of you, located 3000 feet apart along a straight highway that runs east to west, and you know you are north of the highway. Based on the signal delay, it can be determined you are 2050 feet from the first tower, and 1420 feet from the second. Determine the angle, , between your line of sight to the first tower and the highway to the nearest tenth of a degree. (A calculator is needed for this question)
Given x>0 and a natural number n, show that there exists a unique positive real number r such that x=r^n. Usually r is denoted by x^(1/n).
For any given positive real number x and natural number n, there exists a unique positive real number r such that [tex]x = r^n[/tex].
Let x be a positive real number and n be a natural number. We want to show that there exists a unique positive real number r such that [tex]x = r^n[/tex]. This can be expressed mathematically as [tex]x^(1/n) = r[/tex], where r is the unique positive real number we are looking for.
To solve for r, we can take the nth root of both sides of the equation. This gives us [tex]r = x^(1/n)[/tex]. Since x is a positive real number and n is a natural number, it follows that r is also a positive real number.
Furthermore, since x is a fixed value and n is a fixed value, we can conclude that the positive real number r is unique. In other words, for any given x and n, there is only one r that satisfies the equation [tex]x = r^n[/tex].
Therefore, for any given positive real number x and natural number n, there exists a unique positive real number r such that [tex]x = r^n[/tex].
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