Solve these word problems:

Solving linear equations using matrices involves steps ranging from creating an augmented matrix, performing row operations, back substitution, and then interpreting results

To solve a system of linear equations using matrices:

Step 1: Write the system of linear equations in matrix form.

Represent the coefficients of the variables as a matrix (called the coefficient matrix), and the constants on the right side of the equations as another matrix (called the constant matrix).

Step 2: Create the augmented matrix.

Combine the coefficient matrix and the constant matrix into a single matrix by appending the constant matrix as an additional column. This combined matrix is called the augmented matrix.

Step 3: Perform row operations to achieve row-echelon form.

Use row operations (swapping rows, multiplying a row by a constant, or adding/subtracting rows) to manipulate the augmented matrix into row-echelon form. Row-echelon form has zeroes below the diagonal and non-zero elements on the diagonal.

Step 4: Perform back-substitution.

Starting from the last row of the row-echelon form matrix, solve for the variables using back-substitution. Substitute the values of the variables you find into the previous rows to determine the remaining variable values.

Step 5: Interpret the results.

Once you have solved all the variables, you have found the solution to the system of linear equations. If there are infinitely many solutions or no solutions, this will be indicated by the row-echelon form.

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Answer:

  All statements above are false

Step-by-step explanation:

You want to know which of the statements saying translation, rotation, and reflection change the measures of line segments or angles is true.

Translation, rotation, and reflection are referred to as "rigid motion." That means all parts of the transformed figure keep their measures and positions relative to other parts of the figure. No lengths or angle measures are changed.

  All of the (above) listed statements are false.

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The volume of solid figure is,

⇒ V = 113 cm³

We have to given that;

A solid figure is shown with a cylinder and a cone.

Now, We know that;

Volume of cylinder = πr²h

And,

Volume of cone = πr²h / 3

Where, 'r' is radius and 'h' is height.

Here, Height of cylinder = 7 cm

And, Radius of cylinder = 4/2 = 2 cm

Hence, WE get;

Volume of cylinder = πr²h

Volume of cylinder = 3.14 x 2² x 7

Volume of cylinder = 87.9 cm³

And, Height of Cone = 6 cm

Radius of Cone = 4/2 = 2 cm

Hence, WE get;

Volume of Cone = πr²h/3

Volume of Cone = 3.14 x 2² x 6 / 3

Volume of Cone = 25.1 cm³

Thus, The volume of solid figure is,

⇒ V = 87.9 cm³ + 25.1 cm³

⇒ V = 113 cm³

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The designer can choose from 84 possible combinations of tiles, and they can create 6 different sample designs using the 3 chosen tiles when placing them side by side.

To determine the number of possible combinations of tiles that the designer can choose, we can use the concept of combinations.

Since the designer has 9 different kinds of tiles and wants to choose 3 of them, we can calculate the number of combinations using the formula for combinations, which is [tex]nCr = n! / (r! \times (n - r)!).[/tex]

Number of combinations of tiles = 9C3 [tex]= 9! / (3! \times (9 - 3)!)[/tex]

Simplifying further:

[tex]9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1[/tex]

[tex]3! = 3 \times 2 \times 1[/tex]

[tex](9 - 3)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1[/tex]

Plugging these values into the formula:

Number of combinations of tiles[tex]= 9 \times 8 \times 7 / (3 \times 2 \times 1 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)[/tex]

Simplifying the expression:

Number of combinations of tiles = 84

The designer can choose from 84 possible combinations of tiles, and they can create 6 different sample designs using the 3 chosen tiles when placing them side by side.

Therefore, the designer can choose from 84 possible combinations of tiles.

Now, let's calculate the number of different sample designs the designer can make using the 3 chosen tiles.

Since the tiles are placed side by side, the order of the tiles matters.

To calculate the number of different arrangements, we can use the concept of permutations.

Number of sample designs = 3!

Calculating:

[tex]3! = 3 \times 2 \times 1 = 6[/tex]

Therefore, the designer can create 6 different sample designs using the 3 chosen tiles when placing them side by side.

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1. Area of the smaller circle is 100πcm²

2. Area of the bigger circle 800πcm²

The formula for the circumference of a circle is expressed as;

Circumference = 2πr

Substitute the values, we get;

20π = 2πr

Divide by the coefficient of r, we get;

r = 10cm

Now, area of a circle is expressed as;

Area = πr²

Substitute the value of the radius

Area = π × 10²

Find the square

Area = 100πcm²

Area of the big circle = 8(area of the small circle)

substitute the values

Area of the big circle = 8(100π)

expand the bracket

Area of the big circle = 800 πcm²

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The area of the composite shape is 125units²

The area of a figure is the number of unit squares that cover the surface of a closed figure.

