The order of a in (z/(pq))× is exactly (p-1)(q-1), as desired.
To prove that (z/(pq))× has order (p − 1)(q − 1), we need to show that the least positive integer n such that (z/(pq))×n = 1 is (p − 1)(q − 1).
First, let's define (z/(pq))× as the set of all integers a such that gcd(a,pq) = 1 (i.e., a is relatively prime to pq) and a mod pq is also relatively prime to pq.
Now, we know that the order of an element a in a group is the smallest positive integer n such that a^n = 1. Therefore, we need to find the order of an arbitrary element a in (z/(pq))×.
Let's assume that a is an arbitrary element in (z/(pq))×. Since gcd(a,pq) = 1, we know that a has a multiplicative inverse modulo pq, denoted by a^-1. Therefore, we can write:
a * a^-1 ≡ 1 (mod pq)
Now, let's consider the order of a. Since gcd(a,pq) = 1, we know that a^(p-1) is congruent to 1 modulo p by Fermat's Little Theorem. Similarly, we can show that a^(q-1) is congruent to 1 modulo q. Therefore, we have:
a^(p-1) ≡ 1 (mod p)
a^(q-1) ≡ 1 (mod q)
Now, we can use the Chinese Remainder Theorem to combine these congruences and get:
a^(p-1)(q-1) ≡ 1 (mod pq)
Therefore, we know that the order of a must divide (p-1)(q-1).
To show that the order of a is exactly (p-1)(q-1), we need to show that a^k is not congruent to 1 modulo pq for any positive integer k such that 1 ≤ k < (p-1)(q-1).
Assume for contradiction that there exists such a k. Then, we have:
a^k ≡ 1 (mod pq)
This means that a^k is a multiple of pq, which implies that gcd(a^k, pq) ≥ pq. However, since gcd(a,pq) = 1, we know that gcd(a^k, pq) = gcd(a,pq)^k = 1. This is a contradiction, and therefore our assumption must be false.
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The order of a in (z/(pq))× is exactly (p-1)(q-1), as desired.
To prove that (z/(pq))× has order (p − 1)(q − 1), we need to show that the least positive integer n such that (z/(pq))×n = 1 is (p − 1)(q − 1).
First, let's define (z/(pq))× as the set of all integers a such that gcd(a,pq) = 1 (i.e., a is relatively prime to pq) and a mod pq is also relatively prime to pq.
Now, we know that the order of an element a in a group is the smallest positive integer n such that a^n = 1. Therefore, we need to find the order of an arbitrary element a in (z/(pq))×.
Let's assume that a is an arbitrary element in (z/(pq))×. Since gcd(a,pq) = 1, we know that a has a multiplicative inverse modulo pq, denoted by a^-1. Therefore, we can write:
a * a^-1 ≡ 1 (mod pq)
Now, let's consider the order of a. Since gcd(a,pq) = 1, we know that a^(p-1) is congruent to 1 modulo p by Fermat's Little Theorem. Similarly, we can show that a^(q-1) is congruent to 1 modulo q. Therefore, we have:
a^(p-1) ≡ 1 (mod p)
a^(q-1) ≡ 1 (mod q)
Now, we can use the Chinese Remainder Theorem to combine these congruences and get:
a^(p-1)(q-1) ≡ 1 (mod pq)
Therefore, we know that the order of a must divide (p-1)(q-1).
To show that the order of a is exactly (p-1)(q-1), we need to show that a^k is not congruent to 1 modulo pq for any positive integer k such that 1 ≤ k < (p-1)(q-1).
Assume for contradiction that there exists such a k. Then, we have:
a^k ≡ 1 (mod pq)
This means that a^k is a multiple of pq, which implies that gcd(a^k, pq) ≥ pq. However, since gcd(a,pq) = 1, we know that gcd(a^k, pq) = gcd(a,pq)^k = 1. This is a contradiction, and therefore our assumption must be false.
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URGENT ANSWER QUICKLY PLEASE A manufacturer wants to design a cone-shaped container that has a volume of 175 cubic centimeters. Their old container is shown.
Hence,0.72 rather than to increase the radius to meet their requirements.
