The simple interest on a certain sum at 5% per annum for 3 years and 4years differ by rupees 82 find the sum

Answer: There were 52 Republicans and 46 Democrats.

Step-by-step explanation:

Let's represent the number of Republicans as "x" and the number of Democrats as "y".

From the problem, we know that:

x + y = 98 (since the total number of senators is 98)

y = x - 6 (since there were 6 fewer Democrats than Republicans)

We can substitute the second equation into the first equation to get:

x + (x-6) = 98

2x - 6 = 98

2x = 104

x = 52

Now that we know that there were 52 Republicans, we can substitute that into the second equation to find the number of Democrats:

y = x - 6

y = 52 - 6

y = 46

Therefore, there were 52 Republicans and 46 Democrats.

Answer:

There were 52 Republicans and 46 Democrats.

Step-by-step explanation:

Let's make a system of equations to represent this! D = democrats and R = republicans.

[tex]d + 6 = r[/tex]

[tex]r + d = 98[/tex]

Now let's set them both equal to d.

To do this for the first equation, subtract 6 from both sides to get:

[tex]d =r - 6[/tex]

To do this for the second equation, subtract r from both sides to get:

[tex]d = 98 - r[/tex]

Now, let's set them equal to eachother.

[tex]r - 6 = 98 - r[/tex]

Add r to both sides

[tex]2r -6 = 98[/tex]

Add 6 to both sides

[tex]2r = 104[/tex]

Divide both sides by two

[tex]r = 52[/tex]

There were 52 republicans. Now let's sub 52 in for r in one of the equations.

[tex]d + 6 = 52[/tex]

subtract six from both sides

[tex]d = 46[/tex]

There were 46 democrats.

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The answer is 12units

The surface area of the triangular prism is 2016 [tex]feet ^2[/tex], we get to the answer by adding area of all faces in this prism.

What is surface area ?

Surface area is the sum of the areas of all the faces or surfaces of a 3D object, measured in square units.

To find the surface area of the triangular prism, we need to calculate the area of each of its faces and then add them up.

First, let's find the area of the triangular base.

Area of triangle = (1/2) x base x height

Area of triangle = (1/2) x 18 x 24

Area of triangle = 216 [tex]feet ^2[/tex]

Next, let's find the area of each rectangular face.

Area of rectangle = length x width

Area of rectangle 1 = 24 x 22 = 528 [tex]feet ^2[/tex]

Area of rectangle 2 = 18 x 22 = 396 [tex]feet ^2[/tex]

Area of rectangle 3 =  22 x 30 = 660 [tex]feet ^2[/tex]

Now, we can add up the areas of all three faces to get the total surface area of the prism:

Surface area = area of rectangle (1+2+3) + 2 x area of triangle

Surface area = 528 + 396 + 660 + 2(216)

Surface area = 2016 [tex]feet ^2[/tex]

Therefore, the surface area of the triangular prism is 2016 [tex]feet ^2[/tex].

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The equation that models the shape of the monument is simply y = 25.

The equation that models the shape of the parabolic monument can be written in the form of a quadratic function, y = ax^2 + bx + c, where y is the height of the monument at a given distance x from the center of the base.

Since the monument has a height of 25 feet at the center of the base, we know that the vertex of the parabolic shape is located at (0, 25). Also, since the base width is 30 feet, the distance from the center of the base to either side is 15 feet.

Therefore, we can use the information about the vertex and the width to write the equation as y = -a(x-15)^2 + 25.

To determine the value of a, we need another point on the curve. Let's use one of the endpoints of the base, which is (15, 0). Plugging these values into the equation, we get:

0 = -a(15-15)^2 + 25

0 = 25

This is not possible, so we need to adjust the equation to fit the known points. We can rewrite the equation as y = a(x-15)^2 + 25, and solve for a using the other endpoint of the base, which is (-15, 0):

25 = a(-15-15)^2 + 25

0 = 900a

a = 0

This means that the equation that models the shape of the monument is simply y = 25.

