Sue has 18 sweets.
Tony also has 18 sweets.
Sue gives Tony x sweets.
Sue then eats 5 of her sweets.
Tony then eats half of his sweets.
Write expressions for the number of sweets Tony and Sue now have.

The equation of the plane passing P (1,2,1) and orthogonal to the two planes: x-y-z-10 = 0, x-2y + z-2=0 is -3x-2y-z+8=0.

Equation of plane passing through (x1,y1,z1) is given by

A(x-x1)+B(y-y1)+C(z-z1)=0

where, A, B, and C are direction ratios

In the question, it is given that the plane passes through (1,2,1)

So, the equation of the plane will be in the form,

A(x-1)+B(y-2)+C(z-1)=0

It is also given that the plane is perpendicular to give 2 planes.

So, their normal to the plane would be perpendicular to the normal of both planes.

So, the required normal is a cross-product of the normals of planes

x-y-z-10=0 and x-2y+z-2=0

i.e,

-3i-2j-k=0

so, the direction ratios,

A=-3, B=-2, C=-1

putting the direction ratios in the previous equation of the plane,

-3(x-1)-2(y-2)-1(z-1)=0

-3x+3-2y+4-z+1=0

-3x-2y-z+8=0

is the required equation of the plane

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All of the zeros of f(x) algebraically are; x = -3, x = -1/2, x = -5/2

The zeros of a polynomial p(x) are defined as all the x-values that make the polynomial equal to zero.

The polynomial is;

f(x) = 4x³ + 24x² + 41x + 15

To find the x-intercept, we will equate the polynomial to zero to get;

4x³ + 24x² + 41x + 15 = 0

Since (x + 3) is a factor, then we can factorize the polynomial as;

(x + 3)(4x² + 12x + 5) = 0

Rewriting the quadratic term gives;

(x + 3)(4x² + 2x + 10x + 5) = 0

(x + 3)(2x(2x + 1) + 5(2x + 1)) = 0

Thus, the factors are expressed as;

(x + 3)(2x + 1)(2x + 5) = 0

Thus, the zeros are;

x + 3 = 0

x = -3

2x + 1 = 0

x = -1/2

2x + 5 = 0

x = -5/2

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The points that lie on the y-axis of the graph have a x-coordinate of zero.

Hence the format of the point is of:

(0,y).

The general format of an ordered pair is given as follows:

(x,y).

For the x-coordinate, we have that:

On the y-axis, the point does not move horizontally, just vertically, hence the points that lie on the y-axis of the graph have a x-coordinate of zero, and the format of the point is given as follows:

(0,y).

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Answer:

x = 80

Step-by-step explanation:

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Note that x is being divided by 4. To isolate x, do the opposite. Multiply 4 to both sides of the equation:

[tex]\frac{x}{4} = 20\\(\frac{x}{4} ) * 4 = (20) * 4\\x = 20 * 4\\x = 80[/tex]

x = 80 is your answer.

~

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Answer: x=6, y=-6

Step-by-step explanation:

-4x + -2y = -12 (a)

4x + 8y = -24 (b)

(a)+(b)= -4x+4x+8y+(-2y)=-12+(-24)

                                 6y= -36

                                   y=-6

put y=-6 into formula (a)

-4x+12=-12

-4x= -24

x=6

When solving absolute value inequalities involving the greater than symbol (>) or greater than or equal to symbol (≥), we have to consider two cases absolute value expression greater than the constant and less than the constant.

The absolute value expression is greater than the constant:

|x| > c or |x| ≥ c

In this case, we can split the solution set into two parts: x < -c and x > c. The solution is all real numbers less than -c and all real numbers greater than c.

The absolute value expression is less than the constant:

|x| < c or |x| ≤ c

In this case, we can split the solution set into two parts: -c < x < c. The solution is all real numbers between -c and c, excluding -c and c.

For example, consider the inequality |x| > 2. The solution is x < -2 or x > 2, so the solution set is {x | x < -2 or x > 2}.

--The question is incomplete, answering to the question below--

"When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving >. Explain"

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If AB is parallel to CD and angle EAB =50°,angle ECD =100° then Angle AEC is 112∘

Since AB∥CD

∴∠CAB+∠DCA=180 ∘  (Co-interior angles)

The inside angles total 180 degrees and are known as co-interior angles. It indicates that two internal angles that are on the same side of the transversal have a sum that is additional.

∴22 ∘ +∠DCA=180 ∘

⇒∠DCA=180 ∘ −22 ∘ =158 ∘

Also, ∠ECD+∠DCA+∠y=360 ∘  (Angles about a point)

Angles around a point refer to the total number of angles that can be combined to produce a complete turn. A point's surrounding angles add up to 360°. The angles circling a point add up to 360° since they have completed a full turn and are identical in magnitude.

