The cost of 3 cheese sandwiches and 2 bottles of water is £7. 80. The cost of 5 cheese sandwiches and 4 bottles of water is £14. 20. What is the cost of: (a) one cheese sandwich? (b) one bottle of water?

x(t) = [tex]c_1e^1^4^t+c_2e^-^t[/tex]  , y(t) = [tex]2c_1e^1^4^t-c_2e^-^t[/tex]   is the general solution of the sytem of differention eqution dx/dt = 4x+ 5y , dy/dt = 10x + 9y .

given  dx/dt = 4x+ 5y  and dy/dt = 10x + 9y

X'(t) = [tex]\left[\begin{array}{ccc}4&5\\10&9\end{array}\right][/tex]  X

where X = [tex]\left[\begin{array}{ccc}x\\y\\\end{array}\right][/tex]

so A = [tex]\left[\begin{array}{ccc}4&5\\10&9\end{array}\right][/tex]

now we need to find eigen value of the matrix and the corresponding eigen vector.

| A- λI | = 0

so after equation the value to zero

λ = 14 and λ = -1

now for the  λ = 14 corresponding eigen vector = [tex]\left[\begin{array}{ccc}1\\2\\\end{array}\right][/tex]

for λ = -1 corresponding eigen vector =  [tex]\left[\begin{array}{ccc}1\\-1\\\end{array}\right][/tex]

So general equation is given by:

X(t) = [tex]c_1e^1^4^t[/tex]  [tex]\left[\begin{array}{ccc}1\\2\\\end{array}\right][/tex]  +  [tex]c_2e^-^t[/tex]   [tex]\left[\begin{array}{ccc}1\\-1\\\end{array}\right][/tex]

after solving this

x(t) = [tex]c_1e^1^4^t+c_2e^-^t[/tex]

y(t) = [tex]2c_1e^1^4^t-c_2e^-^t[/tex]

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Answer:

x = - 2 or x = - 4

Step-by-step explanation:

Via completing the square:

x^2 + 6x + 8  = x^2 + 6x + (6/2)^2 - (6/2)^2 + 8

                     = (x + 3)^2 - 3^2 + 8

                     = (x + 3)^2 - 9 + 8

                     = (x + 3)^2 - 1

Since x^2+6x+8=0,

(x + 3)^2 - 1 = 0

(x + 3)^2 = 1

x + 3 = +- sqrt(1)

x + 3 = +- 1

x + 3 = 1 or x + 3 = -1

x = -2 or x = -4

On solving the provided question, we can say that - the value of the cube root is 19683 =  27

In mathematics, y3 = x since the cube root of a number x equals a number y. Each non-zero real integer has precisely one real cube root, one set of complex conjugate cube roots, and three different complex cube roots. When a number is multiplied three times, the result is the cube root of that number. Three cube roots, the final cube root, is the symbol for the cube root. The opposite of cubing an integer is finding the cube root of that number. Prime factorization is a fairly straightforward technique that may be used to get a number's cube root. The sign stands for a cube root.

here,

we have f(27)

cube will be [tex]( 27 )^3[/tex] = 19683

and f(-343) = -7

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The union of the intervals is (-∞,∞)
The intersection of the intervals is [0,2)

Hope this helped!

The total cost of an item that sells for $12 is $12.96

From the question, we have the following parameters that can be used in our computation:

Selling price = $12

Sales tax = 8%

The total cost of an item that sells for $12 is then calculated as

Total cost = Selling price * (1 + Sales tax)

Substitute the known values in the above equation, so, we have the following representation

Total cost = 12 * (1 + 8%)

Evaluate

Total cost = 12.96

Hence, the total cost is $12.96

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The polynomial in terms of its simplest factors is (x + 3)(x − 6)

From the question, we have the following parameters that can be used in our computation:

x2 − 3x − 18.