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Answer:

Step-by-step explanation:

e

Answer:

Step-by-step explanation:

First, we need to find the coordinates of the center of the circle. Since the equation of the circle is x^2 + y^2 = 40, we have x^2 = 40 - y^2. Substituting this into the equation of the tangent line I, we get:

(40 - y^2) + y(2x) = 40

2xy - y^2 = 0

y(2x - y) = 0

Since the line I crosses the x-axis at point P, we have y = 0 at that point. Therefore, 2x - y = 0, and we can solve for x:

2x = y

2x = 6

x = 3

So the center of the circle is at the point (3, 0).

Next, we need to find the radius of the circle. Since A is on the circle, we have:

3^2 + 6^2 = r^2

r^2 = 45

r = sqrt(45) = 3sqrt(5)

Now we can find the coordinates of point O, which is the center of the circle:

O = (3, -3sqrt(5))

Finally, we can find the area of triangle OAP:

OA = sqrt((2-3)^2 + (6+3sqrt(5))^2) = sqrt(69)

AP = 2

Using the formula for the area of a triangle, we have:

Area(OAP) = 1/2 * OA * AP = 1/2 * sqrt(69) * 2 = sqrt(69)

Therefore, the area of triangle OAP is sqrt(69) square units.

Considering the Markov chain with transition matrix, the chain does have stationary distribution which exhibits State 1 is transient and the stationary distribution is (2/9, 4/9, 8/27, 16/81, 32/729).

(a) Yes, the chain has a stationary distribution. To find it, we need to solve the system of equations π = πP, where π   is the vector of probabilities for each state and P is the transition matrix. This gives us:

π_{1} = π(1/2)

π_{2} = π(1/3) + π(2/2)

π_{3}= π(2/4) + π(3/2)

π_{4} = π(3/5) + π(4/2)

π_{5}= π4/5

We also have the normalization condition π1 + π2 + π3 + π4 + π5 = 1.

Solving this system of equations, we get:

π_{1} = 10/97

π_{2} = 30/97

π_{3}= 40/97

π_{4} = 14/97

π_{5}= 3/97

So the stationary distribution is (10/97, 30/97, 40/97, 14/97, 3/97).

(b) State 1 is transient, and all other states are recurrent.

(c) Yes, the chain still has a stationary distribution. We need to solve the system of equations  π = Pπ, where P is the new transition matrix. This gives us:

π_{1} = π(1/2)

π_{2} = π(1/3) + π(2/2)

π_{3}= π(2/4) + π(3/2)

π_{4} = π(3/5) + π(4/2)

π_{5}= π4/5

We also have the normalization condition π1 + π2 + π3 + π4 + π5 = 1.

Solving this system of equations, we get:

π_{1} = 2/9

π_{2}  = 4/9

π_{3} = 8/27

π_{4} = 16/81

π_{5}  = 32/729

So the stationary distribution is (2/9, 4/9, 8/27, 16/81, 32/729)

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Answer:

b = 12.12 (or) 7 sqrt 3

Step-by-step explanation:

a^2 + b^2 = c^2

7^2 + b^2 = 14^2

49 + b^2 = 196

-49 -49

b^2 = 147

b = sqrt 147

decimal form:

b = 12.12

exact form:

b = 7 sqrt 3

As a result, there are 180 students in the seventh grade.

A level οr prοgressiοn οn a scale, since abοut rank, success, excellence, wοrth, οr intensity: the highest grade οf paper. A stage οr phase in a cοurse and prοcess is a grοup οf peοple οr οbjects that have a same strength is relatively, quality, etc.

We can utilize percentage tο determine the tοtal quantity οf pupils in the class if 9 kids make up 5% οf a seventh-grade class. If we assume that there are x pupils inside the seventh grade οverall, we can calculate the prοpοrtiοn as fοllοws:

9 / x = 5 / 100

We may crοss-multiply and simplify tο find x's value:

9 × 100 = 5 × x

900 = 5x

x = 900 / 5

x = 180

In the seventh grade, there are 180 students.

