3. The Tilley household wants to build a patio deck in the shape of a 45° - 45 - 90 triangle in a nice

corner section of their backyard. They have enough room for a triangular deck with a leg

measuring 36 feet. What will be the length of the longest side? Round to the ten-thousandth (4

decimal places).

The figure below is made of 222 rectangular prisms.

Completed question :

Part D

Wendell plans to paint the doghouse after it’s built. He wants to know what the surface area of the outside of the doghouse will be. To find the surface area, first break up any composite shapes into rectangles and triangles. Which shape is considered a composite shape? When the shape is decomposed, what are the dimensions of the resulting shapes necessary for finding surface area?

Given data :

The volume of a rectangular prism is given as follows.

Volume of a rectangular prism = length x breadth x height

Therefore, the volume of 222 rectangular

prism

= 2 x 2 x 2

= 8 cubic units

Volume of 222 rectangular prisms = 222

×Volume of a rectangular prism

=222× 8

= 1776 cubic units

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Answer:

The surface area is 5,052 inches

if  RS=x+4, ST=8 and RT=3x then value of RS is 10 units.

Two or more expressions with an Equal sign is called as Equation.

Given that RS=x+4, ST=8 and RT=3x

We have to find RS

RS=RT-ST

x+4=3x-8

Take the variable terms on one side and constants on other side

12=2x

Divide both sides by 2

x=6

So RS is 6+4=10 units

Hence, if  RS=x+4, ST=8 and RT=3x then value of RS is 10 units.

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The pricing for each pizza is given as follows:

The pricing for each pizza is obtained applying the proportions in the context of this problem.

A pizza shop sells a medium pizza with a diameter of 12 inches for 15 dollars, hence the price per inch of the diameter is given as follows:

15/12 = $1.25.

The small pizza has a diameter of 4 inches, hence the price is given as follows:

4 x 1.25 = $5.

The large pizza has a diameter of 16 inches, hence the price is given as follows:

16 x 1.25 = $20.

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The standard form from (-7x - 5) - (-9[tex]x^{2}[/tex] + 8x + 7) is B) 9[tex]x^{2}[/tex] - 15x - 12.

This is a quadratic equation. To simplify the equation, we need to arrange the equation to be standard form or quadratic equation which is

a[tex]x^{2}[/tex] + bx + c

(-7x - 5) - (-9[tex]x^{2}[/tex] + 8x + 7) = -7x -5 + 9[tex]x^{2}[/tex] - 8x -7

Group of a[tex]x^{2}[/tex] --> It will be 9[tex]x^{2}[/tex]

Group of bx ---> It will be -7x-8x = -15x

Group of c ---> It will be -5-7 = -12

We just combine the group into standard equation

(-7x - 5) - (-9[tex]x^{2}[/tex] + 8x + 7) = -7x -5 + 9[tex]x^{2}[/tex] - 8x -7

                                      = 9[tex]x^{2}[/tex] -15x - 12

So the simplification of equation (-7x - 5) - (-9[tex]x^{2}[/tex] + 8x + 7) is 9[tex]x^{2}[/tex] - 15x - 12, or from the answer, it's B)

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Answer: In statistics, the term linear model is used in different ways according to the context.

Step-by-step explanation: The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term is also used in time series analysis with a different meaning.

The angle of reflection for the ray leaving mirror 2 will be 41 degrees, as this is equal to the angle of incidence for the ray striking mirror 2.

The angle of incidence is the angle between the incoming ray and the surface of the mirror, and the angle of reflection is the angle between the reflected ray and the surface of the mirror.

In the given scenario, a ray of light strikes mirror 1 with an angle of incidence of 41 degrees.

The angle of reflection can be calculated using the law of reflection, which states that the angle of incidence is equal to the angle of reflection.

Therefore, the angle of reflection of the ray when it leaves mirror 1 will be 41 degrees.

However, the ray of light continues its journey and strikes mirror 2.

When this happens, the angle of incidence is the angle between the reflected ray from mirror 1 and the surface of mirror 2.

The angle of reflection for the ray leaving mirror 2 can be calculated using the same law of reflection.

The angle of reflection will be equal to the angle of incidence, which is 41 degrees.