Rewrite as

x² − 3x − 18

Expand

x² − 3x − 18 = x² − 6x + 3x − 18

Factorize the expression

This gives

x² − 3x − 18 = x(x − 6) + 3(x − 6)

Factor out x - 6

x² − 3x − 18 = (x + 3)(x − 6)

Hence, the expression is (x + 3)(x − 6)

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The resulting polynomial in standard form (b)  –4g^4 –3g³ +4g² +5g +3

Suppose the considered polynomial is of only one variable.

Then, the standard form of that polynomial is the one in which all the terms with higher exponents are written on left side to those which have lower exponents.

The simplification process is described and partially carried out. We are to finish by combining like terms and writing the result in standard form.

Given

Add: (g² – 4g^4 + 5g + 9) + (–3g³ + 3g² – 6)

Combine Like Terms

g²  + (–4g^4) + 5g + 9 + (–3g³) + 3g²  + (–6)

The like terms are the g² terms and the constants;

= (1 +3)g² + (–4g^4) +5g +(9 +(–6)) +(–3g³)

 = 4g² +(–4g^4) +5g +3 +(–3g³)

Write in Standard Form

In this form, the terms are written in order of decreasing degree.

= –4g^4 –3g³ +4g² +5g +3

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The expression which represents the value of the area could be the six times the circumference of the circle.

What is area of circle ?

area of circle can be defined as the product of pi(3.14) and square of radius of the circle.

Given,

helicopter pad which is in the shape of a circle has a circumference 72.48

and the diameter of 24 feet.

radius = diameter/2

radius = 24/2

radius = 12

circumference of circle = 72.48

area of circle =  π*r*r

area of circle = 22/7 * 12 * 12

area of circle = 452.5

area of circle is approximately 6 times the circumference of circle.

Hence , The expression which represents the value of the area could be the six times the circumference of the circle.

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In the expression 52 + 6 + ya,

The terms are the individual parts of the expression that are being added together. The coefficient is the numerical factor that is multiplied by a variable in a term, and the constant is a term that has no variable.

Any expression with variables will either add or delete one or more variables. Any one of a variety of numbers to represent a generic idea. An absolute number is a constant. A constant is a number, whereas a coefficient is a number that is "in front of" a variable.

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Draw a line and locate the origin '0', mark the positive numbers on the right and the negative number on the left of '0' at an equal distance.

A number line is a horizontal straight line in which the integers are placed in equal intervals. All the numbers can be represented in a number line. This line can be extended indefinitely at both ends. 

A number line is a graphical representation of numbers on a straight line. It is used for comparing and ordering numbers. It can be used for representing any real number that includes every whole number and natural number.

Draw a line and locate the point ‘0’. This point is known as the origin. If the given number is positive, draw it on the right side of the origin. If it is a negative number, draw it on the left side of zero. Divide each unit into the values equal to the denominator of the given fraction.

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Kaj bought 3 bananas and 5 apples.

From these three pieces of information, we can set up the following two equations:

We can substitute the third equation into the first equation and get:

y+2 + y = 8

So we know that y = 3

Now we can substitute this value into the second equation and get:

x + 0.5(3) = 8

So we know that x = 5

So Kaj bought 3 bananas and 5 apples.

And also we can substitute y = 3 into the third equation and get:

x = y + 2

So we know that x = 5

So Kaj bought 3 bananas and 5 apples.

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Step-by-step explanation:

(8 + 2)/(6 - p)= -10

10/6 - p = -10

-60 + 10p = 10

10p= 70

p= 7

The rate exists water being poured into the cup when the water level is 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

To determine the derivative of a composite function in differential calculus, apply the chain rule formula. If y = f(g(x)), then according to the chain rule, the instantaneous rates of change of the functions f and g with respect to x produce the instantaneous rate of change of f with respect to x.

A composition of functions, such as the composition of the functions f and g, f(g(x)), can be used to determine the derivative using the chain rule.