⇒90 ∘ +158 ∘ +∠y=360 ∘

⇒∠y=360 ∘ −(90 ∘ +158 ∘ )

⇒∠AEC=y=360 ∘ −248 ∘ =112∘

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The focal radius of the ellipse of which the two foci seat are:  

An ellipse can be represented in the standard form as (x/a)² + (y/b)² = 1, where (a,b) is the center of the ellipse and a and b are the semi-major and semi-minor axes respectively.

Given the equation of the ellipse:

16x² + y² - 128x - 20y + 292 = 0

To put it in the standard form, we have to complete the square and then divide both sides by the constant on the right-hand side.

First, we complete the square by adding and subtracting (128/2)² and (20/2)² respectively:

16x² - 128x + (128/2)² + y² - 20y + (20/2)² = 292 + (128/2)² + (20/2)²

Then, we divide both sides by 292:

(16x² - 128x + (128/2)²)/292 + (y² - 20y + (20/2)²)/292 = 1

On simplifying we get:

(4x - 8)²/144 + (y - 10)²/36 = 1

Now we have the standard form of the equation of the ellipse and we can find the semi-major and semi-minor axis and the center of the ellipse.

The semi-major axis is equal to the square root of the coefficient of x squared (144) and the semi-minor axis is equal to the square root of the coefficient of y squared (36).

The center of the ellipse is (-8, 10) and the semi-major and semi-minor axis are 12, 6 respectively.

The focal radius is the distance between the center and the focus. The focal radius is equal to the square root of the semi-major axis squared minus the semi-minor axis squared.

Focal radius = √(a² - b²) = √(144-36) = √108 = 6√3

The two foci of the ellipse sit on the x-axis, symmetric about the center of the ellipse. The foci are located at (-8 ± 6√3, 10).

Therefore, the two foci of the ellipse are at (-8 + 6√3, 10) and (-8 - 6√3, 10).

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48.76%

Compound interest describes interest on the principal plus interest.

Compound Interest Formula

The formula to find compound interest is [tex]A=P(1+\frac{r}{n})^{nt}[/tex]. In the formula:

We can plug in our information to find the rate.

Solving For Interest Rate

First, let's rewrite the equation with our information. Note that for simplicity I will write rounded values for each step, but in reality, no rounding should be done until the last step.

Then, simplify the exponent and divide both sides by 4700.

Take the 60th root of both sides.

Subtract 1 from both sides.

Finally, multiply both sides by 12.

This means that the rate is 0.04876. We can multiply this by 100 to get the rate as a percentage. The interest rate is 48.76%.

Answer:

I'm doing this to complete one of the steps. good luck my friend

Step-by-step explanation:

Answer:

The correc answer is Option A.

The difference between the number of expected outcomes and the number of actual outcomes is 1/30.

What is the difference?

The expected outcome of flipping heads and rolling a number less than 3 = probability of flipping a head x probability of rolling a number less than 3

Probability of flipping a head = number of sides that are heads / total number of sides = 1/2

Probability of rolling a number less than 3 = number of sides that are less than 3 / total number of sides = 2/6 = 1/3

The expected outcome = 1/2 x 1/3 = 1 /6

The difference =

[tex]\frac{1}{6} - \frac{2}{15}[/tex]

[tex]\frac{15 - 12}{90}[/tex] = [tex]\frac{3}{90}[/tex] = [tex]\frac{1}{30}[/tex]

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The total interest that will be paid over the life is $5,28,000. The solution has been obtained using arithmetic operations.

What are arithmetic operations?

For all the real numbers, there are four basic mathematical operations which are:

1. Addition(‘+’) wherein the sum of the numbers is obtained.

2. Subtraction(‘-’) wherein the difference of the numbers is obtained.

3. Multiplication(‘×’) wherein the product of the numbers is obtained.

4. Division(‘÷’) wherein the quotient of the numbers is obtained.

We are given 30 year loan amounting $220,000 at interest rate of 8 percent with monthly payments of $1,614.28.

The annual interest comes out to be

8% of $220,000 = $17,600

The total interest = $17,600 * 30 = $5,28,000

Hence, the total interest that will be paid over the life is $5,28,000.

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If the daily prices are $1.50 for children and $2.00 for adults , then the number of adults swam at public pool is 302  and number of children who swam at public pool is 258  .

let the number of children swam at swimming pool be = c ;

let the number of adults swam at swimming pool be = a ;

the total number of person who used the swimming pool is = 560 ;

the equation is ⇒ c + a = 560 ...equation(1) ;

the price for 1 child is $1.50 and price for 1 adult is = $2 ;

the total money collected can be written as :

= (price of children tickets)×(number of children that swam)+(price of adults tickets)×(number of adults that swam) ;

⇒ 1.5c + 2(560 - c) = 991  ;

⇒ 1.5c + 1120 - 2c = 991 ;

⇒ -0.5c = 991 - 1120  ;

⇒ -0.5c = -129 ;

⇒ c = 129/0.5 = 258 ;

So , adults (a) = 560 - 258 = 302 ;

Therefore , the number of adults are 302 and number of children are 258 .