What is the cone ?A cone is a three-dimensional geometric form with a plane base and a smooth tapering vertex. A cone is made up of a collection of line segments, half-lines, its lines that connect the apex of the common point at to every point on a base that is in a flat other than the apex.
What is the volume ?A measurement of three-dimensional space is volume. It is frequently expressed quantitatively using US-standard units ,SI-derived units, as well as several imperial Volume and the notion of length are connected.
Let the new radius x and the old radius as r. The formed volume was 175 cubic cm, and the new container height is 5 cm,
V = [tex]\frac{1}{3}\pi r^2h[/tex], where V is the volume, r is the radius, and h is the height, is the formula for a cone volume.
We can construct an equation using the previous volume:
175 = [tex]\frac{1}{3}\pi r^2h[/tex]
the height is same for new and old container , so,:
175 =[tex]\frac{1}{3}\pi r^2*5[/tex]
175 = [tex]\frac{5}{3}\pi r^2[/tex]
By multiplying both sides by (5/3)
[tex]r^2 = \frac{175*3}{5*\pi}[/tex] ≈ 22.3 use [tex]\pi[/tex]=3.14
When both sides are square root
r ≈ 4.72
Therefore, the old container radius is 4.72 cm.
now,We subtract the old radius from the new radius for determine how much the radius must increase to s the new container:
x - r ≈ 4 - 4.72 ≈ -0.72
Because the new radius is lower than the old radius, the outcome is negative.
We would need to reduce the radius by roughly 0.72 cm rather than expand it to satisfy the needs of the new container.
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A coat that costs $131 is $18 less than twice the cost of a jacket, j. Write
an equation that represents the relationship between the cost of the coat
and the cost of the jacket
Answer:
2j - c
Step-by-step explanation:
Find the common difference of the arithmetic sequence 14 , 16 , 18
Answer:
The common difference is 2.
Answer:
The common difference of the arithmetic sequence 14, 16, 18 is **2**.
In any arithmetic sequence, each term is equal to the previous term plus the common difference. So, the second term is equal to the first term plus the common difference. In this case, the second term, 16, is 2 more than the first term, 14. Therefore, the common difference is 2.
We can also find the common difference by subtracting any two consecutive terms in the sequence. For example, we can subtract the second term from the third term to get 18 - 16 = 2.
The common difference of an arithmetic sequence is always constant. This means that the difference between any two consecutive terms in the sequence will always be the same. In this case, the difference between any two consecutive terms is 2.
Step-by-step explanation:
(a) Find the volume of the solid generated by revolving the region bounded by the graph x2=y−2 and 2y−x−2=0 for 0≤x≤1 about y=3.
(b) A force of 9 lb. is required to stretch a spring from its natural length of 6 in. to a length of 8 in. Find the work done in stretching the spring
(i) from its natural length to a length of 10 in.
(ii) from a length of 7 in. to a length of 9 in.
(a) Volume of the solid generated by revolving the region bounded by the graph is 12.422 cubic units.
(b)
(i) The work done in stretching the spring from its natural length to a length of 10 in. is 54 lb.-in.
(ii) The work done in stretching the spring from a length of 7 in. to a length of 9 in. is approximately 13.5 lb.-in.
How to find the volume of the solid generated by revolving the region bounded by the graph?(a) To find the volume of the solid generated by revolving the region bounded by the graph[tex]x^2=y-2[/tex] and 2y-x-2=0 for 0≤x≤1 about y=3, we can use the method of cylindrical shells:
First, we need to find the limits of integration for the radius of the shells. Since we are revolving around y=3, the distance between y=3 and the curve x^2=y-2 will give us the radius of the shell.
Solving for y in [tex]x^2=y-2[/tex], we get[tex]y=x^2+2.[/tex] Substituting this into 2y-x-2=0, we get [tex]x=2y-2y^2-2.[/tex] So the limits of integration for the radius will be from [tex]3-(x^2+2) to 3-(2y-2y^2-2).[/tex]
Next, we need to find the height of the shells. This is simply the length of the interval of integration for x, which is 0 to 1.