Let us set up the following variables:

R, Radius of conical cup (cm) = 30 cm

H, Height of the conical cup (cm) = 10 cm

Our aim is to find dV/dt when h = 9 and we are given dh/dt = 2

By similar triangles we have

r : h = R : H

r/h = 30/10

⇒ r = 3h

The volume of a cone is 1/3π r²h, So:

V = 1/3π r²h

simplifying the equation, we get

= 1/3π (3h²)h

= 1/3π 9h²h

= 3π h³

Differentiating wrt h we get:

dV/dh = 9π h²

Applying the chain rule we have:

dV/dt = (dV/dh)(dh/dt)

= 9π h² × 2

= 18π h²

If h = 9 then

[tex]$\left[\frac{d V}{d t}\right]_{h=9}=18 \pi\left(9^2\right)=1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

Therefore, the rate exists water being poured into the cup when the water level 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

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the main three types of trigonometry are ,

1- sine angle (sin Q)

2- cosine angle (cos Q)

3- tangent angle (tan Q)

all the other types of trigonometry are related to these three types only.

sine angle

we can find out the sine angle by taking perpendicular of the triangle divided by hypotenuse of the triangle. sine angle is a acute angle.

cosine angle

we can find out the cosine angle by taking base of the triangle divided by hypotenuse of the triangle. cosine angle is a acute angle.

tangent angle

we can find out the sine angle by taking perpendicular of the triangle divided by base of the triangle.

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The system of equations .5(5x + 4) = 17.5x + 14 have Infinite solutions.

We know two simultaneous equations have a unique solution when they intersect at a point,

when they are parallel they have no solution and when they are coinciding they have an infinite no. of solutions.

Given, A system of equation 3.5(5x + 4) = 17.5x + 14.

17.5x + 14 = 17.5x + 14.

As the LSH and RHS are equal the equation of the two lines is the same and hence they coincide and have an infinite number of solutions.

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Answer:

51.50004959055205

Step-by-step explanation:

[tex]sin (x) = \frac{18}{23}[/tex]

The response to the given domain range questions is [go f oh]. (9) = 5(18^2 - 1 )

Domain and range. A function's authorised range of values is known as its domain. The x values in a function like f make up this set (x). A function's range is the collection of values that it can take as input. When we enter an x value, the function outputs this set of values.

f(x) = x^5 – 1, g(x) = 5x^2, and h(x) = 2x

[g o f o h](9)=g(f(h(9))

Let's find h(9) first

h(x)= 2x

h(9)= 2(9) =18

Now plug in 18 for h(9)

[g o f o h](9)=g(f(h(9)) = g(f(18))

now we find f(18)

f(x) = x^5 – 1, f(18)= 18^5 -1

Now we plug in 18^5 -1 for f(18)

[g o f o h](9)=g(f(h(9)) = g(18^5 -1)

Plug it in g(x)

g(x)= 5x^2

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well, maybe Pam has been eating the wrong chips or something, but if she's correct, that can only happen if both equations are exactly equal, are they?

[tex]\boxed{\cfrac{x}{16}=4}\implies x=(16)(4)\implies \cfrac{x}{4}=16 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{x}{4}=16\implies x=(4)(16)\implies \boxed{\cfrac{x}{16}=4}\qquad \begin{array}{llll} \textit{yeap, Pam is correct}\\ \textit{she's Da Bomb} \end{array}[/tex]

Area of quadrilateral is 24 units and it is known as parallelogram.

A point in the plane is one-to-one connected to an ordered pair of real numbers (x, y) using point plotting. The 2-dimensional Cartesian coordinate system is as a result obtained.

A parallelogram results from plotting the four points.

the lower side is between (-6,-3) and (-2,-3)

0 to 3 is at the top (4,3)

As a result, the base length is 4 because the distance

between (-6,-3) and (-2,-3) = 4

the separation between (0,3) and (4,3)

The sides are the lines that connect (0,3) to (-6,-3)

along with (4,3) to (-2,-3).

Vertical height ranges from -3 to 3 or 6.

The parallelogram's area is given by base*height = 4(6) = 24.

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The solution of given triangle is 32°.

