Find the smallest part when £80 is
shared in the ratio 7:3

Answer:

(5,-1/2)

Step-by-step explanation:

Midpoint formula is x1 +x2 /2 is equal to the x-value, y2+y1 /2 is the y-value. Simply plug it into the formula to get the answer

S satisfies all the conditions of being a subring of R, and we can conclude that S is indeed a subring of R.

To show that S is a subring of R, we need to verify the following three conditions:

1. S is closed under addition: Let x, y belong to S. Then, we have ax = 0 and ay = 0. Adding these equations, we get a(x + y) = ax + ay = 0 + 0 = 0. Thus, x + y belongs to S.

2. S is closed under multiplication: Let x, y belong to S. Then, we have ax = 0 and ay = 0. Multiplying these equations, we get a(xy) = (ax)(ay) = 0. Thus, xy belongs to S.

3. S contains the additive identity and additive inverses: Since R is a ring, it has an additive identity element 0. Since a0 = 0, we have 0 belongs to S. Also, if x belongs to S, then ax = 0, so -ax = 0, and (-1)x = -(ax) = 0. Thus, -x belongs to S.

Therefore, S satisfies all the conditions of being a subring of R, and we can conclude that S is indeed a subring of R.

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The formula for dy/dx is the derivative of y with respect to x. It represents the rate of change of y with respect to x.

In calculus, the derivative of a function with respect to a variable x is a measure of how much the function changes as the variable x changes. The notation for the derivative of a function y with respect to x is dy/dx, read as "delta y over delta x". This notation is called a "derivative" and it represents the rate of change of the function y with respect to x.

The derivative of a function can be calculated using the limit definition of the derivative or with the use of different rules such as power rule, product rule, quotient rule, chain rule, and more. The derivative can also be represented graphically as the slope of a tangent line to the function's graph.

Derivatives are widely used in mathematics, physics, engineering, and other sciences to model various phenomena.

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Answer:

86, 65

Step-by-step explanation:

Quotient is 86

Remainder is 65

If one item is chosen from m items, a second item is chosen from n items, and a third item is chosen from p items, the total number of three-item choices is MNP.

Probability  can be regarded as a  measure of the likelihood of an event  that can take place.

It should be noted that  Many events cannot be predicted with total certainty, hence , If one item is chosen from m items, a second item is chosen from n items, and a third item is chosen from p items, the total number of three-item choices is MNP.

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Answer:

16x^4

Step-by-step explanation:

Answer:16x

Step-by-step explanation:

Area of square A = 16 cm

side of square A a1 = 4cm

side of square A = 2/3(2/3a3)

a1= 4/9a3

a3 = 9cm

Area of square C = 9×9= 81 Cm²

Answer:

you have 26 cups so i think its the 4th one or the 5th one

Step-by-step explanation:

Hope this helps!

Answer:

Sara is 9 years old while Yusuf's is 11 years old

Step-by-step explanation:

The given parameters are;

The ratio Sara and Yusuf share the money they receive = The ratio of their ages

The amount Sara and Yusuf received from their grandmother this year = R1,500

The amount of money Sara receives = R675

Therefore, the amount of money Yusuf receives = R1,500 - R675 = R825

Let 'x', and 'y' represent the ages of Sara and Yusuf

Therefore, the ratio of their ages, x/y = R675/R825) = 9/11

x·11 = y·9

x, and y < 18

When x = 9, y = 11, we  get;

9·11 = 11·9

Therefore a possible solution is Sara's age = x = 9 and a possible solution for Yusuf's age, y = 9

Answer:

Step-by-step explanation:

6+3=9/50

9/50 x  100=18 percent

Henry could only see one line since both lines had the same slope, which means that the graphs of both equations will be identical and hence overlap.

Linear equations in a system 3x + 2y = 4, and 9x + 6y = 12

We must demonstrate why Henry could only make out one line when he plotted the equations 3x-2y=4 and 9x-6y=12 on a graph.

Take the provided linear equation system into consideration.

3x - 2y = 4 ................(1)

9x - 6y = 12 ..................(2)

Due to the fact that equation (2) is a multiple of equation (1), 3 (3x - 2y = 4) = 9x - 6y = 12

The slopes of the provided equations are also same.

Difference with regard to x for equation (1) yields,

additional to equation (2),

With regard to x, we can differentiate to get,

The graphs of both equations will overlap since both lines have the same slope and hence have the same appearance on the graph.

