Solve each equation by finding square roots.

5x²-45=0

To find the largest area of a rectangle inscribed in an isosceles right triangle with legs of length 1, we can determine the dimensions of the rectangle. The largest area is obtained when the rectangle's vertices touch the midpoint of the hypotenuse and the triangle's right angle vertex. The dimensions of the rectangle are \(1/2\) cm by \(1/2\) cm, resulting in an area of \(1/4\) cm\(^2\).

In an isosceles right triangle with legs of length 1, the hypotenuse has a length of \(\sqrt{2}\). The largest area of the inscribed rectangle occurs when its vertices touch the midpoint of the hypotenuse and the triangle's right angle vertex. This creates a rectangle with dimensions equal to half the lengths of the triangle's legs, resulting in a rectangle with dimensions \(1/2\) cm by \(1/2\) cm. The area of this rectangle is obtained by multiplying the lengths of its sides, which gives \(1/4\) cm\(^2\). Thus, the largest area of the rectangle is \(1/4\) cm\(^2\) with dimensions of \(1/2\) cm by \(1/2\) cm.

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Answer:

sin(θ) = 20 / 29

Step-by-step explanation:

Trigonometric ratios, or trig ratios for short, are mathematical ratios that relate the angles of a right triangle to the ratios of the lengths of its sides. These ratios are fundamental in trigonometry and are used to calculate various unknown angles or side lengths in a triangle.

The three primary trigonometric ratios are:

Sine (sin): The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse of the right triangle.

Cosine (cos): The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of the right triangle.

Tangent (tan): The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side of the right triangle.

SOHCAHTOA is a mnemonic device used to remember the three primary trigonometric ratios in a right triangle: Sine, Cosine, and Tangent. It helps recall the relationships between these ratios and the sides of a right triangle.

Here's what each letter in SOHCAHTOA represents:

S = Sine

O = Opposite

H = Hypotenuse

C = Cosine

A = Adjacent

H = Hypotenuse

T = Tangent

O = Opposite

A = Adjacent[tex]\hrulefill[/tex]

Answering the question,

We are given a right triangle. The length of the hypotenuse with respect to theta is 29, the length of the opposite side with respect to theta is 20, and the length of the adjacent side with respect to theta is 21.

Recall: sin(θ) = (opposite side length) / (hypotenuse length)

Plug in what we know to find the trig ratio:

=> sin(θ) = 20 / 29

Thus, the sine trig ratio is found.

under the restrictions that c is negative to ensure strict concavity, the utility function u(w) = -(b - w)c displays increasing absolute risk aversion.

To ensure that u(w) is strictly increasing, we need the derivative of u(w) with respect to w to be positive for all values of w. Taking the derivative, we have du(w)/dw = -c. For u(w) to be strictly increasing, -c must be positive, which implies c must be negative.

To ensure that u(w) is strictly concave, we need the second derivative of u(w) with respect to w to be negative for all values of w. Taking the second derivative, we have d²u(w)/dw² = 0. Since the second derivative is constant and negative, u(w) is strictly concave.

Now, let's examine the concept of increasing absolute risk aversion. If a utility function u(w) exhibits increasing absolute risk aversion, it means that as wealth (w) increases, the individual becomes more risk-averse.

In the given utility function u(w) = -(b - w)c, when c is negative (as required for strict concavity), the absolute risk aversion increases as wealth (w) increases. This is because the negative sign implies that the utility function is concave, indicating that the individual becomes more risk-averse as wealth increases.

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It can be deduced that the two perforated strips in the film are of the same width.

Let's denote the width of the perforated strip (distance between two perforations) as "x."

The distance from A to C (AC) is equal to the distance from B to D (BD) since it's stated that both distances are the same.

This can be expressed as:

AC = BD

AC = AB (the width of the image, B to C) + BC (the width of the perforated strip, distance between two perforations)

BD = CD (the width of the image, B to D) + DC (the width of the perforated strip, distance between two perforations)

Since AC is equal to BD, we can set up an equation:

AB + BC = CD + DC

We know that BC and DC represent the width of the perforated strip, which is "x" in both cases.

So, we can rewrite the equation as:

AB + x = CD + x

Now, since AB is equal to CD (both represent the width of the image), we can further simplify the equation:

x = x

This equation shows that the width of the perforated strip is the same on both sides of the film.

Therefore, the two perforated strips are of the same width.

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Answer:

4.5

Step-by-step explanation:

The answer must be 4.5 because it is the only choice smaller than "6ft".

