Kevin needs the following ingredients for the recipe for 3 people 90g strawberry jelly, 6 sponge fingers, 315 ml custard, and 135g tinned fruit.
How to find the amount of ingredients for three people?To find the amounts of each ingredient to make this recipe for three people we must carry out the following procedure:
1. We must make a rule of three to find the exact amounts of each ingredient as shown below:
4 - 120g strawberry jelly3 - ?3*120g/4=90g strawberry jelly.4 - 83 - ?3 * 8 / 4 = 6 sponge fingers4 - 420 ml custard3 - ?3*420ml/4=315ml custard.4 - 180g tinned fruit3 - ?3 * 180g / 4 = 135g tinned fruitAccording to the above, the following ingredients are needed for this recipe:
90g strawberry jelly.6 sponge fingers315 ml custard135g tinned fruitNote: This question is incomplete because there is some information missing. Here is the complete information:
120g strawberry jelly
8 sponge fingers
420ml custard
180 g tinned fruit
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(1 point) select all statements that are correct. there may be more than one correct answer. the statements may appear in what seems to be a random order. a. if a system has three unknowns and the reduced row echelon form of the augmented matrix has two nonzero rows, then the system can have an infinite number of solutions. b. a system of linear equations with more unknowns than equations will have an infinite number of solutions. c. every matrix a is row equivalent to a matrix in (row) echelon form. d. every matrix has a unique reduced row echelon form. e. adding one row to a multiple of another row is one type of elementary row operation. f. if a system has three unknowns and its reduced row echelon form of the augmented matrix has three nonzero rows, then the system has at least one solution. g. adding a multiple of one row to another row is one type of elementary row operation. h. none of the above
The answer to all the following options is given below along with the suitable reasoning.
a. Incorrect. If a system of linear equations has more unknowns than equations and the reduced row echelon form of the augmented matrix has two nonzero rows, the system is inconsistent and has no solutions.
b. Incorrect. A system of linear equations with more unknowns than equations is underdetermined and can have either no solutions or an infinite number of solutions.
c. Correct. A matrix can be row-reduced to an echelon form, which is a simpler form that makes it easier to find the solution to the system of linear equations represented by the matrix.
d. Correct. Every matrix can be row-reduced to a unique reduced row echelon form by performing a sequence of elementary row operations, which are reversible and preserve the solutions of the system of linear equations.
e. Correct. Adding a multiple of one row to another row is one type of elementary row operation used in row-reducing a matrix to its reduced row echelon form.
f. Correct. If a system of linear equations has three unknowns and the reduced row echelon form of the augmented matrix has three nonzero rows, then the system is consistent and has a unique solution.
g. Correct. Adding a multiple of one row to another row is one type of elementary row operation used in row-reducing a matrix to its reduced row echelon form.
h. Incorrect. Some of the statements are correct.
Therefore, The answer to all the following options is given above along with the suitable reasoning.
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The answer to all the following options is given below along with the suitable reasoning.
a. Incorrect. If a system of linear equations has more unknowns than equations and the reduced row echelon form of the augmented matrix has two nonzero rows, the system is inconsistent and has no solutions.
b. Incorrect. A system of linear equations with more unknowns than equations is underdetermined and can have either no solutions or an infinite number of solutions.
c. Correct. A matrix can be row-reduced to an echelon form, which is a simpler form that makes it easier to find the solution to the system of linear equations represented by the matrix.
d. Correct. Every matrix can be row-reduced to a unique reduced row echelon form by performing a sequence of elementary row operations, which are reversible and preserve the solutions of the system of linear equations.
e. Correct. Adding a multiple of one row to another row is one type of elementary row operation used in row-reducing a matrix to its reduced row echelon form.
f. Correct. If a system of linear equations has three unknowns and the reduced row echelon form of the augmented matrix has three nonzero rows, then the system is consistent and has a unique solution.
g. Correct. Adding a multiple of one row to another row is one type of elementary row operation used in row-reducing a matrix to its reduced row echelon form.
h. Incorrect. Some of the statements are correct.