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Answer:

y=3 you have to use PEMDAS

Answer:

y=3

Step-by-step explanation:

Combine multiplied terms into a single fraction

2.4y-3.2=4

12y/5 -3.2=4

Add 3.2 to both sides of the equation

12y/5- 3.2+3.2=4+3.2

Simplify

add the numbers 12y/5=7.2

Solution

y=3

Answer:

(fog)( -8) = -327.

Explanation:

Given the functions, f(x) and g(x):

To evaluate (fog)( -8), first, we rewrite it as follows:

First, evaluate g(-8).

Next, substitute g(-8):

Thus, the value of (fog)( -8) is -327.

Answer:

Step-by-step explanation:

I think 17.5

The HCF is (a-2) and the LCM is 288(a+2)(a+1)(a+5) of 4(a²-4), 6(a²-a-2) and 12(a² + 3a-10)

To find the highest common factor (HCF) and least common multiple (LCM) of the given expressions, let's factorize each expression first:

4(a²-4) = 4(a+2)(a-2)

6(a²-a-2) = 6(a-2)(a+1)

12(a²+3a-10) = 12(a+5)(a-2)

The common factors among these expressions are (a-2). So the HCF is (a-2).

To find the LCM, we multiply all the distinct factors from the factorizations:

LCM = 4 × 6 × 12 × (a+2)(a+1)(a+5) = 288(a+2)(a+1)(a+5)

Therefore, the HCF is (a-2) and the LCM is 288(a+2)(a+1)(a+5).

Let's factorize each expression:

a(c + a) - b(b + c) = a² + ac - b² - bc

b(a + b) - c(c + a) = ab + b² - c² - ac

c(b + c) - a(a + b) = cb + c² - a² - ab

The common factors among these expressions are (a+b+c). So the HCF is (a+b+c).

To find the LCM, we multiply all the distinct factors from the factorizations:

LCM = (a+b+c)(a² + ac - b² - bc)(ab + b² - c² - ac)

Therefore, the HCF is (a+b+c) and the LCM is (a+b+c)(a² + ac - b² - bc)(ab + b² - c² - ac).

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Answer:

In 2007 it was 450 and 2011 it was 300

Step-by-step explanation:

So the change was 300 I think this correct

The cross sectional area of the cargo trailer floor, which is a composite figure consisting of a square and an isosceles triangle, indicates that the volume of the inside of the trailer is about 3,952 ft³.

A composite figure is a figure comprising of two or more regular figures.

The possible cross section of the trailer, obtained from a similar question on the internet, includes a composite figure, which includes a rectangle and an isosceles triangle.

Please find attached the cross section of the Cargo Trailer Floor created with MS Word.

The dimensions of the rectangle are; Width = 6 ft, length = 10 ft

The dimensions of the triangle are; Base length 6 ft, leg length = 4 ft

Height of the triangular cross section = √(4² - (6/2)²) = √(7)

The cross sectional area of the trailer, A = 6 × 10 + (1/2) × 6 × √(7)

A = 60 + 3·√7

Volume of the trailer, V = Cross sectional area × Height

V = (60 + 3·√7) × 6.5 = 3900 + 19.5·√7

Volume of the trailer = (3,900 + 19.5·√(7)) ft³ ≈ 3952 ft³

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To prove that the sum of the exterior angles of any polygon is equal to 360 degrees, the Sum of interior angles = (n - 2) x 180 degrees.

we can use the following steps:

Draw any polygon and mark a point outside the polygon for each vertex.

Draw a line segment connecting each vertex to the point outside the polygon that corresponds to it, creating an exterior angle at each vertex.

Measure each exterior angle using a protractor and record the measurements.

Sum all the exterior angles together and observe the result.

We can see that the sum of all the exterior angles is always equal to 360 degrees, regardless of the number of sides or the shape of the polygon. This can be written as:

The sum of exterior angles = 360 degrees

To prove this algebraically, we can use the fact that the sum of the interior angles of any polygon is given by the formula:

Sum of interior angles = (n - 2) x 180 degrees

where n is the number of sides of the polygon. Each exterior angle is supplementary to the corresponding interior angle, so we can write:

The sum of exterior angles = Sum of interior angles (supplementary angles)

Substituting the formula for the sum of interior angles, we get:

Sum of exterior angles = (n - 2) x 180 degrees (supplementary angles)

= 180n - 360 degrees

= 360 degrees - 360 degrees + 180n

= 360 degrees - Sum of interior angles

= 360 degrees - (n - 2) x 180 degrees

= 360 degrees - 180n + 360 degrees

= 720 degrees - 180n

Since the sum of the exterior angles must be a positive value, we can take the absolute value and simplify:

|Sum of exterior angles| = |720 degrees - 180n|

= 180|4 - n|

Since n is always a positive integer greater than or equal to 3 (since a polygon must have at least three sides), we know that 4 - n is always a negative integer between -1 and -n+1. Therefore, |4 - n| is equal to n - 4. Substituting this into the equation above, we get:

|Sum of exterior angles| = 180|n - 4|

= 180(n - 4)

= 180n - 720 degrees

= 360 degrees - 360 degrees + 180n - 720 degrees

= 360 degrees - Sum of interior angles

Therefore, we have shown that the sum of the exterior angles of any polygon is equal to 360 degrees.