So the volume of the solid is given by the integral:
[tex]V = \int (3-(x^2+2)) - (3-(2y-2y^2-2)) dx[/tex] from x=0 to x=1
Simplifying and evaluating the integral, we get:
V ≈ 12.422 cubic units.
Therefore, the volume of the solid generated by revolving the region bounded by the graph [tex]x^2=y-2[/tex] and [tex]2y-x-2=0[/tex] for 0≤x≤1 about y=3 is approximately 12.422 cubic units.
How to find the work done in stretching the spring from its natural length to a length of 10 in?(b) (i) The work done in stretching the spring from its natural length of 6 in. to a length of 10 in. can be found using the formula:
W =[tex](1/2)k(d2^2 - d1^2)[/tex]
where k is the spring constant, d1 is the initial length, and d2 is the final length.
Given that the force required to stretch the spring from its natural length of 6 in. to a length of 8 in. is 9 lb., we can find the spring constant as follows:
k = F/(d2 - d1) = 9/(8-6) = 4.5 lb/in
So the work done in stretching the spring from its natural length of 6 in. to a length of 10 in. is:
W = [tex](1/2)(4.5)(10^2 - 6^2)[/tex]= 54 lb.-in.
Therefore, the work done in stretching the spring from its natural length to a length of 10 in. is 54 lb.-in.
How to find the work done in stretching the spring from a length of 7 in. to a length of 9 in?(ii) To find the work done in stretching the spring from a length of 7 in. to a length of 9 in., we can use the same formula:
W =[tex](1/2)k(d2^2 - d1^2)[/tex]
Using the same spring constant of 4.5 lb/in, the work done is:
W = [tex](1/2)(4.5)(9^2 - 7^2)[/tex]≈ 13.5 lb.-in.
Therefore, the work done in stretching the spring from a length of 7 in. to a length of 9 in. is approximately 13.5 lb.-in.
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I NEED HELP ON THIS ASAP! PLEASE, IT'S DUE TONIGHT!!!!
The distance travelled by the jet in 25 minutes found using area covered under the graph is 3.7 miles.
What is area?
The size of a surface or the area that any two-dimensional object or figure covers is known as its area.
Area of triangle = [tex]\frac{bh}{2}[/tex]
Area of rectangle=l x w
The area covered under graph= area of triangle+ area of rectangle
= [tex]\frac{bh}{2}[/tex] + l x w
Dimension of triangle:
base(time on x-axis)=5 seconds
height(speed on y-axis)= 600 miles per hour
=600÷3600 miles per seconds
=0.167 miles per seconds
Dimensions of rectangle:
length(time on x-axis):25-5 =20 seconds
width(speed on y-axis)= 600 miles per hour
=600÷3600 miles per seconds
=0.167 miles per seconds
Distance = The area covered under graph
= area of triangle+ area of rectangle
= [tex]\frac{bh}{2}[/tex] + l x w
=[tex]\frac{5(0.167)}{2}[/tex] + 20(0.167)
=0.4175 + 3.34
=3.7575
Distance ≈3.7 miles
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Factor each completely if possible
(1) x^2 - 11x + 28
(2) 2x^2 + 8x + 6
(3) k^2 - 25
(4) a^2 - 9a + 20
(5) 7x^2 - 11x - 6
(6) 14x^2 - 52x + 30
(7) 6n^3 - 8n^2 + 3n - 4
(8) 15y^3 - 3v^2 + 20v - 4
Factors (1) x² - 11x + 28 = (x-4)(x-7), (2) 2x² + 8x + 6 = 2(x+1)(x+3), (3) k² - 25 = (k+5)(k-5), (4) a² - 9a + 20 = (a-5)(a-4), (5) 7x² - 11x - 6 = (7x+2)(x-3) , (6) 14x² - 52x + 30 = 2(7x-3)(x-5), (7) 6n³ - 8n² + 3n - 4 = (2n-1)(3n²-2n+4), (8) 15y³ - 3v² + 20v - 4 = (5y-1)(3y²+1)(4-v)
Describe Factorization?Factorization is a process of finding the factors of a given mathematical expression, which can be a number, polynomial, or algebraic expression. In other words, factorization involves breaking down a mathematical expression into simpler terms that multiply together to give the original expression. For example, the factors of the expression x^2 - 4 are (x + 2)(x - 2).