Therefore, The answer to all the following options is given above along with the suitable reasoning.
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True or False : the null space of a is the solution set of the equation ax = 0.
The statement is True: the null space of a is the solution set of the equation ax = 0.
What is null space?The collection of all solutions to the homogeneous equation makes up a matrix's null space. Any matrix's null space contains the zero vector at all times. The null space is denoted as Null A.
We know that The null space of an m × n matrix A, written as Nul A, is the set of all solutions to the homogeneous equation Ax = 0.
Hence, the statement is True: the null space of a is the solution set of the equation ax = 0.
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2/9 of the students in a school are in sixth grade.
Type the answers to each question in the boxes below.
a. How many sixth graders are there if the school has 90 students?
b. How many sixth graders are there if the school has 27 students?
c. If the school has a students, write an expression for the number of sixth graders in terms of x.
If 2/9 of the students in a school are in sixth grade.
then 20 students sixth graders are there if the school has 90 students.
6 students sixth graders are there if the school has 90 students
If the school has a students then 2x/9 the number of sixth graders in terms of x.
what are fractions?Part of a full number and a means of dividing a numbers into equal pieces are represented by fractions. The denominator is the total number of components in the whole, while the numerator is the fraction of equal portions being counted.
a. From the details of the question, we have 2/9 of the students in the school in sixth grade. Hence;
Number of students in sixth grade = 2/9 × 90 = 20 sixth grade students
b. If the school has 27 students, number of sixth graders = 2/9 × 27 = 6 sixth grade students
c. If we have x students in the school, the number of sixth graders = 2/9 × x = 2/9x students
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A box contains 5 plain pencils and 5 pens. A second box contains 3 color pencils and 1 crayon. One item from each box is chosen at random. What is the probability that a plain pencil from the first box and a color pencil from the second box are selected?
The probability that a plain pencil from the first box and a color pencil from the second box are selected is 3/8.
What is the probability?The Probability in mathematics is the possibility of an event in time. In simple words, how many times that incident is happening in any given time interval.
Given:
A box contains 5 plain pencils and 5 pens.
A second box contains 3 color pencils and 1 crayon.
One item from each box is chosen at random.
The probability that a plain pencil from the first box and a color pencil from the second box are selected,
= 5/10 x 3/4
= 3/8
Therefore, the probability is 3/8.
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7(2y)= Complete the tatement uing the pecified property. Aociative Property of Multiplication
The statement using the specified property i.e Associative Property of Multiplication will be 14y.
One's identity property is its capacity for multiplication.
7(2y)
The phrase is given as follows:
7(2y)
=14y
The one has the multiplication property, which says:
Any number or expression that is multiplied by one doesn't alter in any way.
The result is: 1xa=a
The identification property is another name for this particular attribute.
The general rule is shown as follows:
Examples include
1 * 2 = 2
1 * x = x
1 * ab = ab
and so forth
So, the property of multiplying by one is an identity property.
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How can you find the slope of the line that passes through the points (0,0) and (2,4)? Explain
Answer:
2
Step-by-step explanation:
Let ( x1, y1) represent the first point (0,0) and (x2, y2) represent the second point (2,4). Slope (m) (y2 - y1) / ( x2 - x1)= (4 - 0) / (2-0) =2
The scale on a map is stated as 1 : 200 000 A main road connects two towns which are 34 kilometres apart. How far will it be on the map, in centimetres, between the two towns along the main road?
On the map, the two towns are 17 centimeters apart.
What is a scale factor?The ratio of the scale of an original thing to a new object that is a representation of it but of a different size is known as a scale factor (bigger or smaller).
Given:
The scale on a map is stated as 1 : 200000.
A main road connects two towns which are 34 kilometers apart.
34 kilometers in centimeters,
= 3400000
So, the road on the map = 3400000/200000
The road on the map = 17 centimeters.
Therefore, the value is 17 centimeters.