Therefore, an isosceles triangle has two equal sides as well as two equal angles. The Greek words iso (same) and skelos are the source of the name (leg). An equilateral triangle is one in which all of its sides are equal, and a scalene triangle is one in which none of its sides are equal.

Given,

∠BAC=52°

since, ΔBAC is isosceles triangle hence opposite angle will be equal.

∠BAC+∠ABC+∠ACB=180°

2∠BCA=180°-52°

∠BCA=128/2

∠BCA=64°

also,

∠BCA+∠DCA=180°

∠ACD=180°-64°

∠ACD=116°

since,

ΔACD is isosceles triangle, hence opposite angle is equal

∠ACD+∠CAD∠ADC=180°

2∠CAD=180°-116°

∠CAD=64/2

∠CAD=32°

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Therefore ,the answer to the above trigonometry issue is therefore that the river is 70 feet wide.

The study of the connections between the angles and sides of triangles is known as trigonometry. Due to the fact that each straight-sided figure may be reduced to a collection of triangles, trigonometry is commonly utilized in geometry. There are a mind-boggling number of connections between trigonometry and other branches of mathematics, especially algebra, real or complex, integrals, logarithms, and infinite series.

Here,

Figure depicts a right triangle which, in relation to angle, its next side has length d or its wrong side is equal to the river's width, y. As a result,

tanθ= d/ y

=> y=dtanθ

=> y = (100m)tan(35.00)=70ft

The river measures 70 feet in width.

As a result, the trigonometry problem's answer indicates that the river's width is 70 feet.

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The statements that are always true regarding the diagram are:

i. m∠5 + m∠6 =180°

ii. m∠2 + m∠3 = m∠6

iii. m∠2 + m∠3 + m∠5 = 180°

A triangle is a 2 dimensional plane figure which is formed by three straight lines and has three interior angles. It has both interior and exterior angles. Some examples of a triangle are; isosceles, equilateral, scalene, right angled etc.

The interior angles of a triangle are the measure of the three internal angles of the triangle, while exterior angles are external to the interior angles of the triangle.

Note that; the exterior angle of a triangle is equal to the sum of the two adjacent interior angles.

Therefore considering the given diagram, the required statements that are always true regarding the diagram are:

i. m∠5 + m∠6 =180°

ii. m∠2 + m∠3 = m∠6

iii. m∠2 + m∠3 + m∠5 = 180°

A sketch of the diagram is herewith attached to this answer for clarity.

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Answer: To find the area of the pre-image and the image, you can use the coordinates of the vertices of the shapes to calculate the lengths of the sides and use the formula for the area of a triangle:

Area = (base * height)/2

For the pre-image, the coordinates are X(3,18) Y(18,18) and Z(18,9). The base of the triangle is the distance between X and Z, which is 18 - 3 = 15. The height of the triangle is the distance between Y and Z, which is 18 - 9 = 9. The area of the pre-image is therefore:

Area = (15 * 9)/2 = 67.5

For the image, the coordinates are X’(1,6) Y’(6,6) Z’ (6,3). The base of the triangle is the distance between X’ and Z’, which is 6 - 1 = 5. The height of the triangle is the distance between Y’ and Z’, which is 6 - 3 = 3. The area of the image is therefore:

Area = (5 * 3)/2 = 7.5

The relationship between the two areas is that the area of the image is smaller than the area of the pre-image. This is because the image has been scaled down in size relative to the pre-image. The amount by which the image has been scaled down can be calculated by dividing the area of the image by the area of the pre-image:

Scale factor = area of image / area of pre-image = 7.5 / 67.5 = 0.11

This means that the image has been scaled down by a factor of 0.11 relative to the pre-image.

Step-by-step explanation:

Answer:

Step-by-step explanation:

You want to know the areas and their relationship of the preimage with coordinates X(3,18), Y(18,18), and Z(18,9), and the image with coordinates X’(1,6), Y’(6,6), Z’ (6,3).

When you plot the points on a graph, you see they define right triangles. Each has sides aligned with the x- and y-axes, so it is easy to find their dimensions by counting grid squares or subtracting coordinates.

