a. dilate triangle using center and scale factor . b. what do the properties of dilations tell you about angle ?

Answer: C

Step-by-step explanation:

Yes, the conditions for constructing a t confidence interval are met. The sample size is 10, which is greater than 30, and the sample is a random sample of 10 deliveries from a large number of deliveries made.

1. Constructing a t confidence interval requires that the sample size is greater than 30, so the first condition that must be met is that the sample size is greater than 30.

2. The second condition that must be met is that the sample is a random sample of the population of interest. In this case, the sample is a random sample of 10 deliveries from a large number of deliveries made.

Since both conditions are met, the conditions for constructing a t confidence interval are met.

Yes, the conditions for constructing a t confidence interval are met. The sample size is 10, which is greater than 30, and the sample is a random sample of 10 deliveries from a large number of deliveries made.

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Two types of transformations that can be used to transform f(x) to g(x) are horizontal and vertical stretching/shrinking.

What is transformation ?

A transformation is a function that changes the position, size, or shape of an object or a graph. In mathematics, transformations are often used to map a function from one coordinate space to another. There are several types of transformations, including translations, rotations, reflections, and scalings.

Part A: Two types of transformations that can be used to transform f(x) to g(x) are horizontal and vertical stretching/shrinking.

Part B:

Horizontal stretching/shrinking: k is the horizontal scale factor.

Vertical stretching/shrinking: k is the vertical scale factor.

Part C:

Horizontal stretching/shrinking: g(x) = f(kx)

Vertical stretching/shrinking: g(x) = kf(x)

Two types of transformations that can be used to transform f(x) to g(x) are horizontal and vertical stretching/shrinking.

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The first partial derivatives of the function z = x sin(xy) is x²cos(xy)

The term partial derivatives is defined as the rate of change of a function with respect to a variable and the derivatives are fundamental to the solution of problems in calculus and differential equations.

Here we have given that the function z = x sin(xy).

And as per the definition of partial derivative the value is calculated as,

Here we have given that

=> f(x, y) = x sin(xy)

And then here we need to find fx​ we treat y as constant and differentiate with respect to x, then we get

=> fx = sin(x y) + xy cos(xy)

Similarly now we have to find fy​ we treat x as constant and differentiate with respect to y

=> fy​ = x²cos(xy)

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On solving the provided question we can say that any triangle's perimeter is the sum of its three sides' lengths = [tex]\x = 360/6 = > x = 60[/tex]

A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.

Triangle's sides are x, 2x, and 3x Plus 2 ft. Respectively.

Radius is 362 feet.

Any triangle's perimeter is the sum of its three sides' lengths.

[tex]362 = x + 2 x + 3x +2\\6 x +2=362\\6 x= 362-2\\6 x=360\\x = 360/6\\x = 60[/tex]

Triangle's shortest side measures 60 feet.

Middle side = 2 x 2 x 60 ft = second side

Third side equals 3 x + 2 and is measured as 360+2=180+2=182 ft.

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On solving the provided question, we can say that by linear equation we have 2.5x * cereals = y

A linear equation is one that has the form y=mx+b in algebra. B is the slope, and m is the y-intercept. It's usual to refer to the previous clause as a "linear equation with two variables" because y and x are variables. The two-variable linear equations known as bivariate linear equations. There are several instances of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3.

2.5 cups of this particular cereal,

the following linear equation, you could calculate how many calories and grams of fiber you were eating.

let x = cereals we eat y = calories

so, equation - 2.5x * cereals = y

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The value of r in the given equation is 0.04 by the process of solving the equation for the unknown.

The given equation: 816 = 600(1 + r)

Where it represents the amount of money earned on a simple interest savings account.

We need to find out the value of r in the equation.

To find the value of unknown r, both the LHS and RHS of the equation must be divided by 600.

Therefore, 816/600 = [600(1 + r)]/600

816/600 = 1(1 + r)

816/600 = 1 + r

r = 816/600 - 1

r = 0.04

Therefore, the amount of money earned on a simple interest savings account using the given equation is 0.04

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The hypotenuse side of the right triangle is 7 units.

The value of x in the triangle is 5.92 units.

A right triangle is a triangle with one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

The side of a right triangle can be found using Pythagoras's theorem as follows:

using Pythagoras's theorem,

c² = a² + b²

where

Therefore,

hypotenuse = 7

Let's find x.

7² - (√14)² = x²

49 - 14 = x²

x² = 35

square root both sides

x = √35

x = 5.9160797831

x = 5.92 units

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Answer:

C. y = 30x + 130    D. 5 people  Q3. The 1st number is 25

     y = 20x +180                                 The 2cnd number is 7

Step-by-step explanation:

C. The x represents the amount of people attending.

     30 and 20 are the price per person.