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Select the correct answer. What is the approximate value of the correlation coefficient for the given graph? Graph shows line segment plotted on coordinate plane with number of years on horizontal axis and price of the car in thousand dollars on vertical axis. Line segment begins at (0, 35), goes through (1, 30), (2, 25), (3, 20), (4, 15), (5, 10), (6, 5). A. 1 B. 5 C. 3 D. -1
The approximate value of the correlation coefficient for the given graph is D. -1.
The correlation coefficient is a statistical measure that indicates the strength and direction of the linear relationship between two variables. A correlation coefficient of -1 indicates a perfect negative linear relationship, meaning that as the value of one variable increases, the value of the other variable decreases at a constant rate. In this case, as the number of years increases, the price of the car decreases, so the correlation coefficient is negative.
What are the Types of Corresponding Angles Based on their Sum?
Their total determines which angles are equivalent. Alternative Corresponding Angles Parallel Corresponding angles
What is meant by Corresponding Angles?Whenever a transversal line crosses two parallel lines, corresponding angles are created. Examples of comparable angles that can be found in daily life include the opening and closing of a lunchbox, completing a Rubik's cube, and endless parallel railroad tracks.Corresponding objects are ones that show up at the same location in two related circumstances. In the example below, angles are a common cause. Given that they are in the same spot in the two related shapes, angle A on the left and angle K on the right are equivalent. As we say, A is equivalent to K.Angles that match up are congruent. Corresponding pairs are all angles that are situated in the same location with respect to the parallel and transversal lines.To learn more about Corresponding Angles, refer to:
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what is the sum of the expression 3+256
Answer:259
Step-by-step explanation:
3+256=269
Answer: 259
Step-by-step explanation: 3 + 256 = 259
1. Researchers who were interested in the types of movies pre-
ferred by children of different age groups asked students in the
sixth grade and in the eighth grade if they would prefer to see
an animated feature like The Lion King or an action feature like
The Avengers. The results are summarized in the table:
Sixth Grade
Eighth Grade
Animated Action
45 35
40 60
Which proportions represent the conditional distribution of
grade for children who preferred an action feature?
A) 0. 194 and 0. 333
B) 0. 368 and 0. 632
C) 0. 40 and 0. 60
D) 0. 444 and 0. 556
E) 0. 529 and 0. 471
Sixth grade students prefer action movie is 0.194 & Eight grade students prefer action movie is 0.333
Given that the information below:
Sixth grade students prefer animated movie - 45
Eight grade students prefer animated movie - 40
Sixth grade students prefer action movie - 35
Eight grade students prefer action movie - 60
Total sixth grade students - 80
Total eight grade students - 100
Total students prefer animated movie - 85
Total students prefer action movie - 95
Proportion of sixth grade students prefer action movie =
Sixth grade prefer action movie/Total students
= 35/180 = 0.194
Proportion of eight grade students prefer action movie =
Eight grade prefer action movie/Total students
= 60/180 = 0.333
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hey i need this reallly
pls just help me out
the type of reasoning used to prove a conjecture is called
Answer:
deductive reasoning
Step-by-step explanation:
4.5a−1.5(a+5) what expression is equivelant to it
3a+5
3a-7.5
6a+5
6a-7.5
Answer:
3a - 7.5
Step-by-step explanation:
4.5a−1.5(a+5)
4.5a - 1.5a - 7.5
3a - 7.5
So, the answer is 3a-7.5
Find the lengths of the diagonals of rectangle $WXYZ$ . $WY=14x+10$ $XZ=11x+22$
The lengths of the diagonals of rectangle is 66 units
How to determine the valuesThe properties of a rectangle is given as;
A rectangle is a known as a quadrilateralThe opposite sides are parallel and equal to each other Interior angle is equal to 90 degreesSum of all the interior angles is equal to 360 degreesThe diagonals bisect each other at right angleDiagonals have the same lengthFrom the information given, we have that;
$WY=14x+10
$XZ=11x+22
Since the diagonals are of the same length, we have;
WY = XZ
14x + 10 = 11x + 22
collect like terms
14x - 11x = 22 - 10
subtract the like terms
3x = 12
Make 'x' the subject of the formula
x = 4
WY = 54 + 10 = 66
XZ = 66
Hence, the values are 66
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a triangular fence is being built to surround a garden. if two of the side lengths must be 4 feet and 12 feet, which inequality could be solved to determine the minimum length of the third side?