The angle of reflection is a crucial concept in understanding the reflection of light and is used in various applications such as mirrors, prisms, and lenses.

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Answer: $2171.79

Step-by-step explanation:

The cost of the materials for the cheapest rectangular storage container with an open top to have a volume of 24 cubic meters is $288.

Let the width of the base be x, then the length of the base is 2x. The height of the container is

[tex]24/(x2x) = 24/2x^2[/tex].

The total area of the base is[tex]x2x = 2x^2[/tex] and the total area of the sides is

[tex]2(x*24/2x^2) = 24x/x^2[/tex]= 24/x.

The cost of the base is

[tex]2x^2[/tex] * $12 = $24[tex]x^2[/tex] and the cost of the sides is

24/x * $6 = $144/x.

The total cost is $24[tex]x^2[/tex] + $144/x.

To find the cheapest container, we want to minimize the cost, which is equivalent to finding the minimum value of $24[tex]x^2[/tex] + $144/x.

We can use calculus to find the minimum, or we can use trial and error to find a reasonable value for x that minimizes the cost.

For x = 2, the cost is

$24 * [tex]2^2[/tex] + $144/2 = $96 + $72 = $168

which is less than $288.

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Mateo will be able to choose a gym membership plan that best meets his needs and budget.

To help Mateo decide on the best gym membership plan, the following information would be important:

By considering these factors, Mateo will be able to choose a gym membership plan that best meets his needs and budget.

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The no of gallon of a 14%-salt solution must be mixed with 6 gallon of a 25%-salt solution to obtain a 20%-salt solution is 5 gallon

What is Gallon?

The gallon is a volume unit used in imperial and customary units in the United States. There are now three versions in use: the imperial gallon, which is equal to 4.54609 litres

Given

14%-alt solution must be mixed with 6 gallon of a 25%-salt solution to obtain a 20%-salt solution

.14x +.25*6 = .20(x+6)

.14x + 1.5 = .20x + 1.2

.14x - .20x = 1.2 - 1.5

-0.06x = -0.3

x = 5 gallon

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The no of gallon of a 14%-salt solution must be mixed with 6 gallon of a 25%-salt solution to obtain a 20%-salt solution is 5 gallon

What is Gallon?

A gallon is a volume unit used in imperial and customary units in the United States. There are now three versions in use: the imperial gallon, which is equal to 4.54609 litres

Given

14%-alt solution must be mixed with 6 gallon of a 25%-salt solution to obtain a 20%-salt solution

.14x +.25*6 = .20(x+6)

.14x + 1.5 = .20x + 1.2

.14x - .20x = 1.2 - 1.5

-0.06x = -0.3

x = 5 gallon

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x=6 m<ABD=20 *I think*

Step-by-step explanation:

Find x

plug in your answer

The sum of the n distinct nth roots of unity is equal to 0, i.e. ω0 + ω1 + ... + ωn-1 = 0.

To prove this:

Consider a complex number z = cos(2π/n) + i sin(2π/n). It is a primitive nth root of unity, i.e.[tex]z^n[/tex] = 1.

Let w = [tex]z^k[/tex] for k = 1, 2, ..., n-1, then w is also a primitive nth root of unity.

So the sum of all the nth roots of unity can be expressed as:

[tex]1 + w + w^2 + ... + w^{(n-1)} = \frac{(1 + w + w^2 + ... + w^{(n-1)}}{z^{(n-1)}} = \frac{(z^n - 1)}{(z - 1) }= 0[/tex]

Hence, the sum of the n distinct nth roots of unity is equal to 0.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. It is the combination of a real number and an imaginary number. Complex numbers are used in many areas of mathematics, including algebra, calculus, and number theory.

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1 micrometer equals 1e-6 meters. Dividing by 1000000 will convert your number from micrometers to meters.

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The circumference of the circle with the diagram of 2.25 cm is 14.13 cm.

The circumference, also known as the length of a circle, is its circumference. By using the radius or diameter, we can calculate the circumference of a circle.

An irregular plane figure is a circle. Every point on the circle is equally distant from the circle's fixed centre. It has a radius of 1, making it a 2D shape. The Latin word "circulus," which means "small ring," is the source of the English word "circle."