D. 30 x 5 + 130 = 280

    20 x 5 = 180 = 280

Q3. 32 + 18 = 50

50/2 = 25

25 + y = 32

y = 32 - 25

y = 7

hope this helps =)

y = 30x + 130 and y = 20x + 180 is the  system of equations that represents the model and The two numbers are 25 and 7 in second question.

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

Let y represents the total cost of the party, and x represents the number of the people attending the party.

The hockey suite costs $130 to rent the room and $30 per person.

y=130+30x

The basketball suite costs $180 to rent the room and $20 per person.

y=180+20x

So option c is correct and

If there was going to be a total of 4 people attending the party,

y=130+30(4)=250

y=180+20(4)=260

basket ball is most cost efficient.

If there was an amount of people that could attend the party so that both games would cost the same

130+30x=180+20x

10x=50

x=5

So 5 people could attend the party.

The sum of two numbers is 32

The difference is 18

x+y=32

x-y=18

x+y+x-y=32+18

2x=50

Divide both sides by 2

x=25

y=7

The two numbers are 25 and 7.

Hence, y = 30x + 130 and y = 20x + 180 is the  system of equations that represents the model and The two numbers are 25 and 7 in second question.

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Answer: a = 9

Step-by-step explanation:

The given figure formed is a rectangle.

Area of given region (Yellow region) = 24 [tex]unit^{2}[/tex]

Length (l) = 6-2 = 4 units

Breadth (b) = a-3 units

Area = l×b

24 = 4 x (a-3)

6 = a-3

a = 9 units

Answer:

Perpendicular Bisector

Therefore , the solution of the given problem of function comes out to be the function f(x) = x³ + 2x² + 7x - 11  = -1 and g(x) = 3f(x) = -3

The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A connection between inputs and outputs in which each input leads to a single, distinct result is known as a function. Each function is given a domain and a codomain, or scope.

Here,

Given:

we have been provided with -

f(x) = x³ + 2x² + 7x - 11  

so, x = 1

f(x) = 1 +2 +7 -11= -1

g(x) = 3f(x) = -3

Therefore , the solution of the given problem of function comes out to be the function f(x) = x³ + 2x² + 7x - 11  = -1 and g(x) = 3f(x) = -3

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Answer:

|1|

Step-by-step explanation:

Because absolute value measures the positive distance from the numeral to 1, the answer could be negative or positive one, which is what the graph shows. So, |1| is the answer.

The eight term of the geometric sequence 125/9, 25/3,5,3,.... is 243/625.

These steps to answer:

Question above is a geometric sequence consisting of 4 terms to find the 8th term, then we can use the geometric sequence formula below

Tn= a. r^n-1

is known

a= 125/9

r=T4/T2

r=3/5

then to find T4

Tn= a. r^n-1

T8=125/9. (3/5^8-1)

T8=125/9. (3/5)^7)

T8=125/9. (2187/78125)

T8=243/625

A geometric sequence is a sequence that satisfies the quotient of a term with the preceding terms which are of course consecutive. This thing has a constant value. Not only that, geometric sequences are also known as 'measurable sequences' which are still closely related to arithmetic sequences and series.

Examples of geometric sequences are a, b, and c. Then c/b = b/a = constant, this is where the quotient of adjacent terms will be obtained and then it is said to be the ratio of the geometric sequence which is given the symbol "r".

Another example, which is much easier to understand, is for example, if you have a sequence and a series: 2, 4, 8, 16, 32, ….. etc., then from the sequence and series it can be seen between the first and second terms and so on, have the same multiplier.

So, to find the nth term, you can easily find the ratio first. By knowing 'r', then you will easily find Tn.

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Answer:

Step-by-step explanation:

substitute p for 8, because p=8

question 1 :    8+4=12    ----Yes

question 2: 8x8 =64-------Yes

question 3  5x8=35, not 8----No

question 4 2+8=10, not 12----No

Answer:

Probably  c:  f(x) = y + 3

Step-by-step explanation:

See attached worksheet

The general solution of the system x' (t) is[tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex]

The given system of equation is X'A=X

where

[tex]$$A=\left[\begin{array}{cc}-15 & -39 \\6 & 15\end{array}\right]$$[/tex]

Eigenvector of a square matrix is defined as a non-vector in which when a given matrix is multiplied,  it is equal to a scalar multiple of that vector. Let us suppose that A is an n x n square matrix, and if v be a non-zero vector, then the product of matrix A, and vector v is defined as the product of a scalar quantity λ and the given vector, such that:

Where

v = Eigenvector and λ be the scalar quantity that is termed as eigenvalue associated with given matrix A

The values of A are given by

[tex]$|A-\lambda I|=0 \\\left|\begin{array}{cc}-15-\lambda & -39 \\6 & 15-\lambda\end{array}\right|=0 \\[/tex]

[tex](-15-\lambda)(15-\lambda)-(6)(-39)=0 \\[/tex]

[tex]\Rightarrow-(15+\lambda)(15-\lambda)+(6)(39)=0 \\[/tex]

[tex]\\\Rightarrow(15+\lambda)(15-\lambda)-(6)(39)=0 \\\\\Rightarrow 225-\lambda^2-234=0 \\[/tex]

[tex]\Rightarrow-\lambda^2-9=0 \\[/tex]

[tex]\Rightarrow \lambda^2=-9 \\[/tex]

[tex]\Rightarrow \lambda=\pm 3 i[/tex]

Now, eigen vector u corresponding to [tex]$\lambda=3 i$[/tex] is given by

[tex]$$\begin{gathered}{[A-3 i I] u=O} \\{\left[\begin{array}{cc}-15-3 i & -39 \\6 & 15-3 i\end{array}\right]\left[\begin{array}{l}u_1 \\u_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\\\\end{gathered}$$[/tex]

Applying [tex]R_2 \rightarrow R_2-6 R_1[/tex]

[tex]& {\left[\begin{array}{cc}1 & \frac{-39}{-15-3 i} \\0 & 0\end{array}\right]\left[\begin{array}{l}u_1 \\u_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

[tex]& \Rightarrow u_1+\frac{39}{15+3 i} u_2=0 \\[/tex]

[tex]& \Rightarrow u_1=-\frac{13}{5+i} u_2=-\frac{13}{5+i} * \frac{5-i}{5-i} u_2=-\frac{13(5-i)}{26} u_2=-\frac{5-i}{2} u_2 \\[/tex]

Thus, by choosing [tex]$u_2[/tex]=1 eigenvector corresponding to [tex]$\lambda=3 i$[/tex] is [tex]$$u=\left[\begin{array}{c}-\frac{5-i}{2} \\1\end{array}\right]$$[/tex]

[tex]R_1 \rightarrow \frac{1}{-15+3 i} R_1 \\[/tex]

[tex]{\left[\begin{array}{cc}1 & \frac{-39}{-15+3 i} \\6 & 15+3 i\end{array}\right]\left[\begin{array}{l}v_1 \\v_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

Applying [tex]R_2 \rightarrow R_2-6 R_1 \\[/tex]

[tex]{\left[\begin{array}{cc}1 & \frac{-39}{-15+3 i} \\0 & 0\end{array}\right]\left[\begin{array}{l}v_1 \\v_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

[tex]\Rightarrow v_1+\frac{39}{15-3 i} v_2=0 \\\Rightarrow v_1=-\frac{39}{15-3 i}[/tex]

[tex]\begin{gathered}\Rightarrow v_1+\frac{39}{15-3 i} v_2=0 \\\v_2=-\frac{13(5+i)}{26} \\\Rightarrow v_1==-\frac{5+i}{2}\end{gathered}$$[/tex]

Thus, by choosing [tex]$v_2=1$[/tex] eigenvector corresponding to [tex]$\lambda=-3 i$[/tex] is

[tex]$$v=\left[\begin{array}{c}-\frac{5+i}{2} \\1\end{array}\right]$$[/tex]

Hence, the general solution is given by

[tex]X(t)=c_1 e^{3 i t} u+c_2 e^{-3 i t} v \text { [using } e^{i t}=\cos t+i \sin t \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5-i}{2}(\cos 3 t+i \sin 3 t) \\(\cos 3 t+i \sin 3 t)\end{array}\right]+c_2\left[\begin{array}{c}-\frac{5+i}{2}(\cos 3 t-i \sin 3 t) \\(\cos 3 t-i \sin 3 t)\end{array}\right] \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5}{2}(\cos 3 t+i \sin 3 t)+\frac{i}{2}(\cos 3 t+i \sin 3 t) \\(\cos 3 t+i \sin 3 t)\end{array}\right][/tex][tex]+c_2\left[\begin{array}{c}-\frac{5}{2}(\cos 3 t-i \sin 3 t)-\frac{i}{2}(\cos 3 t-i \sin 3 t) \\(\cos 3 t-i \sin 3 t)\end{array}\right] \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5}{2} \cos 3 t-\frac{5 i}{2} \sin 3 t+\frac{i}{2} \cos 3 t-\frac{1}{2} \sin 3 t \\\cos 3 t+i \sin 3 t\end{array}\right][/tex][tex]+c_2\left[\begin{array}{c}-\frac{5}{2} \cos 3 t+\frac{5}{2} i \sin 3 t-\frac{i}{2} \cos 3 t-\frac{1}{2} \sin 3 t \\\cos 3 t-i \sin 3 t\end{array}\right ][/tex]

[tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex]

Therefore, the general solution of the system X(t) is [tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex].