The minimum length of the third side must be greater than 16 feet.
The minimum length of the third side can be determined using the Triangle inequality theorem. This theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. The Triangle Inequality Theorem can be expressed by the following inequality: a + b > c, where a, b, and c are the lengths of the three sides of the triangle. In this case, we have two sides of 4 feet and 12 feet, so the inequality can be written as 4 + 12 > c, which simplifies to 16 > c. Solving for c yields c > 16, which means that the minimum length of the third side must be greater than 16 feet.
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Three side lengths of
23, 32, and 45 units will
form exactly one
unique triangle.
No, the length of any pair of sides added together is not necessarily longer than the length of the third side.
Can a triangle be built with the provided side lengths of 23 32 45? No, the length of any pair of sides added together is not necessarily longer than the length of the third side.Triangle Unevenness Theorem.According to the Triangle Inequality Theorem, a triangle can only be formed if its two sides are greater than its third side when added together.If and only if the side lengths a, b, and c fulfill the formula a2 + b2 = c2, a right triangle is formed.We state that a Pythagorean triple exists between these integers.Triangles are singular because you can only ever draw one.There may be more than one triangle obtained if a triangle can be formed by using the two side lengths provided and the angle measure of an omitted angle.
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It is true three side lengths of 23, 32, and 45 units will form exactly one unique triangle.
How many triangles are possible with three different side lengths?One unique triangle, You create the same triangle again and over again. Therefore, if a triangle can be formed by your three side lengths, it can only be one specific triangle.If you are provided all three sides (Side-Side-Side), two sides and the included angle (Side-Angle-Side), two angles, and the included side, a unique triangle will be created (Angle-Side-Angle)Any two triangles that have the same side dimensions will be congruent, or identical, according to the SSS rule. You may so create a distinctive triangle. But only if the lengths of any two of those sides added together exceed the length of the third side can this occur.The complete question is
Three side lengths of 23, 32, and 45 units will form exactly one unique triangle. true /false.
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Need help ASAP. Congruent Triangles Gina Wilson all things algebra
The answers for all triangle problems are:
x = 2°∠S = 36°x = 26°; ∠P = 116°; ∠Q = 21°; ∠R = 43°∠J = 44°; ∠K = 81°; ∠L = 51°x = 11°; ∠B = 108°; ∠C = ∠D = 36°10. ΔABC is a equilateral triangle, then:
∠A = ∠B = ∠C = 8x - 44
3 (8x - 44) = 180
24x - 132 = 180
24x = 48
x = 2
11. ΔRST is an isosceles triangle, then:
∠R = ∠T
15x - 31 = 9x + 11
6x = 42
x = 7
∠S + ∠R + ∠T = 180°
∠S + 2∠T = 180°
∠S + 2( 9x + 11) = 180°
∠S + 2(9(7) + 11) = 180°
∠S + 2 (63 + 11) = 180°
∠S + 2 (74) = 180°
∠S = 180 - 144
∠S = 36°
12. Based on the given data:
∠P = 5x - 14
∠Q = x - 5
∠R = 2x - 9
Since ΔPQR is a triangle, then:
∠P + ∠Q + ∠R = 180°
(5x - 14) + (x - 5) + (2x - 9) = 180°
8x - 28 = 180°
8x = 208
x = 26
∠P = 5x - 14 --> 5(26) - 14 = 116°
∠Q = x - 5 --> (26) - 5 = 21°
∠R = 2x - 9 --> 2(26) - 9 = 43°
13. ΔJKL is a triangle, where:
∠J = ∠L - 7
∠K = 2∠L - 21
∠J + ∠K + ∠L = 180
(∠L - 7) + (2∠L - 21) + ∠L = 180°
4∠L - 28 = 180°
4∠L = 208
∠L = 51°
∠J = ∠L - 7 --> (51) - 7 = 44°
∠K = 2∠L - 21 --> 2(51) - 21 = 81°
14. In a triangle ΔBCD:
BD = BD
∠B = 13x - 35
∠C = 5x - 19
∠D = 2x + 14
Since BC = BD; then:
∠C = ∠D
5x - 19 = 2x + 14
3x = 33
x = 11
∠C = ∠D = 2x + 14 --> 2(11) + 14 = 36°
∠B + ∠C + ∠D = 180°
∠B + 36° + 36° = 180°
∠B = 108°
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Carlos's math teacher finds that there's roughly a linear relationship between the amount of time students spend on their homework and their weekly quiz scores. This relationship can be represented by the equation y=61+7.6xy=61+7.6x, where yy represents the expected quiz score and xx represents hours spent on homework that week. What is the meaning of the xx-value when y=92y=92?