Given that the diameter of the circle is 4.50 cm. The circumference of the circle is calculated as:-

C = 2πr

C = 2 x π x 2.25

C = 14.13 cm

Therefore, the circumference of the circle with the diagram of 2.25 cm is 14.13 cm.

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The set of all vectors in r4 satisfying the equation is,

( x, R , y ) = ( 1 , R , 1/2 ) ( 2 , R , 1 ) ( 3 , R , 3/2 ) ( 4 , R , 2 ) ( 5 , R , 5/2 ) ( 6 , R , 3 ) ( 7 , R , 7/2 ) ( 8 , R , 4 )

A basis for R4 always consists of 4 vectors. (TRUE: Vectors in a basis must be linearly independent AND span.) 4. The union of two subspaces is a subspace.

If S is a subspace of R4, then the zero vector 0=[0000] in R4 must lie in S. 2x+4y+3z+7w+1=0. So 0∉S, and we conclude that S is not subspace of R4.

The space R4 is four-dimensional, and so is the space M of 2 by 2 matrices. Vectors in those spaces are determined by four numbers. The solution space Y is two-dimensional, because second order differential equations have two independent solutions.

The interval of x is known and the relation of x , y is known.

So,

x = 2y  ;   x = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 }

Then,

1 = 2y , 2 = 2y , 3 = 2y , 4 = 2y , 5 = 2y , 6 = 2y , 7 = 2y , 8 = 2y ,

We can write,

y = { 1/2 , 1 , 3/2 , 2 , 5/2 , 3 , 7/2 , 4 }

Therefore,

The set of all vectors in r4 satisfying the equation is,

( x, R , y ) = ( 1 , R , 1/2 ) ( 2 , R , 1 ) ( 3 , R , 3/2 ) ( 4 , R , 2 ) ( 5 , R , 5/2 ) ( 6 , R , 3 ) ( 7 , R , 7/2 ) ( 8 , R , 4 )

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The linear demand equation: x = m(p) + b, where x is the number of occupied units, p is the monthly rent, and m and b are constants and the linear demand equation in this case is:

x = -1.04p + 44.16

To find the linear equation, we need to determine the slope (m) and the y-intercept (b). Using the two points provided, we can write a system of equations:

When p = $780, x = 40 units

40 = m($780) + b

When p = $852, x = 36 units

36 = m($852) + b

Solving for m and b, we get:

m = -1.04 and b = 44.16

So, the linear demand equation is:

x = -1.04p + 44.16

This equation represents the inverse relationship between the monthly rent and the demand, where a higher rent results in a lower demand and vice versa.

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Sister uses 420.76 inches more wire per wall than Poppy.

A circle is a two-dimensional figure with a radius and circumference of 2πr.

We have,

Poppy's diameter = d

Older sister diameter.

D = d + 134

Now,

The circumference of the wall hangings.

= 2πr

= πd

Now,

The circumference of the wall hangings by the older sister.

= πD

= π (d + 134)

The circumference of the wall hangings by the Poppy.

= πd

Now,

The difference between the length of the wire.

= π (d + 134) -  πd

= πd + π x 134 - πd

= 3.14 x 134

= 420.76 inches.

Thus,

Sister uses 420.76 inches more wire.

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The logarithm and exponential functions are inverse of each other. The exponential function raises a number to a power and the logarithm gives the power that the number has to be raised to in order to equal a given value. In other words, the exponential function expresses exponential growth, while the logarithm expresses exponential decay.

For example, if y = 2^x, then x = log2(y). This logarithmic relationship is useful in solving exponential problems and understanding exponential dynamics.

In exponential dynamics, logarithms are used to study and model changes that occur at exponential rates, such as population growth or decay, interest rate calculations, and many other real-world applications. The logarithmic relationship helps to simplify the exponential equations and make them easier to work with.

In summary, the logarithm and exponential functions are important in understanding exponential dynamics because the logarithmic relationship helps to simplify exponential problems and make them easier to solve.

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Answer:

False - The standard deviation is always positive.

A linear function would better approximate the data.