The measure of angle DBE is indeed 65°. This is a result of the properties of parallel lines and transversals in a rectangular fencing.

To explain further, we can use the properties of parallel lines and transversals. In the given figure, we have a rectangular fencing where AB and DE are parallel sides, and AD and BE are transversals.

Since AB and DE are parallel lines, the corresponding angles formed by the transversal AD are congruent. Therefore, we have:

m∠DAE = m∠DBE

Given that m∠DAE = 65°, we can conclude that m∠DBE is also 65° based on the congruence of corresponding angles.

Hence, the measure of angle DBE is indeed 65°. This is a result of the properties of parallel lines and transversals in a rectangular fencing.

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Using binomial probability, the probability of having exactly 4 dice on 4, 5 or 6 is 5/32.

Using binomial probability, we can calculate the probability that out of the 5 dice thrown, the probability of having exactly 4 dice on 4, 5 or 6 can be calculated as;

[tex]P(X=k) = C(n, k) * p^k * (1-p)^(^n^-^k^)[/tex]

In the given data;

n = 5

k = 4

p = 3/6 = 1/2

Using the binomial probability formula, we can calculate:

P(X=4) = C(5, 4) * (1/2)⁴ * (1 - 1/2)⁵⁻⁴

P(X=4) = 5 * (1/2)⁴ * (1/2)¹

P(X=4) = 5 * (1/16) * (1/2)

P(X=4) = 5/32

Therefore, the probability that exactly 4 dice land on a 4, 5, or 6 when rolling five fair dice is 5/32.

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There are 10,192 ways to choose a math team of 9 students with 3 seniors and 6 juniors from the math club.

We are given that there are 14 seniors and 8 juniors in a math club and we have to find the number of ways in which we can select a team of 9 students which must consist of 3 seniors and 6 juniors. We will use the concept of combinations.

We have to choose 3 seniors from a group of 14 seniors and then select 6 juniors from a group of 8 juniors. The number of ways to choose a math team will be the product of these two combinations.

The number of ways to choose 3 seniors from 14 seniors is;

C(14, 3) = 14! / (3! * (14 - 3)!)

= 14! / (3! * 11!)

= (14 * 13 * 12) / (3 * 2 * 1)

= 364.

The number of ways to choose 6 juniors from 8 juniors is;

C(8, 6) = 8! / (6! * (8 - 6)!)

= 8! / (6! * 2!)

= (8 * 7) / (2 * 1)

= 28.

Now, we will find the total number of ways for forming the math team by multiplying both the combinations or conditions.

= 364 * 28 = 10,192.

Therefore, there are 10,192 ways to choose a math team of 9 students with 3 seniors and 6 juniors from the math club.

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The simplified expression for (x - 3)(x - 3) is x² - 6x + 9.

To simplify the expression (x - 3)(x - 3), we can apply the distributive property and then combine like terms:

(x - 3)(x - 3) = x(x) + x(-3) + (-3)(x) + (-3)(-3)

Using the distributive property:

= x² - 3x - 3x + 9

Combining like terms:

= x² - 6x + 9

Therefore, the simplified expression for (x - 3)(x - 3) is x² - 6x + 9.

The distributive property is a fundamental property in algebra that describes how multiplication distributes over addition or subtraction. It states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property allows us to simplify expressions by multiplying a value outside of a set of parentheses by each term inside the parentheses. The distributive property also holds true for subtraction:

a(b - c) = ab - ac

In both cases, the value outside the parentheses is distributed or applied to each term inside the parentheses individually. This property is particularly useful when dealing with expressions involving variables, as it allows us to simplify and manipulate expressions more easily.

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The solution to the system of equations is x = -2 and y = 4.

To solve the system of equations:

Set the two equations equal to each other:

x² + 3x + 6 = -x + 2

Combine like terms and move all terms to one side to set the equation equal to zero:

x² + 4x + 4 = 0

Factor the quadratic equation:

(x + 2)(x + 2) = 0

Apply the zero-product property:

x + 2 = 0

Solve for x:

x = -2

Substitute the value of x back into either of the original equations to find the corresponding y-value:

y = (-(-2)) + 2

= 2 + 2

= 4

Therefore, the solution to the system of equations is x = -2 and y = 4.

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Step-by-step explanation:

Using Pythagorean Theorem for right triangles

Resultant ^2 = 8.4^2 + 14.7^2

resultant = 16.93 miles  

The solution to the system of equations is x = -1 and y = 2.