When y = 92, the value of x is approximately 4.07 hours. This means that if a student spends approximately 4.07 hours on their homework each week, we would expect their quiz score to be 92.
What is the linear equation?
Linear equations are the equations of degree 1. It is the equation for the straight line. The standard form of linear equation is ax+by+c =0, where a ≠ 0 and b ≠ 0.
If y = 92, we can solve for x by substituting 92 for y in the equation y = 61 + 7.6x:
92 = 61 + 7.6x
7.6x = 92 - 61
7.6x = 31
x = 31 / 7.6
x = 4.07
Therefore, when y = 92, the value of x is approximately 4.07 hours. This means that if a student spends approximately 4.07 hours on their homework each week, we would expect their quiz score to be 92.
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Train a take 16 hour to reach chicago from wahington while train b take 20 hour, the ratio
Train A takes 20 hours to get to Chicago from Washington whereas Train B takes 16 hours, making a 20: 16 ratio between the two trains.
How to estimate the ratio between the two trains?Given: Time taken by Train A = 16 hours
Time taken by train B = 20 hours
We need to find, the ratios of speeds of trains A and B
The distance exists same (Chicago to Washington) and to finding the speed of train A.
Let the equation be
Speed [tex]$=\frac{\text { Distance }}{\text { time }}$[/tex]
Let the distance between Chicago and Washington be x
Speed [tex]${(\text {train } \mathrm{A})}=\frac{\mathrm{x}}{16} \mathrm{~km} / \mathrm{h}$[/tex]
Hence, speed of train [tex]$A=\frac{x}{16} \mathrm{~km} / \mathrm{h}$[/tex]
To find the speed of train B
Speed [tex]${({train~B})}=\frac{\mathrm{x}}{20} \mathrm{~km} / \mathrm{h}$[/tex]
[tex]$\mathbf{S}_{(\{train~A})}: \mathrm{S}_{(\{train} B)} \\[/tex]
[tex]$& \longrightarrow \frac{\mathrm{x}}{16}: \frac{\mathrm{x}}{20} \\[/tex]
= 20 : 16
Therefore, the ratio of train A and train B be 20 : 16.
The complete question is:
Train a take 16 hour to reach Chicago from Washington while train b take 20 hour, the ratio of the speed of both the train.
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703 divided by 58
With remainder
An airplane factory made 36 airplanes per month. At this rate, how many airplanes can the factory make per year?
The number of planes manufactured by the aeroplane company will be 432.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that an aeroplane factory made 36 aeroplanes per month. The number of aeroplanes made in a year will be calculated as:-
Number = 36 x Months in a year
Number = 36 x 12
Number = 432
Therefore, the number of planes will be 432.