A composite shape can be defines as a shape created with two or more basic shapes.

The composite shape can be divided into 2 equal squares and a rectangles.

Area of the square = l²

= 5× 5 = 25

For two squares it will be;

25 × 2

= 50 units²

area of the rectangle = l× w

where w is the width

A = 15 × 5

= 75 units²

Therefore the area of the composite shape = 50 + 75 = 125 units²

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The triangle shown in the figure formed by the line connecting the center of the Earth to the Station 1, the Station 1 to the Satellite, and the Satellite to the center of the Earth forms a right triangle.

A right triangle is a triangle that has a 90 degrees interior angle.

The line joining the center of the Earth and Station 1 is perpendicular to the line connecting the Satellite to Station 1, because, the line connecting the Satellite to Station 1 is a tangent and tangents to a circle are perpendicular to a radius of the circle at the point of tangency.

The triangle formed by connecting the center of the Earth to the Station 1, and connecting the Satellite to Station 1, and connecting the Satellite to the center of the Earth forms a right triangle, because the angle formed by the line connecting the center of the Earth to Station 1, forms a right angle with the tangent from the Satellite to the Station 1

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The missing reason in the proof is "mZDEC = 90°" or "Angle DEC is a      right angle."

The missing reason in the proof is the statement "mZDEC = 90°". This statement implies that the angle DEC is a right angle, which is a crucial piece of information in proving that ABCD is a square.

In a square, all angles are right angles, so if we can establish that an angle in the figure is 90°, we can conclude that the figure is a square. In this case, the angle DEC being 90° is the key piece of evidence that supports the claim that ABCD is a square.

The statement "mZDEC = 90°" indicates that the measure of angle DEC is 90 degrees. This is significant because it confirms that one of the angles in the figure is a right angle, meeting the definition of a square.

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Answer:

ougylbhj,

Step-by-step explanation:

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The simplified expression of the expression (x)³(−x³y)² is x⁹y²

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

(x)3(−x3y)2

Express properly

So, we have

(x)³(−x³y)²

Open the brackets

(x)³(−x³y)² = x³ * x⁶y²

Multiply (x)³ and x⁶ in the expression

So, we have the following representation

(x)³(−x³y)² = x⁹y²

Hence, the simplified expression is x⁹y²

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The solution of the system of linear equations, is (1.167, 3.667).

Given that the system of linear equations, y = -2x+6 and y = 4x - 1, we need to find the solution for the same,

y = -2x+6............(i)

y = 4x - 1.......(ii)

Equating the equations since the LHS is same,

-2x+6 = 4x-1

6x = 7

x = 1.167

Put x = 1.16 to find the value of y,

y = 4(1.16)-1

y = 4.66-1

y = 3.667

Therefore, the solution of the system of linear equations, is (1.167, 3.667).

You can also find the solution using the graphical method,

Plot the equations in the graph, the point of the intersection of both the lines will be the solution of the system of linear equations, [attached]

Hence, the solution of the system of linear equations, is (1.167, 3.667).

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Answer:

14

Step-by-step explanation:

14.5 to 1

205/14.5 = 14

Answer:

The phrase "The product of a number and five added to 17" can be represented mathematically as:

5x + 17

In this expression, 'x' represents the unknown number. The product of the number and five is calculated by multiplying the number by 5, and then 17 is added to the result.

54° is the value of the given angle BAC.

As we know that the arc and angles formed by intersecting chords,

θ = (α + β)/2    .....(i)

According to the given figure, we have

θ = 10x+4

α =9x-12

β = 12x +15

Substitute the value in equation (1)

10x+4 = (9x-12+12x +15)/2

20x+8 = 21x +3

x = 8-3

x = 5

Thus,

∠BAC = θ

= 10*5+4

= 50 + 4

=54

Therefore, the value of the given angle will be 54°.

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A) The inequality that the weight of a child who has outgrown the small T-shirt is w > 55. B) The inequality that the weight of a child who is not ready for the small T-shirt yet. Is y < 43

a. To describe the weight (w) of a child who has outgrown the small T-shirt, we can use the inequality:

w > 55

This inequality states that the weight of a child (w) must be greater than 55 pounds for them to have outgrown the small T-shirt.

b. To describe the weight (y) of a child who is not ready for the small T-shirt yet, we can use the inequality:

y < 43

This inequality states that the weight of a child (y) must be less than 43 pounds for them to not be ready for the small T-shirt yet.