In algebra, factorization is an important tool for solving equations and simplifying expressions. By factoring, we can often simplify complex expressions, making them easier to work with and understand. In addition, factorization plays an important role in number theory, where it is used to find prime factors and calculate the greatest common divisor and least common multiple of numbers.
(1) x² - 11x + 28 = (x-4)(x-7)
(2) 2x² + 8x + 6 = 2(x+1)(x+3)
(3) k² - 25 = (k+5)(k-5)
(4) a² - 9a + 20 = (a-5)(a-4)
(5) 7x² - 11x - 6 = (7x+2)(x-3)
(6) 14x² - 52x + 30 = 2(7x-3)(x-5)
(7) 6n³ - 8n² + 3n - 4 = (2n-1)(3n²-2n+4)
(8) 15y³ - 3v² + 20v - 4 = (5y-1)(3y²+1)(4-v)
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construct a 95onfidence interval for the population variance σ2 if a sample of size 25 has standard deviation s = 14. round the answers to two decimal places.
We can say with 95% confidence that the population variance σ2 lies within the interval [155.25, 570.06].
To construct a 95% confidence interval for the population variance σ2, we can use the chi-square distribution.
First, we need to calculate the chi-square values for the upper and lower limits of the confidence interval. We use the formula:
chi-square upper = (n-1)*s^2 / χ^2(α/2, n-1)
chi-square lower = (n-1)*s^2 / χ^2(1-α/2, n-1)
where n is the sample size, s is the sample standard deviation, α is the level of significance (0.05 for 95% confidence interval), and χ^2 is the chi-square distribution function.
Plugging in the values, we get:
chi-square upper = (25-1)*14^2 / χ^2(0.025, 24) = 43.98
chi-square lower = (25-1)*14^2 / χ^2(0.975, 24) = 15.14
Next, we can use these chi-square values to calculate the confidence interval for σ2:
confidence interval = [(n-1)*s^2 / chi-square upper, (n-1)*s^2 / chi-square lower]
Plugging in the values, we get:
confidence interval = [(25-1)*14^2 / 43.98, (25-1)*14^2 / 15.14]
confidence interval = [155.25, 570.06]
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Work out the area of this semicircle. Take to be 3.142 and give your answer to 2 decimal places. Diameter is 8cm.
Answer:
3.142 in 2 decimal is 3.100
Step-by-step explanation:
When we come to diameter of 8cm I don't know
A cartographer at point C sites a prominent rock feature, at point R, East from his location. There is a grassy peak, at point G, at a distance of “y” miles directly North of the cartographer. The angle formed by the cartographer, rock feature, and grassy peak is “x” degrees. See the diagram below. Using complete sentences, explain how the cartographer can use only these two measurements to calculate the distance from the grassy peak to the rock feature.
True: A Chorochromatic map is a type of cartographic map that represents features depending on how they are distributed across the surface in terms of quality.`
We have,
The art and science of cartography involves visually depicting a geographic location, typically on a flat surface like a map or chart. It could include superimposing a region's depiction with non-geographical distinctions like political, cultural, or other ones.
Making and utilizing maps is the theory and application of cartography. Cartography, which combines science, aesthetics, and method, is based on the idea that reality may be described in ways that effectively convey spatial information. The same basic components are included on most maps: the main body, the legend, the title, the scale and orientation indications, the inset map, and the source notes.
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complete question:
a cartographic map style that symbolizes features based on the qualitative surface distribution of a mapped feature is called a chorochromatic map.
Given m
|| n, find the value of x.
m
n
126°
(8x-10)
8 is the value of x in parallel lines.
With an example, what is a parallel line?
Two lines in the same plane that are equally spaced apart and never meet are known as parallel lines in geometry. They can be either vertical or horizontal.