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(2 points) suppose y′=−y t. we write f(t,y) for the right-hand side of the differential equation. (a) determine the isoclines f(t,y)=c for c=−1,0,1. express each answer as y=
For c = 1, we have f(t,y) =-y = 1. Rearranging this equation yields y = 1, which simplifies to y = et. The isoclines for c = -1, 0, 1 are y = e-t, y = 0, and y = et, respectively.
c = -1: y = e-t
c = 0: y = 0
c = 1: y = et
For c = -1, we have f(t,y) = -y = -1. Rearranging this equation yields y = -1, which simplifies to y = e-t.
For c = 0, we have f(t,y) = -y = 0. Rearranging this equation yields y = 0.
For c = 1, we have f(t,y) =-y = 1. Rearranging this equation yields y = 1, which simplifies to y = et.
The isoclines for c = -1, 0, 1 are y = e-t, y = 0, and y = et, respectively.
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Pete is standing 4 feet away from a mirror on the ground. Pete is 6 feet tall and can just see the top of the tree in the mirror from where he is standing. The two triangles formed are similar.
If the mirror is 20 feet away from the base of the tree
A.
22 feet
B.
19 feet
C.
13 feet
D.
30 feet
Pete is standing 4 feet away from a mirror on the ground, if mirror is 20 feet away from the base of the tree the height of tree is 30 feet.
let,
Pete is standing 4 feet away from a mirror on the ground.
Pete is 6 feet tall and can just see the top of the tree in the mirror from where he is standing.
We can say two triangles are formed are similar,
So if the mirror is 20 feet away from the base of the tree
After talkig the similarity theorem we get,
[tex]\frac{6}{4} = \frac{x}{20}[/tex]
So x = 6*5
x = 30 feet.
The height of the mirror is 30 feet.
Height is a measurement of vertical distance, or how high something or someone is (how tall they are) (how "high" a point is).
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On a team, 9 girls and 5 boys scored a total of 51 points. The difference between the number of points scored by the 9 girls and the number of points scored by the 5 boys is 21 . Each girl scored the same number of points and each boy scored the same number of points. Find the number of points scored by each girl and each boy.
The number of points scored by each girl is 4 and each boy is 1 point.
What is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.
here, we have,
9 girls and 5 boys scored a total of 51 points.
The difference between the number of points scored by the 9 girls and the number of points scored by the 5 boys is 21 .
Each girl scored the same number of points and each boy scored the same number of points.
Let x be the number of points the 9 girls scored.
=> 51 - x is the number of points the 5 boys scored.
We have x + 21 = 51 - x
or x - 21 = 51 - x.
=> 2x = 30 or 2x = 72
=> x = 15 or x = 36
We reject x = 15
as 15 is not a multiple of 9
So x = 36,
each girl scored 36/9 = 4 points.
and,
=> 41 - (36) = 5,
each boy scored 5/5 = 1 points.
Hence, the number of points scored by each girl is 4
and each boy is 1 point.
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5. Angela earns £35 240 a year.
She has to pay income tax.
She is allowed to earn £6475 before paying tax.
She pays 20% tax on the rest.
Her employer deducts the income tax each month.
Work out how much income tax Angela gets deducted each month
£479.42 is the income tax Angela gets deducted each month.
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
Given, Angela earns £35 240 a year. She has to pay income tax. She is allowed to earn £6475 before paying tax. She pays 20% tax on the rest.
So,
taxable income = 35,240-6,475
taxable income = 28,765
Since She pays 20% tax
Thus,
The Total tax = 28,765 * 20%
the total tax = 5,753
Since Her employer deducts the income tax each month.
so,
Income tax deducted each month = 5,753/12
Income tax deducted each month = 479.417
Therefore, the income tax Angela gets deducted each month is £479.42.
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If ABCD is dilated by a factor of the
coordinate of C' would be:
-4-3-2
A
3
2
B
-10
-1
-2
PT
2
3
4 5
C
D
6
10
11
C' = ([?], [])
Enter
The coordinate of point C' after transformation is; C'(4, 4)
How to interpret Dilation transformation?Dilation is defined as a transformation, that is used to resize the object. Dilation is used to make the objects either larger or smaller. That is a reduction or an enlargement.