A linear function would better approximate the data described. This is because a linear function has a constant rate of change, while an exponential function has a changing rate of change. For example, if we take the data from 10 K to 15 K, we can calculate the rate of change for each 1 K increase in temperature. The rate of change of volume for a 1 K increase in temperature is calculated by taking the difference between the volumes at the two temperatures and dividing by the difference in the temperatures. With the given data, this would yield a rate of change of 0.7 cm3/K, 0.38 cm3/K, 0.92 cm3/K, 1.4 cm3/K, and 1.6 cm3/K. As we can see, the rate of change does not remain constant, but increases over the temperature range, indicating a nonlinear relationship. Therefore, a linear function would better approximate the data.

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The human heartbeats 2,721,600,000 times in 70 years

Let's say that the human heart beats 1 time in 0.8 sec.

Noe, we need to know the number of seconds in 70 years

Number of months in 70 years = 70 x 12 = 840 months

Number of days in 70 years = 840 x 30 = 25,200 days

Number of hours in 70 years = 25,200 x 24 = 604,800 hours

Number of seconds in 70 years = 604,800 x 3600

                                                     = 2,177,280,000 seconds

Number of times a heartbeat in 0.8 sec = 1 time

Number of times a heartbeat in 2,177,280,000 seconds = (1/0.8) x 2,177,280,000

                                                  = 2,721,600,000 times

                                                  ≈ 2.7 billion times

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the helix r(t) = cos(πt/2), sin (πt/2),t) intersects the sphere x^2 + y^2 + z^2 = 2 in two points.

the angle of intersection at each point: θ = cos⁻¹ (√2/√π²+4)

the angle of intersection at each point: α = cos⁻¹(-√2 /(√π² +4))

Scientists in the fields of physics, engineering, and biology all do research on helices, which are a significant topic in the theory of curves in differential geometry. Helix (or generic helix) is defined as tangent vector field in 3-dimensional Euclidean space establishing a fixed angle with the curve's direction. A curve for which the tangent forms a constant angle with a fixed line is referred to as a helix or coil.

helix: r(t) = < cos(πt/2), sin(πt/2), t >    sphere: x² + y² + z² = 2

put it in the sphere's equation we get:

next, {cos(πt/2)}² + {sin(πt/2)}² + t² = 2

or, 1 + t² = 2

or, t² = 1

or, t = ± 1

thus, points of intersection r(1) = < cos(π/2), sin(π/2), 1 > = < 0, 1, 1 >

r(-1) = < cos(-π/2), sin(-π/2), -1 > = < 0, -1, -1 >

angle intersection is angle between their tangent:

helix:

r(t) = < -π/2 sin(πt/2), π/2 cos(πt/2), 1 >

r'(1) = < -π/2, 0, 1 > .........(i)

r'(-1) = < π/2, 0, 1 > ......... (ii)

sphere:

f = (x² + y² + z² - 2) = 0

Δf = <2x, 2y, 2z>

Δf (0,1,1) = <0,2,2> ................ (iii)

Δf (0,-1,-1) = <0,-2,-2> .............(iv)

angle between (i) and (iii) (at, t = 1)

cosθ = {r'(1) Δf (0,1,1)} / {|r'(1)| |Δf (0,1,1)|]

θ = cos⁻¹ (√2/√π²+4)

angle between (ii) and (iv) (at, t = 1)

cosα = r'(-1)Δf (0,-1,-1) / ||r'(-1)|| ||Δf (0,-1,-1) ||

cos α = -√2 /(√π² +4)

α = cos⁻¹(-√2 /(√π² +4))

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Factory should produce 100 high-grade shoes and 200 low-grade shoes for maximum profit.

Let x be the number of high-grade shoes produced, then the number of low-grade shoes produced is at least 2x. The maximum number of shoes the factory can produce is 500, so we have the following constraints:

x + 2x ≥ 500 (production constraint)

x ≥ 100 (delivery constraint)

x ≥ 0 (non-negative constraint)

The profit for each type of shoe is given as birr 2.00 for high-grade shoes and birr 1.00 for low-grade shoes, so the total profit is:

Profit = 2x * 2.00 + 1.00 * 2x = 4.00x + 2x = 6.00x

The objective is to maximize the profit, subject to the constraints above. So, the feasible solution is x = 100, which means the factory should produce 100 high-grade shoes and 200 low-grade shoes. This is the maximum possible profit of 600 birr.