Examples of parallel lines in our everyday lives include zebra crossings, notebook lines, and railway tracks all around us. No matter how far apart they are on either side, two lines on the same plane are considered parallel if they never cross.
given
m ║n
8x - 10 + 126 = 180°
8x + 116 = 180°
8x = 180° - 116
8x = 64
x = 64/8
x = 8
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c×cxdxd×d divided by
cxcxcxdxd
Answer:
Please leave more information regarding the question thank you
The following linear differential equation models the charge on the capacitor, q(t), at time t in an RLC series circuit. L d^2q/dt^2 + R dq/dt + 1/C q = E(t) Find the charge on the capacitor when L = 10 henry, R = 20 ohms, C = (6260)^-1 farad, and E(t) = 100 volts, with the initial conditions q(0) = 0 coulombs and i(0) = 0 amperes.
The charge on the capacitor at time t is given by q(t) = -21292.5e^(-0.00878t) + 21292.5e^(-314.53t) + 626000 coulombs.
How to find the charge on the capacitor?To find the charge on the capacitor, with the initial conditions q(0) = 0 coulombs and i(0) = 0 amperes, we use the given linear differential equation:
L d^2q/dt^2 + R dq/dt + 1/C q = E(t)
We can solve for q(t) by finding the roots of the characteristic equation, and assuming a particular solution. Then we use the initial conditions to solve for the constants in the general solution.
The solution to the differential equation is:
q(t) = -21292.5e^(-0.00878t) + 21292.5e^(-314.53t) + 626000
Therefore, the charge on the capacitor at time t is given by q(t) = -21292.5e^(-0.00878t) + 21292.5e^(-314.53t) + 626000 coulombs.
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For a population with a proportion equal to 0.32, calculate the standard error of the proportion for the following sample sizes of 40,80,120. round to 4 decimal places
The standard errors of the proportion for sample sizes of 40, 80, and 120 are 0.0733, 0.0518, and 0.0422, respectively,
How to calculate the standard error of the proportion?
To calculate the standard error of the proportion for a population with a proportion equal to 0.32 and sample sizes of 40, 80, and 120, we can use the formula:
Standard Error (SE) = √[(p * (1 - p)) / n]
where p is the proportion (0.32), and n is the sample size.
For a sample size of 40:
SE = √[(0.32 * (1 - 0.32)) / 40]
SE ≈ 0.0733 (rounded to 4 decimal places)
For a sample size of 80:
SE = √[(0.32 * (1 - 0.32)) / 80]
SE ≈ 0.0518 (rounded to 4 decimal places)
For a sample size of 120:
SE = √[(0.32 * (1 - 0.32)) / 120]
SE ≈ 0.0422 (rounded to 4 decimal places)
So, for a population with a proportion equal to 0.32, the standard errors of the proportion for sample sizes of 40, 80, and 120 are approximately 0.0733, 0.0518, and 0.0422, respectively, when rounded to 4 decimal places.
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Given that cos2α-5 and α terminates in quadrant I, find the exact value of sino. sina- (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
The result cos²(α) = -2 is not possible, as the square of cosine must be between 0 and 1. It appears there might be an error in the given information. Please double-check the values and try again.
Given that cos(2α) = -5 and that it terminates in quadrant I, we need to find the exact value of sin(α).
First, let's recall the Pythagorean identity: sin²(α) + cos²(α) = 1.
Since cos(2α) = -5, we need to find the value of cos(α). In order to do this, we'll use the double-angle formula for cosine: cos(2α) = 2cos²(α) - 1.
Now, we can plug in the given value of cos(2α) and solve for cos(α):
-5 = 2cos²(α) - 1
-4 = 2cos²(α)
cos²(α) = -2
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At the city museum, child admission is $5.60. and adult admission is $9.40. On wensday, 177 tickets were sold for a total of $1352.20. how many adult tickets were sold that day?
Let's use variables to represent the number of child and adult tickets sold on Wednesday.
Let c be the number of child tickets sold, and let a be the number of adult tickets sold.
We know that the price of a child ticket is $5.60, and the price of an adult ticket is $9.40.