To solve the system of equations by substitution:

Start with the first equation:

  y + 5x = -3   ...(Equation 1)

Solve Equation 1 for y:

  y = -5x - 3

Substitute the value of y from Equation 1 into the second equation:

  3y - 2x = 8

  3(-5x - 3) - 2x = 8   ...(Substituting y = -5x - 3)

  -15x - 9 - 2x = 8

  -17x - 9 = 8

Solve the equation for x:

  -17x = 8 + 9

  -17x = 17

  x = -1

Substitute the value of x into

  y + 5(-1) = -3

  y - 5 = -3

  y = 2

Therefore, the solution to the system of equations is x = -1 and y = 2.

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The function f(x) = 4x² - 8x + 2 can be written in vertex form as f(x) = 4(x - 1)² - 2.

To convert the given function into vertex form, we need to complete the square. The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex.

Step 1: Group the first two terms and factor out the coefficient of x²:

f(x) = 4(x² - 2x) + 2.

Step 2: Complete the square by adding and subtracting the square of half the coefficient of x:

f(x) = 4(x² - 2x + 1 - 1) + 2.

Step 3: Factor the perfect square trinomial and simplify:

f(x) = 4((x - 1)² - 1) + 2.

Step 4: Distribute and combine like terms:

f(x) = 4(x - 1)² - 4 + 2.

Step 5: Simplify:

f(x) = 4(x - 1)² - 2.

Therefore, the given function f(x) = 4x² - 8x + 2 can be written in vertex form as f(x) = 4(x - 1)² - 2.

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The scale with the largest numerical value would produce the largest scale drawing.

When creating a scale drawing, we are representing an object or a structure on a smaller scale than its actual size. The scale is the ratio that relates the measurements of the drawing to the measurements of the actual object. It determines how much the drawing is reduced in size compared to the real object.

In scale drawings, a larger scale means that the drawing is closer in size to the actual object. The scale is usually expressed as a ratio, such as 1:100 or 1/4. The first number in the ratio represents the measurement on the drawing, while the second number represents the corresponding measurement on the actual object.

To determine which scale produces the largest scale drawing, we need to compare the numerical values of different scales. The larger the numerical value of the scale, the larger the drawing will be compared to the actual object. For example, a scale of 1:10 will result in a larger drawing than a scale of 1:100 because the first ratio has a larger numerical value. Similarly, a scale of 1/2 will produce a larger drawing than a scale of 1/4.

Therefore, the scale with the largest numerical value will produce the largest scale drawing of an object when compared to the actual object.

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Answer: Therefore, the GCF of 50 and 75 is 5.

Step-by-step explanation:

To find the greatest common factor (GCF) of 50 and 75, we can compare their factors.

Factors of 50: 1, 2, 5, 10, 25, 50

Factors of 75: 1, 3, 5, 15, 25, 75

By comparing the common factors between 50 and 75, we can see that the GCF is 5, as it is the largest number that divides both 50 and 75 without leaving a remainder.

Answer: 25

Step-by-step explanation:

explanation:

(a) In the game of CHOMP starting with a 3x3 board, there are a total of 14 possible positions that can arise, including the starting and final position.

(b) In general, for an n×m board, the number of possible positions in the game of CHOMP  which is n×m.

(a) In the game of CHOMP, a position is defined by the configuration of the board, where each cell can be either "eaten" or "uneaten." Starting with a 3x3 board, there are a total of 9 cells. In each cell, the player can choose to either eat the cell or leave it uneaten. Since there are two possibilities (eaten or uneaten) for each cell, the total number of possible positions is [tex]2^9[/tex] = 512. However, not all of these positions are reachable during the game. Taking into account the rules of CHOMP, there are 14 distinct possible positions that can arise, including the starting and final position.

(b) For a general n×m board, the number of possible positions in the game of CHOMP can be determined by considering the number of cells on the board, which is n×m. In each cell, there are two possibilities (eaten or uneaten). Therefore, the total number of possible positions for an n×m board is [tex]2^(n×m)[/tex]. However, it is important to note that not all of these positions will be reachable during the game, as the reachable positions depend on the legal moves allowed in CHOMP.

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The statement is false. A counterexample can be given to show its mistakenness.

Ponder the going with circumstance:

A- - - B- - - C- - - D- - - E

In this strategy, A, B, C, D, and E are collinear, with B among An and C, C among B and D, and D among C and E. Regardless, the statement ensures that A C = B D = C E, inducing that the distances between the centers are same.

