The floor of Demetri’s house is rectangular in shape and is 20 feet longer than it is wide. A scale model of the floor has a length of 15 inches and a width of 10 inches. What is the width, in feet, of the floor of Demetri’s house?

By using matrix, it can be calculated that:

[tex]\frac{1}{4}C = \begin{bmatrix}3 & 4 & -5 \\ 1 & -6 & 7\end{bmatrix}[/tex]

The term "matrix" refers to any configuration of numbers in the form of rows and columns. a collection of numbers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix are the integers.

In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics. Solving linear equations is made easier by it. Matrices are incredibly priceless items that are used in a variety of contexts. In addition to mathematical applications, matrices are employed in a wide range of scientific disciplines. Nearly every element of our life uses engineering mathematics.

Here,

[tex]C = \begin{bmatrix} 12 & 16 & -20 \\ 4 & -24 & 28 \end{bmatrix}[/tex]

[tex]\frac{1}{4}C = \begin{bmatrix} 12\times \frac{1}{4} & 16 \times \frac{1}{4} & -20 \times \frac{1}{4} \\ 4 \times \frac{1}{4} & -24 \times \frac{1}{4}& 28 \times \frac{1}{4}\end{bmatrix}\\\\\frac{1}{4}C = \begin{bmatrix}3 & 4 & -5 \\ 1 & -6 & 7\end{bmatrix}[/tex]

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The average rate of change of the given function from x equals -7 to x equals 2 is calculated as: B. 101/90.

If we are given a function, f(x), to find the average rate of change of the function within the interval from a to b, the formula to use is:

Average rate of change = f(b) - f(a) / b - a.

Given the table that represents  function, we have the following:

a = -7

b = 2

f(a) = f(-7) = -2.7

f(b) = f(2) = 7.4

Plug in the values into the formula:

Average rate of change = (7.4 - (-2.7)) / (2 - (-7))

= (7.4 + 2.7) / 2 + 7)

= 10.1 / 9

= 101/90

Therefore, the average rate of change is: B. 101/90.

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Let the new board after final cutting is x. So the new board that is fit for the house will be x= 8.0625

A method for calculating the length of a psychological process that involves timing the reactions to two different tasks—one that involves the psychological process in issue and the other—and then subtracting the second reaction time from the first.

Taking away, part whole and comparison are three types of subtraction.

Let the old house =y= 13.375 inches

Cutting from old house=z= 5.3125 inches

New board=x= y-z

                 = 13.375-5.3125

                = 8.0625

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Let the new board after final cutting is x. So the new board that is fit for the house will be x= 8.0625

What is subtraction method?

A method for calculating the length of a psychological process that involves timing the reactions to two different tasks—one that involves the psychological process in issue and the other—and then subtracting the second reaction time from the first.

What are three types of subtraction?

Taking away, part whole and comparison are three types of subtraction.

Let the old house =y= 13.375 inches

Cutting from old house=z= 5.3125 inches

New board=x= y-z

  = 13.375-5.3125

              = 8.0625

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The measures of the lengths of the sides and angles of the triangles found using the law of cosines and the law of sines are presented as follows;

Question 1

∠A = 47.35°

∠B = 78.65°

c = 3.3

Question  2

∠B = 78.24°

∠C = 29.76°

a = 67.03

Question 3

∠A = 45.77°

∠B = 86.417°

∠C = 47.813°

Question 4

∠A = 36.15°

∠B = 60.48°

∠C = 93.37°

Question 5

a = 258.98

b = 327.41

∠C = 53°

Question 6

a = 34.11

b = 73.987

∠C = 53°

The law of cosines is a relationship between two sides (b and c) and the included angle, (∠A) and the third side (a) of the triangle.