From the problem statement, we know that 177 tickets were sold in total, so:
c + a = 177
We also know that the total revenue from ticket sales was $1352.20, so:
5.60c + 9.40a = 1352.20
Now we have a system of two equations with two variables. We can solve for a by using the first equation to express c in terms of a, and then substituting into the second equation:
c + a = 177 --> c = 177 - a
5.60c + 9.40a = 1352.20
Substituting c = 177 - a into the second equation, we get:
5.60(177 - a) + 9.40a = 1352.20
Expanding and simplifying:
992.20 - 5.60a + 9.40a = 1352.20
3.80a = 360
a = 95
Therefore, 95 adult tickets were sold on Wednesday.
Its bugging out but I got 95 tickets I would add explanation if it didn't act out.
X=Adult tickets
Y=Child tickets
X+Y=117
Y=117-X
9.40X+5.60Y=1352.20
9.40X+5.60(117-X)=1352.20
9.40X+991.20-5.60x=1352.20
3.80X=361
X=95
Unit 10: Circles
Homework 5: Inscribed Angles
** This is a 2-page document! **
Directions: Find each angle or arc measure.
The measure of arc FE is 27degrees,angle m<B is 112degrees, <GHJ = <GIJ = 73⁰ , m<S = 90 degrees in the given circles
The sum of angle in the triangle DEF is 180 degrees
mFE = <D
Recall that <D+<E+<F = 180⁰
<D+63+90 = 180
<D = 180-153
<D = 27 degrees
Hence the measure of arc FE is 27degrees
6) For this circle geometry, we will use the theorem
The sum of Opposite side of a cyclic quadrilateral is 180 degrees.
A + C = 180
m<A + 101 = 180
m<A = 180-101
m<A = 79degrees
Similarly
B + D = 180
m<B + 68 = 180
m<B = 180-68
m<B = 112degrees
7) The sum of angle in a circle is 360, hence;
arcGJ+68+31+115 = 36p
arcGJ = 360 - 214
arcGJ = 146⁰
Since the angle at the centre is twice angle at the circumference, then;
<GHJ = 1/2 arcGJ
<GHJ = 1/2(146)
<GHJ = 73⁰
<GHJ = <GIJ = 73⁰ (angle in the same segment of the circle are equal)
8) Recall that the sum of Opposite side of a cyclic quadrilateral is 180 degrees.
P + R = 180
57 + <R = 180
m<R = 180-57
m<R = 123degrees
Similarly, m<Q+m<S = 180⁰
Since the triangle in a semi circle is a right angled triangle, hence m<Q = 90 degrees (triangle PQR is a right angled triangle)
m<S = 180 - 90
m<S = 90 degrees
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George is randomly selecting an outfit from his dresser. He has two pair of blue pants and four pair of black pants. In his closet there are two blue shirts, four green shirts, and one red shirt.
1. what is the probability he selects black pants and a green shirt?
2. what is the probability he selects a green or blue shirt?
3. what is the probability he does not choose a blue shirt?
Answer:
Probability of George picking black pants and a green shirt is 2/13
Probability of George picking a green shirt is 6/7
Probability of George not choosing a blue shirt 5/7
A farmer builds a water though to fit in a corner. The water though is made of two rectangular prisms
A) Length A = 5 ft and width B= 4 ft
B) Volume of the water though = 88 [tex]ft^3[/tex]
What is volume?
The space taken up by any three-dimensional solid constitutes a volume, to put it simply. A cube, cuboid, cone, cylinder, or sphere can be one of these solids. Cubic units are used to measure the volume of solids. The volume will be given in cubic metres, for instance, if the dimensions are given in metres.
Here consider the prism plane figure the dotted lines are equal to the width,
Then width B= 8-4 = 4 ft
Length A = 8-3 = 5 ft
B) Now volume of rectangular prism = lwh cubic unit.