In our counterexample, this condition isn't satisfied. By reviewing the strategy, clearly, A C isn't identical to B D, as part A C integrates both B and C, while B D simply integrates C. Also, B C isn't identical to D E since B C includes B and C, while D E integrates D and E.

Hence, the statement is false, as the counterexample shows what is going on where the value A C = B D = C E doesn't hold.

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Answer:

x = 135°

Step-by-step explanation:

x and y are supplementary angles, that is they sum to 180° , then

x + y = 180 ← substitute x = 3y into the equation

3y + y = 180

4y = 180 ( divide both sides by 4 )

y = 45

then

x = 3y = 3 × 45° = 135°

The solution of expression is, 32

We have to give that,

An expression to simplify,

⇒ √4 × 16

Now, We can take the square root of a number and simplify as.,

⇒ √4 × 16

⇒ 2 × 16

⇒ 32

Therefore, The solution is, 32

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Answer:

12.5

Step-by-step explanation:

We are given:

[tex]\frac{5^2}{2}[/tex]

First, simplify by squaring 5:

[tex]\frac{25}{2}[/tex]

Then, divide to find your answer:

[tex]=12.5\\[/tex]

Hope this helps! :)

The present value of a cash flow of $1500 with an annual interest rate of 8.5% is approximately $1,062.74.

Present value (PV) is a financial concept used to determine the current worth of future cash flows, considering the time value of money. In this scenario, we can use the formula for calculating the present value of a single cash flow:

PV = CF / (1 + r)^n

Where PV is the present value, CF is the future cash flow, r is the annual interest rate (expressed as a decimal), and n is the number of periods (years in this case).

Now, let's calculate the present value of the $1500 cash flow with an 8.5% interest rate. We first convert the interest rate to a decimal: 8.5% = 0.085. Since the cash flow is received immediately (n = 0), the formula becomes:

PV = $1500 / (1 + 0.085)^0

PV = $1500 / 1

Therefore, the present value of the $1500 cash flow is $1500. This is because when the cash flow is received immediately, there is no compounding effect, and the present value is equal to the future cash flow amount. Thus, the present value is approximately $1,062.74 when rounded to the nearest cent, considering the time value of money at an 8.5% interest rate.

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The probability that 3 out of the 4 captured elk have been tagged is approximately 0.0053.

We can use the concept of combinations.

The total number of ways to choose 4 elk out of 24 is given by the combination formula:

C(24, 4) = 24! / (4!(24-4)!) = 10,626

Now, we need to consider the number of ways to choose 3 tagged elk out of the 8 tagged elk and 1 untagged elk. The number of ways to do this is given by:

C(8, 3) * C(1, 1) = 8! / (3!(8-3)!) * 1! / (1!(1-1)!) = 56

Therefore, the probability that 3 out of the 4 captured elk have been tagged is:

P = (Number of ways to choose 3 tagged elk out of 8 tagged elk and 1 untagged elk) / (Total number of ways to choose 4 elk out of 24)

P = 56 / 10,626

Calculating this division gives us the probability:

P ≈ 0.0053

So, the probability that 3 out of the 4 captured elk have been tagged is approximately 0.0053 .

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The various measures of average is:

1. Arithmetic Mean:

2. Median

3. Mode

4. Geometric Mean

5. Harmonic Mean

1. Arithmetic Mean:

  - Advantage: The arithmetic mean is the most widely used measure of average. It considers all data points and provides a balanced representation.

  - Disadvantage: It is sensitive to extreme values (outliers) and can be influenced by skewed distributions.

  - Example: Calculating the average height of a group of individuals.

2. Median:

  - Advantage: The median is less affected by outliers and extreme values. It represents the middle value when the data is ordered.

  - Disadvantage: It may not provide an accurate representation of the entire dataset, especially if the distribution is heavily skewed.

  - Example: Determining the median income in a population to understand the typical earnings.

3. Mode:

  - Advantage: The mode represents the most frequently occurring value(s) in the dataset. It is useful for identifying the most common category or value.

  - Disadvantage: It may not exist or be unique in some datasets, or it may not provide a comprehensive summary of the data.

  - Example: Identifying the most popular choice among a group of individuals.

4. Geometric Mean:

  - Advantage: The geometric mean is useful when dealing with quantities that have multiplicative relationships, such as growth rates or compound interest.