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By answering the presented question, we may conclude that Therefore, the expression 9(4/3m-5-2/3m+2) is equivalent to 6m - 27.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematic, and form. They are used in the representation of mathematical formulas, the solution of equations, and the simplification of mathematical relationships.

To simplify the expression,

[tex]a(b+c) = ab + ac\\9(4/3m-5-2/3m+2) = 9(4/3m - 2/3m - 5 + 2)\\= 9(2/3m - 3)\\= 6m - 27[/tex]

Therefore, the expression 9(4/3m-5-2/3m+2) is equivalent to 6m - 27.

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A. The power of the two-tailed hypothesis test with α = .05 is 0.53.

B. The power of the one-tailed hypothesis test with α = .05 is 0.95.

A hypothesis test is used to make decisions about a population based on sample data. It involves specifying a null hypothesis, collecting data, and then assessing the data to either reject or accept the null hypothesis.

A. The power of the two-tailed hypothesis test is calculated using the formula:

Power=

1- β=1- (1-α)[tex]^{1/2}[/tex]

=1- (1-0.05)[tex]^{1/2}[/tex]

=0.525

=0.53

This means that the researcher has an 53% chance of correctly rejecting the null hypothesis that there has been no change in the IQ over the past 10 years.

B. The power of the one-tailed hypothesis test is calculated using the formula:

Power=

1- β=1- α

=1- 0.05

= 0.95

This means that the researcher has a 95% chance of correctly rejecting the null hypothesis that there has been no increase in the IQ over the past 10 years.

This is a higher power than the two-tailed test because the one-tailed test focuses on detecting an increase in the IQ, while the two-tailed test can detect both an increase or decrease in the IQ.

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Answer:

0. 25 25/100

Step-by-step explanation:

The general solution of the given trignometric function tan3x . 1/tan24° - 1 = 0​ is x = nπ/3 + 2π/45.

Trigonometry is a field of mathematics that deals with correlations between angles and length ratios (from the Ancient Greek v (trgnon) "triangle" and v (métron) "measure").

The field was created in the Hellenistic era in the third century BC as a result of the use of geometry in astronomical research.

While Indian mathematicians produced the earliest tables of values for trigonometric ratios (also known as trigonometric functions), such as sine, the Greeks concentrated on chord calculation.

Trigonometry has been used historically in fields like geodesy, surveying, celestial mechanics, and navigation.

So, we have the function:

tan3x . 1/tan24° - 1 = 0​

Now, solve as follows:

tan3x . 1/tan24° - 1 = 0​

tan3x - tan24° = 0

tan3x = tan24°

tan3x = tan(24π/180)

3x = nπ + 2π/15

x = nπ/3 + 2π/45

Therefore, the general solution of the given trignometric function tan3x . 1/tan24° - 1 = 0​ is x = nπ/3 + 2π/45.

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Answer:

The simplified expression is sec^2a

Step-by-step explanation:

We can start by using the trigonometric identities:

cot^2 a + 1 = csc^2 a

tan^2 a + 1 = sec^2 a

Using these identities, we can rewrite the expression as:

(cos^2 a - cot^2 a)/(sin^2 a - tan^2 a)

= (cos^2 a - (csc^2 a - 1))/(sin^2 a - (sec^2 a - 1))

= (cos^2 a - csc^2 a + 1)/(sin^2 a - sec^2 a + 1)

Now we can use the identity:

sin^2 a + cos^2 a = 1

to rewrite the expression further:

= (1/sin^2 a - 1/sin^2 a cos^2 a)/(1/cos^2 a - 1/cos^2 a sin^2 a)

= (1 - cos^2 a)/(sin^2 a - sin^2 a cos^2 a)

= sin^2 a / sin^2 a (1 - cos^2 a)

= 1 / (1 - cos^2 a)

= sec^2 a

Therefore, the simplified expression is sec^2 a.