Mathematically; a² = b² + c² - 2·b·c·cos(A)

Question 1

The dimensions of the triangle ΔABC are;

a = 3.0, b = 4.0, ∠C = 54°

The law of cosines indicates that we get;

c² = b² + a² - 2·b·a·cos(∠C)

Therefore;

c² = 3.0² + 4.0² - 2 × 3.0 × 4.0 × cos(54°) ≈ 10.893

The law of sines indicates that we get;

sin(54°)/3.3 = sin(∠A)/3.0

Question 2

b = 69, c = 35, ∠A = 72°

a² = 69² + 35² - 2 × 69 × 35 × cos(72°) ≈ 4493.45

The law of sines indicates that we get;

sin(72°)/67.03 = sin(∠B)/69

∠B = arcsine(69 × sin(72°)/67.03) ≈ 78·24°

∠C = 180° - 72° - 78.24° ≈ 29.76°

Question 3

a = 28, b = 39, c = 29

a² = b² + c² - 2·b·c·cos(A)

cos(A) = (a² - (b² + c²)) ÷ (2·b·c)

Therefore; cos(A) = (28² - (39² + 29²)) ÷ (-2 × 39 × 29) ≈ 0.6976

sin(45.77)/28 = sin(B)/39

sin(B) = 39 × sin(45.77)/28 ≈ 0.998

Question 4

a = 13, b = 17, c = 22

cos(A) = (13² - (17² + 22²)) ÷ (-2 × 17 × 22) ≈ 0.807

sin(36.15)°/13 = sin(∠B)/17

sin(∠B) = 17 × sin(36.15)°/13

Question 5

The parameters of the triangle are; ∠A = 50°, ∠B = 77°, c = 270

Please find attached the sketch of the triangle in the correct option created with MS Word

a/sin(50°) = 270/sin(53°)

a = sin(50°) × 270/sin(53°) ≈ 258.98

b = sin(77°) × 270/sin(53°) ≈ 329.41

Question 6

The parameters of the triangle are;

∠A = 27°, ∠B = 100°, c = 60

Please find attached the drawing of the correct triangle

∠C = 180° - 27° - 100° = 53°

∠C = 53°

60/sin(53°) = a/sin(27°)

a = sin(27°) × 60/sin(53°) ≈ 34.11

b = sin(100°) × 60/sin(53°) ≈ 73.987

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Answer: O B. Quadrilateral PQRS is a rectangle.

Step-by-step explanation: The image should help you picture my answer.  Good luck with your end of semester test!

Have a great day! :)

Answer:

non-violence and kindness with animals

Step-by-step explanation:

The area of the painted face with vertices at (-4, 2), (-4, -11), (4, 2), and (4, -11) which represents a rectangle is 104 square inches

A rectangle is a quadrilateral with the four interior angles that are each right angles, and opposite sides that are congruent and parallel.

The specified vertices of the face of the brick are;

(-4, 2), (-4, -11), (4, 2), and (4, -11)

The graph of the above point used to draw the quadrilateral representing the shape of the face in the question, created with MS Word, is attached

From the attached graph and drawing, we get;

Shape of the face of the doorstop = A rectangle

Length of the sides of the quadrilateral (rectangle)

Length = 2 - (-11) = 13

Width = 4 - (-4) = 8

Area of a rectangle  Length × Width

Area of the painted face of the brick, A = 13 × 8 = 104

The area of the rectangular painted face of the brick = 104 square inches

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The equation of given graph is y = 2x + 6.

What is equation of line?

The formula for a straight line is y = mx + c where c is the height at which the line intersects the y-axis, also known as the y-intercept, and m is the gradient.

Given:

The graph of the line is given.

From graph we have to find the equation of line.

Let the graph passes through the points (0, 6) and (2, 10).

From these two points to find the slope.

Slope = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex]

Here, [tex](x_1, y_1) = (0, 6), (x_2, y_2) = (2, 10)[/tex]

⇒ Slope = m = [tex]\frac{10-6}{2-0}= \frac{4}{2} = 2[/tex]

So, the slope is 2.

Now to find the equation of line.