The preimage triangle is 18-3 = 15 units in the x-direction, and 18-9 = 9 units in the y-direction.

The image triangle is 6-1 = 5 units in the x-direction, and 6-3 = 3 units in the y-direction.

We notice that both the coordinates and the dimensions of the image triangle are 1/3 those of the preimage triangle.

The area of each triangle is given by the formula ...

  A = 1/2bh

Using the dimensions we found above, the areas are ...

  preimage area: 1/2(15)(9) = 135/2 = 67.5 . . . . square units

  image area: 1/2(5)(3) = 15/2 = 7.5 . . . . square units

We have observed that the image dimensions are 1/3 those of the preimage. The image area is 7.5/67.5 = 1/9 of that of the preimage. This value is the square of the scale factor of the dimensions,

The value of x and y in triangles AEB and BCD is 40° and 50°.

A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.

We know the sum of all the interior angles in a triangle is 180°.

Therefore, In ΔAEB,

(2x + 5)° + 50° + 45° = 180°.

2x + 100° = 180°.

2x = 180° - 100°.

2x = 80°.

x = 40°.

Now, ΔBCD,

x + y + 90° = 180°.

40° + y + 90° = 180°.

y = 180° - 130°.

y = 50°.

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K must be equal to 0, for the pair of equations to have no solution.

For the pair of equations KX + 3y - K = 0 and 12x + Ky - K = 0 to have no solution, the value of K must be such that the two equations have no common solution.

This can be done by equating the two equations and solving for K.

KX + 3y - K = 0

12x + Ky - K = 0

Subtracting equation (1) from equation (2)

11x + 3y = 0

Divide both sides by 3

x + y = 0

Therefore, K must be equal to 0, for the pair of equations to have no solution.

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The degree of the polynomial 7x³ 5x 4x³ 2x² is 3.

The degree of a polynomial is the highest power of x in the polynomial.

In this case, the polynomial is 7x³ 5x 4x³ 2x², and the highest power of x is x³. This means that the degree of this polynomial is 3.

It's important to note that when working with polynomials it's important to pay attention to the signs of the coefficients. In this case, we have 7x³, 5x, 4x³, 2x². The x³ term has the highest degree and it's coefficient is 7, the x term has the second highest degree and it's coefficient is 5, the x³ term has the same degree as the first x³ term, but the coefficient is different, and the x² term has the fourth highest degree and it's coefficient is 2.

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Answer:

absolute deviation means distance between each data value and the mean of the data set.

Step-by-step explanation:

(MAD)

or

(Average)

The factory can make 744 quarts of ice cream per day.

To find out how many quarts of ice cream could be made in 48 hours, we can use the principle of proportionality. Since the factory makes 310 quarts of ice cream in 10 hours, we can set up a proportion:

310 quarts / 10 hours = x quarts / 48 hours

To solve for x, we can cross multiply:

310 quarts * 48 hours = 10 hours * x quarts

x = (310 quarts * 48 hours) / 10 hours = 1480 quarts

So the factory could make 1480 quarts of ice cream in 48 hours.

To find the rate per day, we need to divide the number of quarts made per hour by the number of hours in a day. Since there are 24 hours in a day, we can calculate the rate per day as:

rate per day = (310 quarts / 10 hours) * 24 hours = (31 quarts/hour) * 24 hours = 744 quarts/day

So the factory can make 744 quarts of ice cream per day.

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Answer:

The price of a table is rs1720

Step-by-step explanation:

Let C and T be the individual prices of the tables and chairs.

We know that 3C + 4T = rs7540

We are then told that C = rs220, so we can enter that into the equation to find T:

3C + 4T = rs7540  

3(220) + 4T = rs7540  [for C = rs220]

660 + 4T = 7540

4T = rs6880

T = rs1720

Check:

Do 3 chairs at 220 each and 4 tables at 1720 each add to a total of 7540?

Chairs:   rs660

Tables:  rs6880

Total = rs7540  YES