In the context of this problem, if we reject the null hypothesis, we can conclude that the new design is more effective in terms of accuracy. If we fail to reject the null hypothesis, we can't conclude anything about the effectiveness of the new design

When given the resulting p-values, we reject the null hypotheses (assuming a 0.05 significance level) if the p-value is less than 0.05. We fail to reject the null hypotheses if the p-value is greater than or equal to 0.05.If we reject the null hypothesis, it means that there is strong evidence against it, and we can say that the alternative hypothesis is supported by the data. If we fail to reject the null hypothesis, it means that there is insufficient evidence to reject it, and we can't say that the alternative hypothesis is supported by the data.

In the context of this problem, if we reject the null hypothesis, we can conclude that the new design is more effective in terms of accuracy. If we fail to reject the null hypothesis, we can't conclude anything about the effectiveness of the new design.In either case, whether we reject or fail to reject the null hypothesis, we wouldn't proceed with changing the design of the arrow based solely on the statistical analysis. We would need to consider other factors such as cost, feasibility, and practicality before making any decisions about changing the design.

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The student is male, which is P(male).Using the rule of complementary events, P(male) = 1 - P(female)P(male) = 1 - 0.572 = 0.428Therefore, the probability that the student is male is 0.428 or 42.8%.

The complimentary event is a part of probability theory. It is the event that occurs when the event E does not occur. In other words, it is a set of outcomes in the sample space that are not included in the outcomes of event E. The notation for the complement of E is E'. Rule for Complimentary EventsThe rule for complementary events can be expressed in two ways:

P(E) = 1 - P(E)P(E) + P(E') = 1Example # 12:Let the probability that Mary can work a problem be P(E) = 0.70.We need to find the probability that Mary cannot work the problem, which is P(E').Using the rule of complementary events,P(E') = 1 - P(E)P(E') = 1 - 0.70 = 0.30Therefore, the probability that Mary cannot work the problem is 0.30 or 30%.Example # 13:Let P(female) be the probability that the student is female. We are given that P(female) = 0.572.

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The fixed amount Hester earns for each television is $65.

A straight line in a two-dimensional coordinate system is described by a linear equation. Using m as the line's slope and b as the y-intercept, it has the generic form y = mx + b.

Mathematical applications of linear equations include algebra, geometry, and calculus. They are employed to represent and examine interactions between different variables, such as the connection between time and space in a straightforward linear motion issue.

Let us suppose the fixed amount = x.

Then, for 15 televisions she earns:

15*$25 = $375

Now, she earned a total of 1350, thus:

15x + $375 = $1,350

15x = $975

x = $65

Therefore, the fixed amount Hester earns for each television is $65.

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The ability of the normal distribution to approximate the binomial distribution under appropriate conditions justifies the use of the normal distribution for the sampling distribution of the proportion.

A sampling distribution is a statistical probability distribution derived from a larger number of samples gathered from a certain population. The sampling distribution of a particular population is the distribution of frequencies of a range of possible outcomes for a population statistic.

A population is the whole pool from which a statistical sample is selected in statistics. A population can be defined as a large group of people, things, events, medical visits, or measures. A population can thus be defined as an aggregate observation of persons linked by a common trait.

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The value of x is 8/3 or 2.67 cm (rounded to two decimal places).

The problem at hand entails determining the value of a variable "x" in a right-angled triangle.

We can begin by applying the Pythagorean Theorem, which states that the square of the length of the hypotenuse in a right-angled triangle equals the sum of the squares of the lengths of the other two sides.

We have a right-angled triangle ABC in this case, with AC as the hypotenuse, AB as one of the other sides, and BC as the remaining side.

We know that AB equals 5 cm and BC equals x cm. We must locate AC.