Consider, the point - slope form of the line,

[tex]y-y_1=m(x-x_1)[/tex]

Plug [tex]m = 2, (x_1, y_1) = (0, 6)[/tex]

⇒

[tex]y-6=2(x-0)\\y-6=2x\\y=2x+6[/tex]

Hence, the equation of given graph is y = 2x + 6.

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Answer:

x=38

Step-by-step explanation:

linear par

180-134=46

180-84=96

sum of a triangle is 180

96+46+x=180

142+x=180

x=180-142

x=38

(a) C(p) is not defined since we can't divide by 0.

(b) The domain of C(p) as given by the equation is (-∞, 100) ∪ (100, ∞)

(c) The domain of C(p) in the context of the application is [0, 100)

(d) The percentage of particulate pollution removed from an industrial plant can't be negative or higher or equal than 100

Here we have given that Suppose that the cost C (in dollars) of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by

C(p) = 8100p / (100 − p)

And we need to find the following.

(a) In general ma the we know that dividing by zero is not possible one.

Therefore, C(p) is undefined when we try to divide it by 0.

(b) The domain of C(p) is the function is not defined at p =100 the domain would be written as,

=> D = (-∞, 100) ∪ (100, ∞)

(c) The domain of C(p) in the context of the application  is calculated as,

=> D = [0, 100)

Therefore, the percentage of particulate pollution removed from an industrial plant can't be negative or higher or equal than 100.

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(1, 5); (4, 7) - corresponding, (3, 5) - Alternate Interior, (2, 7); (1, 8) - Exterior

(3, 6) are Consecutive Interior Angles.

Any line that crosses two straight lines at different locations is known as a transversal.

Corresponding Angles:

When two parallel lines cross the third one, the angles that are in the same relative position at each junction are referred to as corresponding angles to one another

Alternate Interior Angles:

The alternative interior angles are those that are on the inner side of the parallel lines but on the opposing sides of the transversal.

Alternate Exterior Angles:

Alternate external angles are positioned on the opposing sides of the transversal and are always outside the two lines where the transversal intersects.

Consecutive Interior Angles:

The pair of non-adjacent internal angles that are located on the same side of the transversal are referred to as consecutive interior angles.

Here we have

Two parallel lines intersected and formed the angles, ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, and ∠8.

By the given information,

∠1  and ∠ 5 are corresponding angles

∠4 and ∠ 7 are corresponding angles

∠3 and ∠5 are Alternate Interior Angles

∠2, and ∠7 are Alternate Exterior angles

∠1 and ∠ 8 are Alternative Exterior angles

∠3 and ∠ 6 are Consecutive Interior Angles

Therefore,

(1, 5); (4, 7) - corresponding, (3, 5) - Alternate Interior, (2, 7); (1, 8) - Exterior

(3, 6) are Consecutive Interior Angles.

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Resulting box has the greatest volume for the values  (25 ± 5√7)/6 .

This is a problem that can be solved using derivatives , maxima & minima and common logic.

Hence , going by logic :

Creating a flap of 'a' inches in width, the base of the box will be

 (10 - 2a) by (15 - 2a)

and the depth of the box will be the width of the fold-up flap: a.

Then the volume of the box is

 v = [tex]a(10 -2a)(15 -2a) = 150a -50a^2 +4a^3[/tex]

Using the derivative of the volume will be zero at the maximum volume.

 0 = [tex]dv/da = 150 -100a +12a^2[/tex]

This has roots at

 a = (100 ±√(100² - 4(12)(150)))/(2·12)

 a = (100 ± √2800)/24 = (25 ± 5√7)/6

Only the smaller of these solutions gives a maximum volume.

You should cut (5/6)(5-√7) ≈ 1.962 inches to obtain the greatest volume.

Similarly , replacing the values of 10 by A and 15 by B , a generalized solution can be formed .

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The smallest possible value of f ( t ) = 9 based on the values of p , q , r , s.

Given :

Let f ( x ) be a polynomial with integer coefficients. Suppose there are four distinct integers p , q , r , s such that f ( p ) = f ( q ) = f ( r ) =f ( s ) = 5. If t is an integer and f ( t ) > 5,

Let g(x) = f(x) − 5.

g(x) = (x−p)(x−q)(x−r)(x−s)h(x)

The condition f(t) > 5 translates to g(t) > 0.