Usually when we dilate an object by a scale factor, what we simply do is to multiply the original coordinates by the scale factor to get the new dilated coordinates.
Since the quadrilateral is dilated by a scale factor of 2, we will multiply the original coordinate of point C by 2.
Coordinate of C = (2, 2)
Thus;
Dilated coordinate; C' = (2*2, 2*2) = 4,4
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Seasons entire family took the train to watch the baseball game. The admission to the game is eight dollars per person. The round-trip price for the train cost $4.50 per person. There are six people and seasons family. Write an expression for the total cost of the outing using parentheses. Use the distributive property to find the cost.
Answer: $75
Step-by-step explanation:
Price for one person: $4.50+$8
Price for six people: 6($4.50+$8)=$75
Use polar coordinates to find the volume of the given solid. Below the paraboloid z = 8 - 2x 2 - 2y 2 and above the xy-plane.
The volume of the solid is approximately 54.8 cubic cm.
What do you mean by integration?Integration is a fundamental concept in calculus that deals with finding the area under a curve and calculating the accumulation of values over an interval. Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.
There are two main types of integration: definite and indefinite integration. Definite integration involves finding the exact area between a curve and the x-axis over a specified interval. Indefinite integration involves finding the general antiderivative of a function, which is a function that, when differentiated, will yield the original function.
The process of integration can be represented symbolically as ∫ (the integral symbol) and the result of an integration is typically represented as an antiderivative. The fundamental theorem of calculus states that differentiation and integration are inverse operations, so the derivative of an antiderivative is the original function.
The paraboloid z = 8 - 2x² - 2y² can be written in polar coordinates as z = 8 - 2r². The volume of the solid can be found by using the double integral of the height of the paraboloid over the region defined by the polar coordinates.
In polar coordinates, the region is defined by 0 <= r <= √4 and 0 <= θ <= [tex]\frac{\pi }{2}[/tex]The volume of the solid is given by:
V=[tex]\int\limits \int\limits {(8-2r^{2} )} r\, dr dy[/tex]
= [tex]\int\limits^0_\frac{\pi }{2} {} \, dx\int\limits^0_\sqrt{4} \,}(8-2r^{2})r }dr[/tex]
= [tex]\int\limits^0_\frac{\pi }{2} dx[ {4r^{2} -\frac{2}{3}r^{3} }] \,[/tex] from 0 to sqrt(4)
= [tex]\frac{\pi }{2} [4(4)-(\frac{2}{3}4^{\frac{3}{2} } ][/tex]
= [tex]\frac{\pi }{2}[16-\frac{2}{3}(16^{\frac{1}{2} } )][/tex]
= [tex]\frac{\pi }{2}[16-8({\frac{2^{} \frac{1}{2} }{3} }) ][/tex] cubic cm
So the volume of the solid is approximately 54.8 cubic cm.
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Giorgi paid 1 GEL and 25 tetris for 5 erasers and 2 rulers, Tornike paid 1 GEL and 5 tetris for 2 erasers and 3 rulers.
Let's say the price of an eraser is x white, the price of a ruler is y white. To find these prices
The requried price of an eraser and ruler is given as 0.15 Tetris and 0.25 Tetris respectively.
What are simultaneous linear equations?Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.
1 GEL and 25 tetris = 1.25
1 GEL and 5 tetris = 1.05
We can set up two equations to find the unknown values of x and y:
1.25 = 5x + 2y - - - - - -(1)
1.05 = 2x + 3y - - - - - (2)
To solve the system of equations, we can substitute the value of y in the first equation using the second equation:
y = (1.25 - 5x) / 2
Substituting this value of y in the second equation:
1.05 = 2x + 3((1.25 - 5x) / 2)
x = 0.15
Now,
Put x = 0.15 equation 1,
e get
y = 0.25
Thus, the requried price of an eraser and ruler is given as 0.15 Tetris and 0.25 Tetris respectively.
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