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The number of times one needs to use completely filled cone to completely fill the cylinder with water is 24

The volume of cylinder is equal to the product of the area of the circular base and the height of the cylinder.

Given that radius of the base of a cylinder is 10 centimeters, and its height is 20 centimeters.

The radius of the cone's base is 5 centimeters, and its height is 10 centimeters.

The volume of cylinder is πr²h

For a volume of cone it is 1⁄3πr²h.

So volume of cylinder = 22/7×(10)2×20=3.142×100×20

=6284

volume of cone it is 1⁄3πr²h=1/3×3.142×(5)^2×10=261.16

Number of times one needs to use the completely filled cone to completely fill the cylinder with water =

= Volume of cylinder/Volume of cone

6284 / 261.16 = 24

Hence the number of times one needs to use completely filled cone to completely fill the cylinder with water is 24

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Answer:

The area of this triangle is 825.5 square feet

Step-by-step explanation:

For a given Triangle

Base = 13 ft

Height = 127 ft

Now, Area of Triangle

A = ½ × b × h

A = ½ × 13 × 127 ft

A = ½ × 1751 ft

A = 825.5 square feet

Thus, The area of this triangle is 825.5 square feet.

-TheUnknownScientist 72

The covariance of X and Y is 13/10, correlation coefficient of X and Y is 1,

the Covariance at (3X, 5Y) is also 3/200 of the given continuous random varaible.

a) To find E(XY), we need to calculate the double integral of xy times the joint pdf over the region where the pdf is nonzero:

E(XY) = ∫∫ xy * f(x,y) dx dy

      = ∫0¹ ∫0^(1-x) 24xy * dx dy     (since x + y < 1, the limits of y are from 0 to 1-x)

      = ∫0¹ 12x(1-x)^2 dy

      = ∫0¹ (12x - 24x^2 + 12x^3) dy

      = 4/3 - 6/5 + 3/4

      = 1/5

Therefore, E(XY) = 1/5.

(b) The covariance of X and Y is given by:

Cov(X, Y) = E(XY) - E(X)E(Y)

We have already calculated E(XY) in part (a), so we just need to calculate E(X) and E(Y):

E(X) = ∫∫ x * f(x,y) dx dy

= ∫0¹ ∫0^(1-x) 24xy * x dx dy

= ∫0¹ 6x(1-x)^3 dx

= 1/2 - 1/5

= 3/10

Similarly, we can find that E(Y) = 3/10.

Therefore, Cov(X, Y) = E(XY) - E(X)E(Y) = 1/5 - (3/10)*(3/10) = 1/100.

(c) The correlation coefficient of X and Y is given by:

ρ(X, Y) = Cov(X, Y) / (σ(X)σ(Y))

where σ(X) and σ(Y) are the standard deviations of X and Y, respectively. To find σ(X) and σ(Y), we need to first find the variances of X and Y:

Var(X) = E(X^2) - [E(X)]^2

      = ∫∫ x^2 * f(x,y) dx dy - (3/10)^2

      = ∫0¹ ∫0^(1-x) 24xy * x^2 dx dy - 9/100

      = ∫0¹ 2x(1-x)^3 dx - 9/100

      = 1/18 - 1/40

      = 11/360

Similarly, we can find that Var(Y) = 11/360.

Therefore, σ(X) = sqrt(Var(X)) = sqrt(11/360), and σ(Y) = sqrt(Var(Y)) = sqrt(11/360).

Now, we can find ρ(X, Y) as follows:

ρ(X, Y) = Cov(X, Y) / (σ(X)σ(Y))

= (1/100) / (sqrt(11/360) * sqrt(11/360))

= 1

Therefore, the correlation coefficient of X and Y is 1.