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The old box of cereal contained 24.5 ounces.

Take 30% of the original box of cereal.

To find 30% of a number, multiply the number by 0.3.

35 ounces x 0.3 = 10.5 ounces

Subtract 10.5 ounces from 35 ounces.

35 ounces - 10.5 ounces = 24.5 ounces

Answer: The old box of cereal contained 24.5 ounces.

When you take a single number and multiply it by several, you are multiplying. A 5 multiplied by a 4 results in 20 (5 + 5 + 5 + 5). We multiplied the number five by four times. Due to this, multiplication is occasionally referred to as "times."

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The constant of proportionality is known to be the ratio of two proportional values that is said to be in a constant value.

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The sum of money that can be divided between two parts = Rs 200

Adding three times the smaller part to half of the larger part then its will always be less than Rs 200

Let The smaller part = Rs S

The larger part = Rs L

The sum of parts = Rs 200

Larger + Smaller = Rs 200

L+S=200---------------(1)

And, Adding three times the smaller part to half of the larger part then the result will always be less than Rs 200

(3S+1/2× L)-200 = 0

3S+ L/2=200                    

6S+L=400 --------(2)

Solving eq 1 and eq 2

(6S+L)-(L+S)=400 - 200

(6S-S )+(L-L )=200

5S+0=200

S = 200/4

∴Smaller = S=Rs40

Now, Plug the value of S into eq 1

L+S = Rs 200

L+40 = Rs 200

L = 200 - 40

L = Rs 160

Larger = L= Rs160

Hence, the larger part is Rs 160 and the smaller part is Rs 40      

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Answer:

To factor a trinomial of the form ax^2 + bx + c, you can use the following steps:

Step-by-step explanation:

To factor a trinomial of the form ax^2 + bx + c, you can use the following steps:

Find a factor pair of ac that has the sum of b. For example, if a = 2, b = -7, and c = -21, then the factor pair for -42 (2 * -21) that has the sum of -7 is (-7, -6).

Rewrite bx as a sum of those factors. In the example, -7x = -7x + (-6x).

Factor out the GCF (greatest common factor) from the two groups of terms. In the example, (2x - 7)(x - 6) = 2x(x - 6) - 7(x - 6) = 2x^2 - 14x + 12x - 42 = 2x^2 - 2x - 42.

Write the original trinomial as the product of two binomials. In the example, the trinomial 2x^2 + -7x + -21 can be factored as (2x - 7)(x - 6)

So the final factorization is (2x - 7)(x - 6)

The formula for a cube's volume is L * W * H.But because it is a cube, its length, breadth, and height are all equal and correspond to one of the cube's edges. The answer is 2.7 equals 3 centimeters.

The formula for a cube's volume is L * W * H.But because it is a cube, its length, breadth, and height are all equal and correspond to one of the cube's edges.Finding the cube root of the volume will allow you to determine the length of a cube's edge.

A cube's edge measures 3 centimeters in length and has a volume of 27 cubic centimeters.

Six square faces make up the three-dimensional shape of a cube.

It has 12 edges and 8 vertices.

Volume of a cube is equal to the length of an edge3

In order to calculate the length of an edge, the cube root of the cube's volume must be known if the cube's volume is given.

Find the cube root of 27 to get the length of an edge.

2.7 equals 3 centimeters

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An  inequality represents the situation is: t < 3

The correct answer is an option (a) t < 3

We know that an inequality is a mathematical statement that represents the inequalities between two algebraic expressions with two variables .

It is an unequal relation between two algebraic expressions.

The symbols used between the expression:

less than ‘<’,

greater than ‘>’,

less than or equal to ‘≤’

greater than equal to ‘≥’,

Let t represents the distance between Tori's house and the school.

Tori lives less than 3 miles from the school.

This means, the distance t is less than 3 miles

so, we get an inequality t < 3

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Answer:

The range of a function is the set of all possible outputs (or y-values) of the function. To find the range of h, we can look at the y-values in the table.

The y-values for h(x) are: -27, -13, -3, 3, 5, 3, -3

The range of h is the set of all these y-values. We can see that the lowest y-value is -27 and the highest y-value is 5. So, the range of h is {-27,-13,-3,3,5}

Therefore, the range of h is from -27 to 5.

17 people were fans of the Chiefs or Royals but not Sporting KC.

What is algebra ?