Since p,q,r,s,t are distinct integers, the smallest possible positive value of (t−p)(t−q)(t−r)(t−s) is 4 :

the four numbers in the parentheses are all distinct integers ≠ 0, so the smallest value we can get from the product (−2)⋅(−1)⋅1⋅2. }

The smallest possible positive value of h(t) is 1, since we must have g(t)≠0.

Thus the smallest possible value of g(t) is 4, and therefore the smallest possible value of f(t) is 9, and it is achieved for t=2 if we have

f(x)=x(x−1)(x−3)(x−4)+5

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Full question ;

Let f(x) be a polynomial with integer coefficients. Suppose there are four distinct integers p,q,r,s such that f(p)=f(q)=f(r)=f(s)=5. If t is an integer and f(t)>5, what is the smallest possible value of f(t)?

Answer:

$22.50

Step-by-step explanation:

Answer:

a = 4

Step-by-step explanation:

Step 1: Find two numbers that multiply to give 16 and add to give -8.

t²-8t+16 = 0

t² -4t -4t + 16 = 0

The two numbers are 4 and -4.

Step 2: Rewrite the equation in the form (t + 4)(t - 4).

t² - 8t + 16 = (t + 4)(t - 4)

Step 3: Factor the equation to get (t + 4)².

(t + 4)² = (t + 4)(t + 4)

Therefore, a = 4.

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Answer:

x = 15°

Step-by-step explanation:

⭐the exterior angle of a triangle is equal to the sum of the angles that is opposite of said exterior angle

Thus, to solve for x, we have to make an equation in regards of the exterior angle.

132° = x° + 117°

∴ 15 = x° . . . .  subtract 117 from LHS and RHS

⭐if this response helped you, please mark it the "brainliest"!⭐

The value of width will be;

⇒ 18.5 cm

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The volume of a rectangular prism = 6,618.375 cm³

The height is 13.25 cm and the length is 27 cm.

Now,

We know that,

The volume of rectangular prism = Length x Width x Height

Substitute all the values, we get;

⇒ 6,618.375 = 27 × x × 13.25

⇒ 6,618.375 / 357.75 = x

⇒ x = 18.5 cm

Thus, The value of width = 18.5 cm

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Answer: 8 Nights

Step-by-step explanation:

Cost of a plane ticket to Florida(c) = $1000

cost at the hotel per night (n)

Amount left with Karla after spending on the flight ticket =

$3000-$1000= $2000

To find out how many nights she can stay we should divide the remaining amount she has with (n) = $2000/$250 = 8

Hence Karla can stay for 8 nights

Answer:

[tex]\dfrac{1}{36n^2+6n}[/tex]

Step-by-step explanation:

Given factorial expression:

[tex]\dfrac{(6n-1)!}{(6n+1)!}[/tex]

[tex]\boxed{\begin{minipage}{6cm}\underline{Factorial Rule}\\\\$n!=\:n\cdot \left(n-1\right) \cdot \left(n-2\right) \cdot ... \cdot 3 \cdot 2\cdot 1$\\ \end{minipage}}[/tex]

Apply the factorial rule to the numerator and denominator of the given rational factorial expression:

[tex](6n-1)!=\left(6n-1\right)\cdot \left(6n-2\right)\cdot \left(6n-3\right)\cdot... \cdot 3 \cdot 2\cdot 1[/tex]

[tex]\left(6n+1\right)!=\left(6n+1\right)\cdot \:6n \cdot (6n-1) \cdot...\cdot 3 \cdot 2\cdot 1[/tex]

Therefore:

[tex]\begin{aligned}\implies \dfrac{(6n-1)!}{(6n+1)!}&=\dfrac{\left(6n-1\right)\cdot \left(6n-2\right)\cdot \left(6n-3\right)\cdot... \cdot 3 \cdot 2\cdot 1}{\left(6n+1\right)\cdot \:6n \cdot (6n-1) \cdot...\cdot 3 \cdot 2\cdot 1}\\\\&=\dfrac{1}{(6n+1) \cdot 6n}\\\\&=\dfrac{1}{6n(6n+1)}\\\\&=\dfrac{1}{36n^2+6n}\end{aligned}[/tex]

Answer:

--------------------------------

We know that:

Therefore:

Therefore:

Answer:

The Rate of return would then be 192 / 273 ≈ 70.32%

The lines that carry the rectangle onto itself are x = 0 and y = 1

The graph that completes the question is added as an attachment

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

The rectangular graph

The coordinates of one end of the graph are

(-3, 3) and (-3, -1)

Next, we calculate the midpoint of these ends

So, we have

Midpoint = 1/2(x₁ + x₂, y₁ + y₂)

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

Midpoint = 1/2(-3 + 3, -1 + 3)

Evaluate the like terms

Midpoint = 1/2(0, 2)

So, we have

Midpoint = (0, 1)

So, we have

x = 0 and y = 1

Hence, the reflection lines are x = 0 and y = 1

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a) The number alive after 7 years is given as follows: 48.

b) The percentage of herd that dies each year is of 10%.

The exponential function in the context of this problem is defined as follows:

N(t)=100(0.9)^t.

The parameters of the function are defined as follows:

The amount of herd alive after 7 years is found with the numeric value at t = 7, replacing the lone instance of t in the function by 7, hence:

N(7) = 100 x (0.9)^7 = 48.

(rounding to the nearest whole number).

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The difference of the given polynomials 8x³+5x+6 and 2x²+3x is 8x³-2x²+2x+6. So, the correct answer is D.

To subtract polynomials from another, we should change the signs (from '+' to '-' or from '-' to '+') of all the terms of the expression which is to be subtracted and then the two expressions are added.

Given that, 8x³+5x+6-(2x²+3x)

Group the like terms are perform the addition or subtraction

= 8x³+5x+6-2x²-3x

= 8x³+(5x-3x)-2x²+6 (Here, like terms are 5x and 3x)

= 8x³+2x-2x²+6

= 8x³-2x²+2x+6

So, the standard form of obtained polynomial is 8x³-2x²+2x+6.

The polynomials difference is 8x³-2x²+2x+6. Therefore, option D is the correct answer.

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"Your question is incomplete, probably the complete question/missing part is:"

Which polynomial represents the difference below?

8x³ + 5x+6-(2x² + 3x)

A. 10x+ 8x² +6

B. -2x² + 8x³ + 8x+6

C. 6x¹⁰+2x+6

D. -2x²+ 8x³+2x+6

Answer:196

Step-by-step explanation:

Using trigonometry identity the value of cosθ is 0.942.

What is round to 3 decimal?

Rounding to three decimal places is equivalent to rounding to the closest thousandth. Take 3.3685 as an example and round to the nearest thousandth. It is possible to round up to 3.369 or down to 3.368 because the third digit following the decimal point is 8 instead of 3.

Given that sinθ=3/9

Divide the denominator and numerator by 3:

sinθ=1/3

To find the value of cosθ, use trigonometry identity.

The trigonometry identity is sin²θ + cos²θ = 1

Solve the identity for  cosθ:

Subtract both side by sin²θ:

cos²θ = 1 - sin²θ

Take square root on both sides:

cos θ = √(1 - sin²θ)

Putting sinθ = 1/3 in the above equation:

cos θ = √(1 - (1/3)²)

cos θ = √(1 - 1/9)

cos θ = √((9 - 1)/9)

cos θ = √(8/9)

cos θ = 0.9428

cos θ = 0.943 (round to 3 decimal places)

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A)

A manufacturer of banana chips would like to know whether its bag-filling machine works correctly at the 414-gram setting.

So, Null hypothesis: [tex]H_{0}[/tex] : μ < 414

It is believed that the machine is underfilling the bags.