(d) To find Cov(3X, 5Y), we can use the property that Cov(aX, bY) = ab Cov(X, Y) for any constants a and b:

Cov(3X, 5Y) = 3 * 5 * Cov(X, Y)

            = 15 * 1/100

            = 3/200

Therefore, Cov(3X, 5Y) = 3/200

e)we can find the covariance of X + 1 and Y - 2:

Cov(X+1, Y-2) = E[(X+1)(Y-2)] - E[X+1]E[Y-2]

Using the definition of expected value and the joint pdf, we have:

E[(X+1)(Y-2)] = ∫∫(x+1)(y-2) f(x,y) dx dy

             = ∫[0,1] ∫[0,1-x] (x+1)(y-2) 24xy dy dx

             = ∫[0,1] 24x ∫[0,1-x] (xy-2x-2y+2) dy dx

             = ∫[0,1] 24x [x(1-x)^2 - 2x(1-x) - 2(1-x)^2 + 2(1-x)] dx

             = 2/3

Similarly, we can find:

E[X+1] = E[X] + 1/2 = 1

E[Y-2] = E[Y] - 2/3 = 1/3

Therefore,

Cov(X+1, Y-2) = E[(X+1)(Y-2)] - E[X+1]E[Y-2] = 2/3 - 1*(1/3) = 1/3

f)To find the covariance of X + 1 and 5Y - 2, we can use the same formula as above and compute the required expected values:

E[(X+1)(5Y-2)] = ∫∫(x+1)(5y-2) f(x,y) dx dy

              = ∫[0,1] ∫[0,1-x] (5xy-2x+5y-2) 24xy dy dx

              = ∫[0,1] 24x ∫[0,1-x] (5xy-2x+5y-2) dy dx

              = 4/3

E[X+1] = 1

E[5Y-2] = 5E[Y] - 2/3 = 13/3

g)Cov(3X + 5, X) = E[(3X + 5)X] - E[3X + 5]E[X]

To find E[(3X + 5)X], we'll use the definition of expected value for continuous random variables:

E[(3X + 5)X] = ∫∫ (3x+5)x f(x,y) dy dx

       = ∫∫ (3x^2 + 5x) (24xy) dy dx from y=0 to y=1-x and x=0 to x=1

       = ∫ 0^1 ∫ 0^(1-x) (72x^2y + 120xy) dy dx

       = ∫ 0^1 36x^2(1-x)^2 + 60x(1-x)^3 dx

       = 6/5

Next, we need to find E[3X + 5]:

E[3X + 5] = 3E[X] + 5

To find E[X], we'll integrate the marginal pdf of X:

f_X(x) = ∫ (24xy) dy from y=0 to y=1-x = 12x(1-x) for 0<x<1

E[X] = ∫ x f_X(x) dx from x=0 to x=1

 = ∫ 0^1 x (12x(1-x)) dx

 = 1/2

Putting it all together:

Cov(3X + 5, X) = E[(3X + 5)X] - E[3X + 5]E[X]

           = 6/5 - (3(1/2) + 5)(1/2)

           = -2/5

Therefore, Cov(3X + 5, X) = -2/5.

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The time taken for the cargo to hit the ground will be 0.6 seconds and it travels in the horizontal direction for around 14.69 feet.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign. The point-slope form, standard form, and slope-intercept form are the three main types of linear equations. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Here,

The calculations are attached in images.

To calculate t we will use that is y because at the moment of cargo will have a height 0, so we get,

= −4.9t² +1.75

-1.75=-4.9t²

t²=-1.75/-4.9

t²=√0.3571

=0.6 seconds

Now we will calculate x, then y = 0

because since the cargo will be on the ground it will no have height, so

0= -0.0081x²+1.75

-1.75 =-0.0081x²

x²=-1.75/-0.0081

x= √216.04 = 14.69 feet

It will take the cargo 0.6 seconds to hit the ground, and it will travel 14.69 feet in a horizontal direction.

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Answer:

The first term should be positive

She did not distribute the -4 properly

She added \(\frac{2}{7}\) to negative -4 instead of multiplying

Step-by-step explanation:

In order to tell where she is wrong we should solve the question our selves. we are going to get the following......

\(-4(-3x + \frac{2}{7})= -4(-3x) + (-4)(\frac{2}{7} )= 12x - \frac{8}{7}\)

So now we see that the three things she did wrong are.

The first term should be positive

She did not distribute the -4 properly

She added \(\frac{2}{7}\) to negative -4 instead of multiplying

Hope this helps :)

Answer:

1, 4

Step-by-step explanation:

1. the product of two negatives are positive and they didn't do that, so yes.

2. the second term should NOT be positive, so they are correct on that.

3. they distributed the -4 correctly but added wrong, so they are correct on that.

4. -4+2/7=3 5/7    2/6(-4)=-4/3.  They added instead of multiplying, so they are wrong.

5.  they did simplify completely.  they just did it wrong.

Using five equal intervals and Euler's method, we approximate the value of y(1) to be 3.69 (corrected to 2 decimal places).