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The vertices of the polygon before and after dilation are shown below

preimage         Image

J (-5, 3)            J' (-5/2, 3/2)

K (2, 3)            K' (1, 3/2)

L (2, -3)           L' (1, -3/2)

M (-5, -3)        M' (-5/2, -3/2)

Dilation is a method of transformation that magnify or shrink the preimage depending on the scale factor

The rule used in  transformation is as follows

(x, y) for a scale factor of r → (rx, ry)

Applying the rule to the vertices  of the polygon we have

preimage                                  Image

J (-5, 3)      (1/2 * -5, 1/2 * 3)      J' (-5/2, 3/2)

K (2, 3)       (1/2 * 2, 1/2 * 3)       K' (1, 3/2)

L (2, -3)      (1/2 * 2, 1/2 * -3)       L' (1, -3/2)

M (-5, -3)    (1/2 * -5, 1/2 * -3)     M' (-5/2, -3/2)

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Answer:

16

Step-by-step explanation:

The side lengths of the original square was 2.  2 x 2 = 4

.25 x 2 = .5

The squares are now 1/2 x 1/2.  One of the 4 original squares will now need 4 new smaller size squares to cover the one square.

4 x 4 = 16

Given the polynomial Q(x)=4x^3-3x+1,

P(x) = 8x^3 - 6x + 2

Generally, To find P(x), we can start by substituting

P(x) = 2*(Q(x) + P(x)) into the equation:

P(x) = 2*(Q(x) + P(x))

This simplifies to:

P(x) = 2*(4x^3 - 3x + 1 + P(x))

Expanding and rearranging the equation gives:

P(x) = 8x^3 - 6x + 2 + 2P(x)

Subtracting 2P(x) from both sides of the equation gives:

P(x) - 2P(x) = 8x^3 - 6x + 2

This simplifies to:

-P(x) = 8x^3 - 6x + 2

Adding P(x) to both sides of the equation gives:

0 = 8x^3 - 6x + 2 + P(x)

Therefore, the polynomial P(x) that satisfies the equation

P(x) = 2*(Q(x) + P(x)) is:

P(x) = 8x^3 - 6x + 2

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Answer:

a = 0

(0,3)

Step-by-step explanation:

The possible length of the third side of the triangle is x>52 inches.

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. If the sum of the two sides is equal to the third side, then the two sides will coincide with the third side so a triangle cannot be formed. Hence, the sum of the two sides must be greater than the third side for the triangle to be formed.

Given: Sides of two sides of a triangle as 15 and 37 inches.

let the third side of the triangle be x.

then x>15+37

       x>52 inches.

Hence, The possible length of the third side of the triangle is x>52 inches.

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The algebraic representation of this dilation is: (x', y') = (-0.9x, -1.5y)

A dilation is a transformation that changes the size of a geometric figure, but not its shape. The center of dilation is a fixed point, and all other points in the figure are expanded or contracted in relation to it. In this case, the center of dilation is the origin, and the original figure is m'n'o'p' and the image figure is mnop.

The algebraic representation of a dilation is given by the equation:

(x', y') = (kx, ky)

Where (x', y') are the coordinates of the image point, (x, y) are the coordinates of the original point and k is the scale factor of the dilation.

We can find the scale factor k by comparing the coordinates of any two corresponding vertices of the figures. For example, if we take the vertex m' and m, we have:

m'(-5, -4) and m(4.5, 6)

Here we can see that the x-coordinate of m' is -5 and the x-coordinate of m is 4.5 and the y-coordinate of m' is -4 and the y-coordinate of m is 6.

So, k = mx/mx' = 4.5/-5 = -0.9 and k = my/my' = 6/-4 = -1.5

We can use this scale factor to find the coordinates of any other point in the image figure.

Therefore, the algebraic representation of this dilation is: (x', y') = (-0.9x, -1.5y)

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There are 22 different types of functions based on range, element, equation, and domain.