Volume of big prism = 8*2*3=48 [tex]ft^3[/tex]
Volume of small prism = 5*4*2 = 40 [tex]ft^3[/tex]
Then Total volume = 48+40 = 88 [tex]ft^3[/tex]
Hence in the rectangular prisms,
A) Length A = 5 ft and width B= 4 ft
B) Volume of the water though = 88 [tex]ft^3[/tex]
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which equations are true? Select the four correct answers. A. 3/4=6/8 B. 4/6=10/12 C. 2/3=8/12 D. 8/8=5/5 E. 2/5=4/10 F. 1/4=5/8
Answer:
The Correct answers are
A
C
D
E
Answer:
Correct Answers:
A 3/4=6/8
C 2/3=8/12
D 8/8=5/5
E 2/5=4/10
Step-by-step explanation:
-Where does typology come from
-how does it work
-Defines types and antitypes of typology
-what assumptions do we make about the bible for typology to work
Learn
Dotebook
1. Mario has 12 boxes of pizza He cut each pizza into eights. How mar
pieces of pizza will there be?
Answer: 96 slices
Step-by-step explanation:
double integral of x^2 2y bound by y = x, y = x^3, and x> 0
The double integral of x² 2y over the given region (bounded by y = x, y = x³, and x > 0) is 2/35.
How to evaluate the double integral of the x² 2y?To solve this double integral, we need to integrate the function x² 2y over the given region in the xy-plane.
The region is bounded by the curves y = x and y = x³, and the line x = 0.
First, we need to determine the limits of integration. Since x > 0,
we can integrate from x = 0 to x = 1.
For each value of x in this range, the lower bound of y is given by y = x, and the upper bound is given by y = x³.
Therefore, we need to integrate with respect to y from y = x to y = x³ for each value of x in the range [0, 1].
So, the double integral can be written as:
∫(0 to 1) ∫(x to x³) x² 2y dy dx
Integrating with respect to y first, we get:
∫(0 to 1) [x² y²]x³_x dy dx= ∫(0 to 1) (x⁶ - x⁴) dx= [1/7 x⁷ - 1/5 x⁵]0_1= 1/7 - 1/5= 2/35Therefore, the double integral of x² 2y over the given region is 2/35.
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An artist is creating a scale drawing of a mural in the shape of a right triangle she will paint for the city. Her drawing is 8 inches long and has a hypotenuse of 15 inches. If the mural has a hypotenuse of 96 inches, how long is the mural?
Answer:
51.2 inches
Step-by-step explanation:
You want the length of a mural whose hypotenuse is 96 inches if the scale drawing has a length of 8 inches and a hypotenuse of 15 inches.
RatiosThe ratios of corresponding lengths will be the same:
drawing length / drawing hypotenuse = mural length / mural hypotenuse
8 in / 15 in = mural length / 96 in
SolutionMultiplying the equation by 96 in, we have ...
(96 in)·8/15 = mural length
51.2 in = mural length
The mural is 51.2 inches long.
F(1) = 15 f(n)= f(n-1) x n evaluate the sequences in recursive form
Answer:
f(1) = 15
f(n) = f(n-1) x n
Step-by-step explanation:
The sequence in recursive form is:
f(1) = 15
f(n) = f(n-1) x n
Using this recursive formula, we can find the value of any term in the sequence by calculating the value of the previous term and multiplying it by the index of the current term.
For example, to find the value of f(2), we would use the formula:
f(2) = f(1) x 2
f(2) = 15 x 2
f(2) = 30
Similarly, to find the value of f(3), we would use the formula:
f(3) = f(2) x 3
f(3) = 30 x 3
f(3) = 90
And to find the value of f(4), we would use the formula:
f(4) = f(3) x 4
f(4) = 90 x 4
f(4) = 360
We can continue using this formula to find the values of any term in the sequence.
Q- 6
Triangle NMO is drawn with vertices N(−5, 2), M(−2, 1), O(−3 , 3). Determine the image vertices of N′M′O′ if the preimage is reflected over x = −5.
N′(5, 2), M′(2, 1), O′(3, 3)
N′(2, −5), M′(1, −2), O′(3, −3)
N′(0, 2), M′(3, 1), O′(2, 3)
N′(−5, 2), M′(−8, 1), O′(−7, 3)
The vertices of the triangle after reflection is
D)N′(−5, 2), M′(−8, 1), O′(−7, 3).
What is triangle?
A triangle is a form of polygon with three sides; the intersection of the two longest sides is known as the triangle's vertex. There is an angle created between two sides. One of the crucial elements of geometry is this
Remember that the general rule to reflect over a vertical line in the form
if x = a then,
=> (x , y) -> (-x-2a , y)
For x = 5, we'll have that the general rule is:
=> (x , y) -> (-x-10 , y).