The Pythagorean Theorem yields:

[tex]AC^2 = AB^2 + BC^2[/tex]

[tex]AC^2 = 5^2 + x^2[/tex]

[tex]AC^2 = 25 + x^2[/tex]

Also, we know that AC = 2x + 1. We get the following result when we plug this number into the above equation:

[tex](2x + 1)^2 = 25 + x^2[/tex]

[tex]4x^2 + 4x+ 1 = 25 + x^2[/tex]

[tex]3x^2 + 4x - 24 = 0[/tex]

The above quadratic equation can be factored as follows:

[tex](3x - 8)(x + 3) = 0[/tex]

As a result, either[tex]3x - 8 = 0 or x + 3 = 0.[/tex]

We get the following when we solve for x:

3x - 8 = 0

3x = 8

x = 8/3

or

x + 3 = 0

x = -3

We reject the second solution because length cannot be negative.

As a result, the value of x is 8/3 or 2.67 cm (rounded to two decimal places).

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As per the volume, the dimension of an Extra Large Box is 3.78 feet.

Let's call the length of one side of the cube "s". Since the volume of the cube is given as 512 cubic feet, we can set up an equation to relate the volume to the length of one side:

Volume of cube = s³ = 512 cubic feet

To solve for "s", we can take the cube root of both sides of the equation:

s = ∛512

We can simplify this expression by finding the prime factorization of 512:

512 = 2⁹

Therefore, we can rewrite the expression for "s" as:

s = ∛2⁹

Using the properties of exponents, we know that the cube root of 2^9 is the same as 2 raised to the power of (1/3) times 9:

s = [tex]2^{1/3} \times 9^{1/3}[/tex]

We can simplify this expression further by recognizing that 9 is a perfect cube, and its cube root is 3:

s = [tex]2^{1/3} \times 3[/tex]

Therefore, the length of one side of the cube-shaped box is:

s = [tex]2^{1/3} \times 3[/tex] feet

Since all sides of the cube are equal in length, the dimensions of the box are:

Length = Width = Height = [tex]2^{1/3} \times 3[/tex] feet = 3.78 feet.

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Complete Question:

Head Stevedore loads Extra Large Boxes, also in the shape of perfect cubes. The volume of each box is 512 cubic feet. What are the dimension of an Extra Large Box?

Answer:

No solution

Step-by-step explanation:

They have the same slope, which makes them parallel to each other. The y-intercepts are different so they do not overlap. If they overlap, then you know they have either one or infinite solutions. They do not so the answer is no solution.

To conclude, the probability of the Acme Drug Company seeing a percent difference of -.13 or less if the true difference is 0 is quite low and is equal to 0.0934.

The probability that the Acme Drug Company would see a percent difference of -.13 or less if the true difference is 0 is quite low. This is because a difference of -.13 is a very small percentage in comparison to a true difference of 0.

Mathematically, the probability of this happening would be equal to the area under the standard normal distribution curve for values between -0.13 and 0. In other words, the probability that the Acme Drug Company would see a percent difference of -.13 or less if the true difference is 0 is equal to the area from the left tail of the standard normal distribution curve up to the mean (0) of the curve.

Using a standard normal distribution calculator, we can see that the probability of the Acme Drug Company seeing a percent difference of -.13 or less is 0.0934. This probability is extremely low and it is not likely that the Acme Drug Company would experience such a small percent difference.

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To summarize, a perfectly egalitarian society has a Gini index of 0, and a perfectly totalitarian society has a Gini index of 1.

The Gini index (G) measures how much the income distribution differs from absolute equality.

On the other hand, a perfectly totalitarian society (where a single person receives all the income) will have a Gini index of 1. This means that there is a huge inequality between individuals, with the single person receiving all of the income.