Based on the range it includes Modulus Function, rational function, signum function, even and odd functions, periodic functions, greatest integer function, inverse function, and composite functions. On the basis of elements, it includes One One function, many one functions, onto function, one one and onto function, into function, and constant function. On the basis of the equation, it includes the identity function, linear function, quadratic function, cubic function, and polynomial functions. On the basis of the domain, it has algebraic functions, trigonometric functions, and logarithmic functions.

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The number of pages that the student will read in 7 hours, given the number of pages read in 4 hours, is

The constant variation is 14 pages.

This constant variation is direct.

The quantity that connects two variables that are directly or inversely proportional to one another is known as the constant of variation.

Variation in mathematics demonstrates how one variable fluctuates in respect to another. Typically, a ratio is used to illustrate this relationship. When we remark that a variation is continuous, we are referring to how consistently the ratio changes.

The constant of variation in this instance therefore, is:

= Number of pages read / Number of hours

= 56 / 4

= 14 pages

This constant of variation is direct.

The number of pages read in 7 hours is:

= Constant of variation x number of hours

= 14 x 7

= 98 pages

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To reduce your department by 15%, you would need to let go of 33 workers.

Reducing a department by 15% means that 15% of the current workforce in that department will be let go. To calculate this, you first need to determine the percentage of the total workforce that your department represents. In this case, your department has 220 workers out of 512 workers in the whole company, which represents 43% of the total workforce.

Then, you can calculate the number of workers that need to be let go by multiplying the total number of workers in your department by the percentage that needs to be cut. In this case, that would be

220 x 0.15 = 33 workers

This means that 33 out of the 220 workers in your department would need to be let go in order to achieve a 15% reduction.

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Answer:

8.1

Step-by-step explanation:

12/x = tan56

12/x =1.48

so x = 12/1.48 =8.1

The rate in centimeters per second, at which the radius is increasing at the instant when the radius is 5 centimeters is 0.16 cm/s.

Using a known value of a function at a certain point along with its rate of change at that moment, one application of derivatives is to estimate an unknown value of a function at a particular point.

Given:

Both radius r and area A is a function of time, so in fact:

A(t) = π [r(t)]²

Deriving this we get that the area changes and the radius changes as:

[tex]\frac{dA(t)}{dt} = 2\pi r(t)\frac{dr(t)}{dt}[/tex]

But,

[tex]\frac{dA(t)}{dt} = 5 \frac{cm^{2} }{s}[/tex]

Considering when r(t) = 5cm at the instant we get,

[tex]5 = 2\pi . 5\frac{dr(t)}{dt}[/tex]

Simplify 5 and take 2π to the other side and divide,

The rate of change of radius is given by:

[tex]\frac{dr(t)}{dt} =\frac{1}{2\pi } = 0.16 cm/s[/tex]

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The solutions to the equation y/ (y- 4) - 4/ (y + 4) = 32/ y are approximately 30.948, 4.627 and -3.575.

The given equation is y/ (y- 4) - 4/ (y + 4) = 32/ y

Let us simplify the left hand side (LHS) :

Make the denominator same to (y - 4)(y + 4)

y/ (y- 4) - 4/ (y + 4) = (y/ (y- 4))×((y + 4)/(y + 4)) - (4/ (y + 4))×((y - 4)/(y - 4))

y/ (y- 4) - 4/ (y + 4) = ( y(y + 4) - 4(y - 4))/ (y - 4)(y + 4)

y/ (y- 4) - 4/ (y + 4) = (y² + 4y - 4y + 16)/ (y² - 16)

y/ (y- 4) - 4/ (y + 4) = (y² + 16)/ (y² - 16)

Now plugging this simplified version in LHS

(y² + 16)/ (y² - 16) = 32/ y

y(y² + 16) = 32(y² - 16)

y³ - 32y² + 16y + 32×16 = 0

The solutions are approximately 30.948, 4.627 and -3.575.

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Answer:

b.

y = 3x + 11, y = 3x + 15

Step-by-step explanation:

Both lines have the same slope, so they are parallel.

85.41  cubic inches of the treasure chest is taken up by the coins

a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

Remember that the volume of a cylinder is given by:

volume = (height)×pi×(radius)²

Where pi = 3.14

Then here the volume of each coin is:

V = (0.0625)(3.14)(1.6)²

= 0.5024 in³

And there are 170 coins, so the total volume is:

170×0.5024

85.408 in³

85.41 in³

Hence, 85.41  cubic inches of the treasure chest is taken up by the coins

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Answer:

c or 85.41

Step-by-step explanation: its simple