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Answer:

The function is given by p(x) = x^2 - 5x^2 + x + 15. The potential rational zeros of the function are given by the factors of the constant term (15) divided by the factors of the leading coefficient (1).

So the potential rational zeros are ±1, ±3, ±5, ±15.

The list of potential rational zeros of the function includes all of the options listed except option (b) -2. Therefore, the answer is (b) -2.

Step-by-step explanation:

572 square inches nylon will Lyndon need.

To determine how much nylon Lyndon will need to make a case for his snare drum, we need to calculate the surface area of the drum.

The surface area of a cylinder can be calculated using the formula:

Surface Area = 2π[tex]r^2[/tex] + 2πrh

where r is the radius of the base of the cylinder and h is the height of the cylinder.

Since the diameter of the drum is 14 inches, the radius is 7 inches.

The height of the drum is 6 inches.

So, the surface area of the drum is:

Surface Area = 2π[tex](7)^2[/tex] + 2π(7)(6)

Surface Area = 2π(49) + 2π(42)

Surface Area = 98π + 84π

Surface Area = 182π

Surface Area = 182 pi

Surface Area = 182 x 22/7

Surface Area = 572 squae inches

Therefore, Lyndon will need 182π square inches of nylon to make a case for his snare drum.

This is approximately 572 square inches when rounded to the nearest hundredth.

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1. The equation to be formed is 40 + 5x + 90 = 180

2. The value of x is 10

A straight line's total angles are always 180 degrees. Around the place where two lines join, they create four angles. The total of the angles on either side of a straight line, if those lines are straight and form one, is always 180 degrees.

We know that;

40 + 5x + 90 = 180 (Sum of angles on a straight line)

130 + 5x = 180

5x = 180 - 130

x = 50/5

x = 10

Thus the value of x is given as 10

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Answer:

43.47 feet (2 d.p.)

Step-by-step explanation:

Points A, B and C form a triangle.

We have been given sides a and b, and their included angle C.

The distance between points A and B is side c of triangle ABC.

Therefore, we can solve this problem using the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines (for finding sides)} \\\\$c^2=a^2+b^2-2ab \cos (C)$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

Given values:

Substitute the given values into the Law of Cosines formula and solve for side c:

[tex]\implies c^2=72^2+54^2-2(72)(54)\cos(37^{\circ})[/tex]

[tex]\implies c^2=8100-7776\cos(37^{\circ})[/tex]

[tex]\implies c=\sqrt{8100-7776\cos(37^{\circ})}[/tex]

[tex]\implies c=43.4719481...[/tex]

[tex]\implies c=43.47\; \sf ft\;(2\; d.p.)[/tex]

Therefore, the width of the pond from point A to point B is 43.47 feet, to two decimal places.

The amount in the account after 4 years, rounded to the nearest cent, will be approximately $17,332.8.

Recall the compounding formula:

A = P * [tex]e^{rt}[/tex]

Where:

A = Final amount in the account

P = Initial principal (deposit)

e = Euler's number (approximately 2.71828)

r = Annual interest rate (as a decimal)

t = Time in years

Given:

Initial principal (P) = $16,000

Annual interest rate (r) = 2%  = 0.02

time (t) = 4 years.

Substitute these values into the formula, we get:

A = $16,000 * [tex]e^{0.02 * 4}[/tex]

Using a calculator, we can calculate:

A = $16,000 * [tex]e^{0.08}[/tex]

A = $16,000 * 1.0833

A = $17,332.8

Therefore, the amount in the account after 4 years, rounded to the nearest cent, will be approximately $17,332.8.

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Answer:

-3x + 3

Step-by-step explanation:

To find the equation of a line perpendicular to the line passing through (-2, 1) and (4, 3), we need to determine the slope of the original line first. Then, we can use the negative reciprocal of that slope to find the slope of the perpendicular line. Finally, we can use the point-slope form to write the equation of the perpendicular line.

Step 1: Find the slope of the original line.

Slope (m) = (change in y) / (change in x)

m = (3 - 1) / (4 - (-2))

m = 2 / 6

m = 1/3

Step 2: Determine the slope of the perpendicular line.

The slope of the perpendicular line is the negative reciprocal of the original line's slope.

Perpendicular slope = -1 / (1/3)

Perpendicular slope = -3

Step 3: Use the point-slope form to write the equation.