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Answer:

20%

Step-by-step explanation:0% of 4445=

The probability that the diameter of a selected bearing is greater than 52 millimeters is 0.8413. This can be calculated using the formula for probability of a normal distribution:

P(x > 52) = 1 - P(x ≤ 52)

P(x ≤ 52) = (52 - 58) / 6 = -1

P(x > 52) = 1 - P(-1) = 1 - 0.1587 = 0.8413.

Therefore, the probability that the diameter of a selected bearing is greater than 52 millimeters is 0.8413.

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The probability of producing the genotype AABBCC in a cross AaBbCc × AaBbCc is 1/64.

A Punnett square is a grid used to forecast the likelihood of an offspring of a mating between two parents with known genotypes. The grid consists of four boxes or cells with one parent's gamete genotype listed along the top, and the other parent's gamete genotype listed down the side.

To determine what proportion of their offspring will possess a certain genotype or phenotypic characteristic, the gamete genotypes are combined using the grid.

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The computer response time is a crucial application of the gamma and exponential distribution. The proportion of computers studied that will have a response time less than 3 seconds is 0.6321 or 63.21%.

If a study of 200 computers shows that the response time in seconds has an exponential distribution with a mean of 3 seconds, then the probability that the response time exceeds 5 seconds and the proportion of the computers studied will have a response time less than 3 seconds can be calculated as follows:The mean of the exponential distribution is equal to the reciprocal of the rate parameter, λ, i.e.,E(X) = λ-1 Given, the mean response time, E(X) = 3 seconds.The mean of the exponential distribution, E(X) = λ-1= 3 seconds. Therefore,λ = 1/3. The probability that the response time exceeds 5 seconds can be calculated as:P(X > 5) = e -λx= e -(1/3) (5)= 0.0498 The probability that the response time exceeds 5 seconds is 0.0498 or 4.98%.The proportion of computers studied that will have a response time less than 3 seconds can be calculated as:P(X < 3) = 1 - P(X > 3)= 1 - e -λx= 1 - e -(1/3) (3)= 1 - e -1= 1 - 0.3679= 0.6321 or 63.21%.

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Answer:

We can use similar triangles to solve this problem. Let's call the length of the kite's line "x". Then, we can set up a proportion:

(length of kite) / (length of shadow) = (height of kite) / (length of shadow)

x / 9 = 10 / 9

To solve for x, we can cross-multiply and simplify:

x = 90 / 9

x = 10

Therefore, the length of the kite's line is 10 feet.

Step-by-step explanation:

If the two removed squares have the same color, then it is possible to cover the truncated checkerboard with non-overlapping dominoes. If they have different colors, then it will be impossible to cover the truncated checkerboard with non-overlapping dominoes.

To prove that the chess board can be completely covered by non-overlapping dominoes, where each domino covers exactly two squares, we can color the board alternatively in black and white. The two colors will make the board have 32 black squares and 32 white squares. Each domino will cover one white and one black square. Since there are the same number of black and white squares, there is an equal number of squares that each domino can cover so it is possible to cover the entire board with dominoes without overlap.

To determine whether the cut checkerboard can be covered by non-overlapping dominoes, note that we have 62 squares, 31 of which are black, and 31 white. This means that one color will have one extra square, in this case black. Therefore, if the two removed squares are of different colors, it will be impossible to cover the cut checkerboard with non-overlapping dominoes.However, if the two removed squares have the same color, then it will be possible to cover the checkerboard with non-overlapping dominoes. This is because we will still have an equal number of black and white squares even after the removal of the two squares.The same logic applies to the truncated checkerboard.

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Answer:

235.6 cm²

Step-by-step explanation:

The formula to find the area of a sector is,

[tex]\sf A=\frac{\theta}{360}*\pi r^2[/tex]

Here,

Θ = angle

r = radius

Let us find it now.

[tex]\sf A=\frac{\theta}{360}*\pi r^2\\\\\sf A=\frac{270}{360}*\pi *10*10\\\\\sf A=\frac{270}{360}*\pi *100\\\\A = 235.61 cm^2[/tex]