Algebra is a branch of mathematics in which letters and symbols are used to represent numbers and quantities in equations and formulas. The basic operations of algebra include addition, subtraction, multiplication, and division, as well as the manipulation of exponents and roots.

Have given ,

26 people were fans of the Kansas City Chiefs

36 people were fans of the Kansas City Royals

18 people were fans of both the Chiefs and the Royals

13 people were fans of the Chiefs and Sporting KC

20 people were fans of the Royals and Sporting KC

6 people were fans of none of the teams

The principle of inclusion and exclusion says that the number of people who are fans of the Chiefs or the Royals but not Sporting KC is:

(26 + 36) - (18 + 13 + 20) + 6 = 62 - 51 + 6 = 17

So, 17 people were fans of the Chiefs or Royals but not Sporting KC.

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Step-by-step explanation:

....................

Answer:

Step-by-step explanation:

top=125 degrees

second from the top= 55 degrees

3rd=35 degrees

4th=55 degrees

5th=125 degrees

Answer:

t=1.92938456

Step-by-step explanation:

Isolate the variable by dividing each side by factors that don't contain the variable.

The inequality statements are 1 < 0, x > 0, x = x and x = 5

An empty set

An empty set is a set with no elements.

An inequality statement with no solutions results in an empty set.

One example of an inequality statement with no solutions is 1 < 0.

Some solution but not all real numbers

An inequality statement whose solution is some but not all real numbers is x > 0.

The solution set is the set of all positive real numbers, excluding 0.

The number line is represented as follows

|------------------------------------| 0  |----------------------------- |

All real numbers

A statement whose solution is all real numbers is x = x

The solution set is the set of all real numbers

The number line is represented as follows

|--------------------------- | ...  | -2  | -1  | 0  | 1  | 2  | ... |

Exactly one number

A statement whose solution is exactly one number is x = 5.

The solution set is the set containing only the number 5

The number line is represented as follows

|--------------------------- | ...  | 4  | ... |

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The polynomial that represents the area of the new lot is x² +200x +9600 and the new area of the increase is 25 feet is 15225 ft²

A polynomial is an expression made up of variables (also known as indeterminates) and coefficients that only employs the addition, subtraction, multiplication, and non-negative integer exponents of the variables.

A single variable, x, is used as the only variable in the indeterminate polynomial x² - 4x + 7. Polynomials are used in many areas of science and mathematics.

They are employed in calculus and numerical analysis to approximate other functions, as well as in the definition of polynomial functions, which are used in a variety of fields, from basic chemistry and physics to economics and social science.

Simple word problems to sophisticated scientific conundrums can all be  represented using polynomial equations.

Polynomials are used to create algebraic varieties in higher mathematics.

Area = Length x Breadth

Area =(80 + x)x(120 + x) = x² +200x + 9600

Area = (80 + 25)x(120 + 25) = 105x145

Area = 15225 ft²

Therefore, the new area of the plot is 15225 ft²

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Answer:

RT = 17.9

RS = 9.7

m<T = 33°

Step-by-step explanation:

sin R = opp/hyp

sin R = ST/RT

sin 57° = 15/RT

RT = 15/sin 57°

RT = 17.9

cos R = adj/hyp

cos R = RS/RT

cos 57° = RS/17.88544939

RS = 17.88544939 × cos 57°

RS = 9.7

m<R + m<S + m<T = 180°

57° + 90° + m<T = 180°

m<T = 33°

The region is function determining the total sales and the number of boxes covered to make the desired sale. The inequality representing the function is f(b) ≤ 3.75b + 25.

In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign" to indicate that two values are not equal. But several inequalities are utilized to compare the values and determine whether they are less than or higher than.

a. The region represents the function determining the total sales and the number of boxes covered in order to make the desired sale.

b. The inequality that represents the area under the graph up until the point of intersection is given by:

f(b) ≤ 3.75b + 25

The less than equal to sign is used, as the area used lies below the curve.

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The system of equations is solved and the number of toy cars of each machine of Company A should produce to have equal number of toy cars is t = 375 toys

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the number of toy cars produced by Company A be A

Let the number of toy cars produced by Company B be B

The number of machines for A = 18 machines

The number of machines for B = 15 machines

The number of toy cars produced by A = t cars

The number of toy cars produced by B = 75 + t

And ,

The total number of toy cars of A = 18 x t

The total number of toy cars of B = 15 ( 75 + t )

So , in order for both the company's to have the same toys is

18t = 15 ( 75 + t )

On simplifying the equation , we get

18t = 1125 + 15t

Subtracting 15t on both sides of the equation , we get

3t = 1125

Divide by 3 on both sides of the equation , we get

t = 375 toys

Hence , the equations is A = 18t and B = 15 ( 75 + t )

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