So, Alternate hypothesis: [tex]H_{1}[/tex] : μ < 414

Given,

n= 8

Population standard deviation (б) = 18

x= 407

We will use the t-test since n > 8 and we are given the population standard deviation.

t=x-μ / (б/[tex]\sqrt{n-1}[/tex])

t= [tex]\frac{407-414}{\frac{18}{\sqrt{7} } }[/tex]

t= -1.028

Use the t table to find p value

p-value = 12.706

Level of significance α = 0.025

p-value>α

It is a two-tailed test.

So, we fail to reject the null hypothesis.

So, its bag-filling machine works correctly at the 414-gram setting.

B)

Let μ be the population mean amount of ozone in the upper atmosphere.

As per the given, we have

[tex]H_{0}[/tex]    : μ = 4.8

[tex]H_{1}[/tex] : μ ≠ 4.8

Sample size: n= 26

Sample mean = 4.6

Standard deviation = 1.2

Since population standard deviation is now given, so we use a t-test.

t= [tex]\frac{4.6-4.8}{\frac{1.2}{\sqrt{25} } }[/tex]

t= -0.2/0.24

t= -0.833

It is a two-tailed test.

We are accepting the null hypothesis.

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polynomial function for  x = 4, - 1 & 2. are

 [tex]x^{3}[/tex] - [tex]x^{2}[/tex] - 10x - 8

Given data :

Given the zeroes (roots) of a polynomial are:  4, - 1 & 2.

Here, we are given three roots of the polynomial. That means, the polynomial must be of third degree.

Also, (x - a) is a factor of the polynomial if and only if x = a is a root of the polynomial.

Here, 4, - 1 & 2. are roots. So, the factors are: ( x - 4 ) ( x + 1 ) ( x - 2 )

Multiplying them will result in the polynomial.

= ( x - 4 ) ( x + 1 ) ( x - 2 )

= ( [tex]x^{2}[/tex] - 3x - 4 ) ( x + 2 )

=  Distribute parentheses : [tex]x^{2}[/tex] . x + [tex]x^{2}[/tex] . 2 - 3xx - 3x . 2 - 4x - 4 . 2

= Simplify [tex]x^{2}[/tex] .x - [tex]x^{2}[/tex] . 2 - 3xx - 3x . 2 - 4x - 4 . 2

=  [tex]x^{3}[/tex] - [tex]x^{2}[/tex] - 10x - 8

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The correct matrix for this system of equations in row reduced echelon form amongst the given options is none of the above.

The given system of equations to get the reduced row echelon form is:

x1 - 3x2 - x3 +3x4 = 2

-x1 - 2x2 - x3 +4x4 = 2

-2x1 + 4x2 + 4x3 +4x4 = -3

To get the reduced row echelon form we need to form the matrix:

[tex]\left[\begin{array}{ccccc}1&-3&-1&3&2\\-1&-2&-1&4&2\\-2&4&4&4&-3\end{array}\right][/tex]

Row reduced echelon form of a matrix is one in which the pivot points equal 1, go from top left to bottom right, there are 0's above and below each pivot point.

R2-R1:

[tex]\left[\begin{array}{ccccc}1&-3&-1&3&2\\-2&1&0&1&0\\-2&4&4&4&-3\end{array}\right][/tex]

R3-R2 then R2+2R1 then R1+R3 then R2+2R3 then R3-3R2

[tex]\left[\begin{array}{ccccc}1&0&3&6&-1\\0&1&6&13&-2\\0&0&-14&-36&3\\\end{array}\right]\\[/tex]

(-1/14)R3 then R2-6R3 then R1-3R3

[tex]\left[\begin{array}{ccccc}1&0&0&-12/7&-5/14\\0&1&0&-17/7&-5/7\\0&0&1&18/7&-3/14\\\end{array}\right]\\[/tex]

which is the reduced row echelon form of the given system of equations.

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Answer:

Step-by-step explanation:

3.14 x 3=9.42