Euler's method is a first-order numerical procedure used for solving ordinary differential equations (ODEs) with a given initial value. In simple terms, Euler's method involves using the tangent line to the curve at the initial point to estimate the value of the function at some point.

The formula for Euler's method is:

y_(i+1) = y_i + h*f(x_i, y_i)

where y_i is the estimate of the function at the ith step, f(x_i, y_i) is the slope of the tangent line to the curve at (x_i, y_i), h is the step size, and y_(i+1) is the estimate of the function at the (i+1)th step.

Given that y' = xy and y(0) = 3, we want to approximate the value of y(1) using five equal intervals. To use Euler's method, we first need to calculate the step size. Since we want to use five equal intervals, the step size is:

h = 1/5 = 0.2

Using the initial condition y(0) = 3, the first estimate of the function is:

y_1 = y_0 + hf(x_0, y_0) = 3 + 0.2(0)*(3) = 3

The second estimate is:

y_2 = y_1 + hf(x_1, y_1) = 3 + 0.2(0.2)*(3) = 3.12

The third estimate is:

y_3 = y_2 + hf(x_2, y_2) = 3.12 + 0.2(0.4)*(3.12) = 3.26976

The fourth estimate is:

y_4 = y_3 + hf(x_3, y_3) = 3.26976 + 0.2(0.6)*(3.26976) = 3.4588

The fifth estimate is:

y_5 = y_4 + hf(x_4, y_4) = 3.4588 + 0.2(0.8)*(3.4588) = 3.69244

Therefore , using Euler's approach and five evenly spaced intervals, we arrive at an approximation for the value of y(1) of 3.69 (adjusted to two decimal places).

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All finite subgroups of C* are the trivial subgroup, the subgroup of roots of unity, the cyclic subgroups, and the subgroup consists of all powers of a single nonzero complex number

Let's explore the finite subgroups of C*, the group of nonzero complex numbers under multiplication.

First, let's recall that C is isomorphic to the group (R, +), where R represents the set of real numbers under addition.

Thus, we can view C as the group of real numbers excluding 0 under multiplication.

Now, since we are looking for finite subgroups, let's consider the possible finite subgroups of C*:

The trivial subgroup:

This consists of the identity element 1, which is the only element in this subgroup.

The subgroup of roots of unity:

This subgroup consists of complex numbers of the form \(e^{2\pi i/n}\), where n is a positive integer.

These numbers are all of finite order, with order n.

This subgroup is isomorphic to the group Zn of integers modulo n under addition.

The cyclic subgroups:

These subgroups are generated by a single nonzero complex number.

The order of the generator determines the order of the subgroup.

For example, if we take a complex number z and generate a subgroup, it will consist of z, z², z³, and so on, until we reach zⁿ = 1, where n is the order of the subgroup.

The subgroup consists of all powers of a single nonzero complex number:

This is similar to the cyclic subgroups, but instead of considering the powers of a single element, we consider the powers of all nonzero complex numbers.

This subgroup is isomorphic to the group of integers Z under addition.

These are the main types of finite subgroups of C. Each of these subgroups has its own unique properties and structure, making C a fascinating group to study.

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Answer

A) 4b+11   B) To evaluate the expression make sure you give the value of B because in order to simplify you need to find out the answer to b to then evaluate

Step-by-step explanation:

Part A

8b+8-4b+3

8b-4b+8+3

4b+11

Answer:

4b+11

Step-by-step explanation:

8b + 8 - 4b + 3

8b + 8 - 4b + 3

4b+8+3

4b+8+3

4b+11

The system of linear equations are;

a + b = 5.30

a + b = 7

Expressing the information as a system of linear equations:

Consider that apples = a, oranges = b

If $5.30 is charged for one apple and one orange, then we get the equation as

a + b = 5.30 - - - (1)

If $14 is charged for 2 apples and 2 oranges,  then we get the equation as ;

2a + 2b = 14 - - - - (2)

a + b = 7

Since both equations give varying combined cost for an equal amount of fruit, so a unique cost cannot be obtained for each fruit from the systems of equation using a simultaneous equation process.

From (1)

a = 5.30 - b

Put a, b in (2)

2(5.30 - b) + 2b = 14

10.6 - 2b + 2b = 14

10.6 = 14

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Answer:

ok so x is her currect age y is her age in 20 years

2x+6=y

x+20=y

that means that both sides are equal

2x+6=x+20

-6

2x=x+14

-x

x=14

so her currect age is 14 and lets plug in

14+20=34

her currect age is 14 and her age in 20 years is 34

Hope This Helps!!!