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The probability of selecting a yellow and then a yellow without replacement is 3/10.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

We have,

Number of red marbles = 1

Number of green marbles = 1

Number of yellow marbles = 3

Total number of marbles = 5

The probability of selecting a yellow and then a yellow without replacement.

= 3/5 x 2/4

= 6/20

= 3/10

We are using 2/4 since yellow marbles become 2 and the total marbles become 4 when the condition of without replacement is given.

Thus,

The required probability is 3/10.

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The value that makes the pair of ratios equivalent is x = 7.

To discover the unknown values in a mathematical equation, we cross multiply the numbers in the equation. The method of cross multiplication is frequently employed to identify the missing variable in an equation. We multiply both the numerator and denominator of the provided expression or fractions.

The given pair of ratios is:

12:42 = 2:x

12 /42 = 2/ x

Using cross multiplication we have:

12x = 2(42)

x = 7

Hence the value that makes the pair of ratios equivalent is x = 7.

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The number of months that the tree has been growing is; 8.375 months

The general form for the equation of a line in slope intercept form is;

y = mx + c

where;

m is slope

c is y-intercept

Now, we are told that the tree grows 8 centimeters each month. This means the slope is 8 cm.

Now, since she trimmed 6 cm in the first month, it means that the y-intercept will be -6. Thus;

Equation is;

y = 8x - 6

Since the current height is 61 centimeters, then;

61 = 8x - 6

8x = 61 + 6

8x = 67

x = 67/8

x = 8.375 months

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The smallest superfortunate number is 993

What is super fortunate number?

A Fortunate number, named after Reo Fortune, is the lowest integer m > 1 for which pn# + m is a prime number for a given positive integer n, where the primorial pn# is the product of the first n prime numbers.

What is Prime Number?

A prime number is a natural number larger than one that cannot be calculated as the product of two lesser natural numbers. A composite number is a natural number greater than one that is not prime. 5 is prime, for example, because the only methods to write it as a product, 1 5 or 5 1, involve 5 itself.

993 is the smallest superfortunate number.

You can obtain this by letting the digits of n

be ...a99...9b

where a<9 and there are, say, k 9s. Then look at the results of adding 1,2,3 ... ,12.

The smallest number is 993.

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The smallest superfortunate number is 993

What is super fortunate number?

A Fortunate number, named after Reo Fortune, is the lowest integer m > 1 for which pn# + m is a prime number for a given positive integer n, where the primorial pn# is the product of the first n prime numbers.

What is Prime Number?

A prime number is a natural number larger than one that cannot be calculated as the product of two lesser natural numbers. A composite number is a natural number greater than one that is not prime. 5 is prime, for example, because the only methods to write it as a product, 1 5 or 5 1, involve 5 itself.

993 is the smallest superfortunate number.

You can obtain this by letting the digits of n

be ...a99...9b

where a<9 and there are, say, k 9s. Then look at the results of adding 1,2,3 ... ,12.

The smallest number is 993.

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The value of Sn for the arithmetic series is, 1,600.

An arithmetic sequence is the sequence of numbers where each consecutive numbers have same difference.

We have to given that;

For the arithmetic series,

⇒ a₁ = - 20

⇒ n = 25

⇒ an = 148

Now, We know that;

The nth partial sum of an arithmetic sequence can be calculated using the first and last terms as follows:

⇒ Sn = n(a₁ + an) / 2

Hence, Substitute all the given values, we get;

⇒ Sn = n(a₁ + an) / 2

⇒ Sn = 25(- 20 + 148) / 2

⇒ Sn = 25 × 128 / 2

⇒ Sn = 1600

Thus, The value of Sn for the arithmetic series = 1,600.

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Answer:

x = ± 8

Step-by-step explanation:

x² = 64 ( take square root of both sides )

x = ± [tex]\sqrt{64}[/tex] = ± 8

solutions are x = 8 and x = - 8 ,  since

8 × 8 = 64 and - 8 × - 8 = 64