The point-slope form is given by:

y - y1 = m(x - x1)

Using the point (-2, 9) and the perpendicular slope (-3), we can write the equation as:

y - 9 = -3(x - (-2))

y - 9 = -3(x + 2)

y - 9 = -3x - 6

y = -3x + 3

Therefore, the equation of the line perpendicular to the line passing through (-2, 1) and (4, 3) and passing through (-2, 9) is y = -3x + 3.

Answer:

d. 465 degrees (not my own answer, see below)

Step-by-step explanation:

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8/5 is the value that is not one of the possible rational zeros of the given polynomial.

Using the Rational Root Theorem, we need to consider the factors of the constant term (-8) divided by the factors of the leading coefficient (5).

The factors of -8 are ±1, ±2, ±4, ±8.

The factors of 5 are ±1, ±5.

Since the leading coefficient of the given polynomial is positive (5), the negative factors can be ignored.

So, the possible rational zeros are:

1/1, 2/1, 4/1, 8/1, 1/5, 2/5, 4/5, 8/5

Now, we can substitute each of these values into the polynomial and see if any of them result in a zero.

Upon checking, we find that 8/5 is not a zero of the polynomial 5x³ - 2x² + 20x - 8.

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The probability that the engine will not fail between 55,000 and 65,000 miles, based on normal distribution probabilities, when the mean failure is 60,000 miles and standard deviation of 5,000 miles is e) 68%.

Normal distribution probability is a continuous probability distribution with symmetrical values that mostly cluster around the mean.

We can compute the normal distribution probability using the normal distribution calculator.

Mean number of miles when a car's engine fails = 60,000 miles

Standard deviation = 5,000 miles

Sample  cutoff = between 55,000 and 65,000 miles

The probability of the engine fail between 55,000 and 65,000 miles = 0.6827.

= 68%.

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a) 25%

b) 40%

c) 35%

d) 32%

e) 68%

1. Event A includes the outcomes of H and T,

2. while event AC includes all the possible outcomes of rolling a number cube, which are 1, 2, 3, 4, 5, and 6.

1. Event A is defined as tossing a heads on a coin, regardless of the outcome of rolling a number cube. Therefore, the outcomes in event A are H (heads) and T (tails), since either of these outcomes could occur when rolling a number cube and tossing a coin.

2. Event AC is the complement of event A, i.e., it is the set of outcomes that are not in event A. Since event A contains H and T, the outcomes in event AC are the remaining outcomes that are not in event A, which are all the possible outcomes when rolling a number cube: 1, 2, 3, 4, 5, and 6.

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24. The length RT is 18 units

25. The equation of the circle graphed is x² + (y - 3)² = 16

The length RT can be calculated using the intersecting secants equation

So, we have

11 * (11 + x) = 9 * (9 + 13)

So, we have

11 + x = 18

This gives

RT = 18

The equation of the circle graphed is represented as

(x - a)² + (y - b)² = r²

Where, we have

So, we have

(x - 0)² + (y - 3)² = 4²

Evaluate

x² + (y - 3)² = 16

Hence, the equation of the circle graphed is x² + (y - 3)² = 16

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The exponential function graphed in this problem is given as follows:

[tex]y = e^x[/tex]

An exponential function has the definition presented according to the equation as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The graphed function has an intercept of 1, hence the parameter a is given as follows:

a = 1.

The function has the base e, hence it is given as follows:

[tex]y = e^x[/tex]

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The 91-day T-bill that Nadeen bought at an interest rate of 4.30% p.a. and face value of $5,000 indicates;

a) Nadeen paid about $4,946.24 for the T-bill

b) Barbara's selling price for the T-bill is about $4,962.74

A Treasury bill (T-bill), is a short-term obligation that is issued by the U.S. Department of Treasury and which is backed by the United States government, and has a maturity of less than a year. T-bills are low risk investment as they are backed by the credit and full faith of the U.S. government.

The formula for the price of the T-bill can be calculated with the formula;

Price = Face Value/(1 + (Interest Rate × Days to Maturity/360))

Plugging in the value from the question, we get;

Price = 5000/(1 + (0.043 × 91/360)) ≈ 4946.24

b) The formula for the selling price can be presented as follows;

Selling Price = Face Value/(1 + (Interest Rate × Remaining Days to Maturity/360))

Plugging in the known values, we get;

Selling Price = 5,000/(1 + (0.053 × (91 - 40)/360)) ≈ $4,962.74

Therefore, Barbara sold the T-bill to her friend for $4,962.74

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