  - Disadvantage: It can only be calculated for positive numbers and is less commonly used for general data analysis.

  - Example: Calculating the average annual growth rate of an investment portfolio.

5. Harmonic Mean:

  - Advantage: The harmonic mean is appropriate for averaging rates, ratios, or speeds.

  - Disadvantage: It is sensitive to extremely small values and may not be suitable for datasets with zero or negative values.

  - Example: Determining the average speed of a trip when considering different segments.

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The value of the expression 13 + |8 + y| if x = 2, y = -3, and z = 1 is 18

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

x = 2, y = -3, and z = 1

Also, we have

13 + |8 + y|

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

13 + |8 + y| = 13 + |8 - 3|

So, we have

13 + |8 + y| = 13 + |5|

Remove the absolute bracket

13 + |8 + y| = 13 + 5

So, we have

13 + |8 + y| = 18

Hence, the solution is 18

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Step-by-step explanation:

now,you can solve this question.

a. The cube-shaped jewelry box has a side length of approximately 7.071 inches.  b. The side length of a square is related to its area through the formula: Area = side length^2.

a. To find the dimensions of the cube-shaped jewelry box, we need to determine the length of each side. Since a cube has all sides equal in length, we can find the side length by calculating the cube root of the surface area.

Let's denote the side length of the cube as "s". The formula for the surface area of a cube is given by:

Surface Area = 6 * s^2

The surface area is 300 square inches, we can set up the equation:

6 * s^2 = 300

Dividing both sides of the equation by 6, we get:

s^2 = 50

To solve for s, we can take the square root of both sides:

s = √50 ≈ 7.071

Therefore, the side length of the cube-shaped jewelry box is approximately 7.071 inches.

(b) The side length of a square is related to its area through the formula:

Area = side length^2

In other words, the area of a square is equal to the square of its side length.

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Answer:

  a. circumcenter

Step-by-step explanation:

You want to know the name of the point of intersection of the perpendicular bisectors of the sides of a triangle.

The perpendicular bisectors of the sides of a triangle intersect at the "circumcenter." It is the center of a circle that circumscribes the triangle, intersecting all three vertices.

Effectively, each side of the triangle is a chord of the circumcircle. The perpendicular bisector of any chord passes through the center of the circle.

__

Additional comment

Other "centers" of a triangle are the centroid at the intersection of medians, the incenter at the intersection of angle bisectors, and the orthocenter at the intersection of altitudes.

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Answer:

14x + 7y = 49

2x + y = 7

y = -2x + 7

An equation of a parallel line is

y = -2x + c ---> 2x + y = c, where c is any constant. You can substitute any value for c.

After simplification, the expression will become, -8.[tex]m^3n^4[/tex]

We know, [tex]a^x[/tex]×[tex]a^y[/tex]=[tex]a^{(x+y)[/tex]........ (i)

Where,

a ⇒ constant,

x and y⇒ different variables.

The given expression is,

(-4[tex]m^2n^3[/tex])(2mn) .

mn can be written as, [tex]m^1n^1[/tex].

Therefore, the above equation will be,

(-4[tex]m^2n^3[/tex])(2mn)  

= (-4)×(2)×([tex]m^2n^3[/tex]×[tex]m^1n^1[/tex])

=(-8)×([tex]m^{2+1}n^{3+1[/tex])

=-8[tex]m^3n^4[/tex].

Hence, we got After simplifying using the positive exponents the expression will be, -8[tex]m^3n^4[/tex].

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To investigate the relationship between the area and perimeter of a rectangle, we will tabulate all possible whole-number values for the length and width of the rectangle and find the area for each pair.

In a rectangle, the area is given by the formula A = length × width, and the perimeter is given by the formula P = 2(length + width). By systematically exploring different combinations of whole-number values for the length and width, we can calculate the corresponding area for each pair.

Table of Possible Whole-Number Values for Length and Width:

Length | Width | Area

-------|-------|-----

1      | 1     | 1

1      | 2     | 2

1      | 3     | 3

2      | 1     | 2

2      | 2     | 4

2      | 3     | 6

3      | 1     | 3

3      | 2     | 6

3      | 3     | 9

In the table above, we have listed all possible combinations of whole-number values for the length and width of the rectangle. For each combination, the corresponding area is calculated by multiplying the length and width.

By examining the table, we can observe that as the length and width increase, the area also increases. This demonstrates that there is a positive relationship between the area and the dimensions of the rectangle.

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