Now the triangle vertices are,
N(-5,2) => (-(-5)-10,2) => (5-10 , 2) => N'(-5,2)
M(-2,1) => (-(-2)-10,1) => (2-10,1) => M' (-8,1)
O(-3,3) => (-(-3)-10,3) => (3-10,3) => O'(-7,3)
Hence the vertices of the triangle after reflection is D)N′(−5, 2), M′(−8, 1), O′(−7, 3).
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Complete the square to re-write the Quadratic function in vertex form
The vertex form of the given quadratic equation is.
y = -5(x + 6)² - 356
How to complete squares?To complete squares we need to use the perfect square trinomial:
(a + b)² = a² + 2ab +b²
Here we can rewrite our quadratic as follows:
y = -5x² - 60x - 176
y = -5*( x² + 12x) - 176
y = -5*( x² + 2*6x) - 176
Now we can add and subtract 6² = 36 then we will get:
y = -5*( x² + 2*6x + 36 - 36) - 176
y = -5*( (x + 6)² - 36) - 176
y = -5(x + 6)² - 356
Which means that the vertex is at (-6, -356)
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Can someone pls help me out with this?
Answer:
Every day, 33% of locusts are added to the locust population
Can anyone post fake answers on this website
Answer:
If you're asking if its possible, yes it is
Classify the following triangles as obtuse, acute, or right triangle, using the side- length relationship. a. 15, 16, 17 b. 20, 18, 7 c. 17, 144, 145 d. 24, 32, 40.
a. 15, 16, 17 : The triangle is an acute triangle
b. 20, 18, 7 : The triangle is an obtuse triangle
c. 17, 144, 145 : The triangle is a right triangle
d. 24, 32, 40 : The triangle is a right triangle
Classifying triangles as Obtuse, Acute or RightFrom the question, we are to classify the given triangles as obtuse, acute or right triangles
To classify the triangles, we will consider the longest side of the triangles
If the square of the longest side is lesser than the sum of squares of the other two sides, the triangle is acute If the square of the longest side is equal to the sum of squares of the other two sides, the triangle is rightIf the square of the longest side is greater than the sum of squares of the other two sides, the triangle is obtusea. 15, 16, 17
Is 17² = 15² + 16² ?
17² = 289
15² + 16² = 225 + 256 = 481
NO,
17² < 15² + 16²
Thus,
The triangle is an acute triangle
b. 20, 18, 7
Is 20² = 18² + 7² ?
20² = 400
18² + 7² = 324 + 49 = 373
NO,
20² > 18² + 7²
Thus,
The triangle is an obtuse triangle
a. 17, 144, 145
Is 145² = 144² + 17² ?
145² = 21025
144² + 17² = 20736 + 289 = 21025
YES,
Thus,
The triangle is a right triangle
a. 24, 32, 40
Is 40² = 32² + 24² ?
40² = 1600
32² + 24² = 1024 + 576 = 1600
YES
Hence,
The triangle is a right triangle
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On a recent quiz, the class mean was 73 with a standard deviation of 3.1. Calculate the z-score (to at least 2 decimal places) for a person who received score of 71. Z-Score: ____Is this unusual? A. Unusual B. Not Unusual
The, a z-score of -0.65 is not unusual
To calculate the z-score, we use the formula:
[tex]z =\frac{ (x - μ)}{σ}[/tex]
where x is the individual score, μ is the mean, and σ is the standard deviation.
Plugging in the values given, we get:
[tex]z= \frac{71-73}{3.1}[/tex]
z = -0.65
Rounding to 2 decimal places, the z-score is -0.65.
To determine if this score is unusual or not, we need to compare it to the normal distribution. A z-score of -0.65 means that the individual's score is 0.65 standard deviations below the mean.
According to the empirical rule, about 68% of the data falls within 1 standard deviation of the mean. Therefore, a z-score of -0.65 is not unusual and falls within the normal range of scores.
So, the answer is B. Not Unusual.
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