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1. The surface area of the pool is given as follows: 2,202,000 square feet.

2. The amount of water that the pool holds is of: 10,000,000 cubic feet.

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The dimensions for this problem are given as follows:

l = 10000, w = 100, h = 10.

Hence the surface area is given as follows:

S = 2 x (10000 x 100 + 10000 x 10 + 100 x 10)

S = 2,202,000 square feet.

The volume of a rectangular prism is given by the multiplication of it's dimensions, hence:

V = 10000 x 100 x 10

V = 10,000,000 cubic feet.

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Answer:

Step-by-step explanation:

In a parallelogram, the two opposite sides are parallel and equal

WR = 2 (6x - 7.7)

221 = 2( 6x - 7.7)

Answer:

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As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.

A variable in mathematics is a symbοl οr letter that designates a number that is subject tο variatiοn οr change. Mathematical expressiοns and fοrmulae that can be sοlved tο determine the value οf a variable are written using variables. A, B, C, and οther symbοls are frequently used tο denοte variables, including x, y, and z.

Numerοus different types οf quantities, including integers, functiοns, vectοrs, matrices, and οthers, can be represented by them. X and Y are factοrs in the equatiοn y = 2x + 1, fοr instance. We can determine the cοrrespοnding number οf y by substituting a value fοr x.

given

By dividing 3 by 12 and taking away the cοrrespοnding expοnents οf r and p, we can first simplify the numeratοr οf the expressiοn.

[tex](3m^{(-3)}r^{(4)}p^{(2)})/(12r^{(4)}) = (1/4)m^{(-3)}r^{(4-4)}p^{(2)}[/tex]

Even mοre simply put, we have:

[tex](1/4)m^{(-3){p^{(2)}[/tex]

As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.

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4x + 10 = 30To solve for x, we can start by subtracting 10 from both sides:4x + 10 - 10 = 30 - 10

4x = 20Then, we can divide both sides by 4 to isolate x:4x/4 = 20/4, x = 5

Therefore, the solution to this equation is x = 5.

4x - 8 = 20, To solve for x, we can start by adding 8 to both sides: 4x - 8 + 8 = 20 + 8, 4x = 28

Then, we can divide both sides by 4 to isolate x:

4x/4 = 28/4, x = 7 Therefore, the solution to this equation is x = 7.

5 + 2x = 65, To solve for x, we can start by subtracting 5 from both sides: 5 + 2x - 5 = 65 - 5, 2x = 60

Then, we can divide both sides by 2 to isolate x:

2x/2 = 60/2, x = 30 Therefore, the solution to this equation is x = 30. 9 + 4x = -5, To solve for x, we can start by subtracting 9 from both sides: 9 + 4x - 9 = -5 - 9, 4x = -14

Then, we can divide both sides by 4 to isolate x:

4x/4 = -14/4, x = -3.5, Therefore, the solution to this equation is x = -3.5. 14 + 6x = 2,To solve for x, we can start by subtracting 14 from both sides:14 + 6x - 14 = 2 - 14, 6x = -12Then, we can divide both sides by 6 to isolate x: 6x/6 = -12/6, x = -2

Therefore, the solution to this equation is x = -2.

2x - 3 = -2

To solve for x, we can start by adding 3 to both sides:

2x - 3 + 3 = -2 + 3

2x = 1

Then, we can divide both sides by 2 to isolate x:

2x/2 = 1/2

x = 1/2 or 0.5

Therefore, the solution to this equation is x = 0.5.

5 + 10x = -5

To solve for x, we can start by subtracting 5 from both sides:

5 + 10x - 5 = -5 - 5

10x = -10

Then, we can divide both sides by 10 to isolate x:

10x/10 = -10/10

x = -1

Therefore, the solution to this equation is x = -1. 10 = 7=x ,This equation is not solvable. It appears to be a typographical error, as it does not make sense to say that 10 is equal to both 7 and x at the same time.

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The value of the probability that Y is less than or equal to 9 is approximately 0.893

Given that the moment generating function:

M(t) = (pe⁺ + q)ⁿ

And also

q = 1 - p

When M(t) = (0.8e⁺ + 0.2)¹⁰ and M(t) = (pe⁺ + q)ⁿ are compared, we have

n = 10

p = 0.8

q = 0.2

To find P(Y ≤ 9), we can use the cumulative distribution function (CDF) for the binomial distribution:

[tex]F(k) = P(Y \le k) = \sum\limits^k_{i=0}\left[\begin{array}{c}n&i\end{array}\right] p^i q^{n-i}[/tex]

In this case, we want to find P(Y ≤ 9), so we can evaluate the CDF at k=9:

So, we have

[tex]P(Y \le 9) = \sum\limits^9_{i=0}\left[\begin{array}{c}10&i\end{array}\right] 0.8^i * 0.2^{n-i}[/tex]

Using a calculator to evaluate this sigma notation, we find that

P(Y ≤ 9) ≈ 0.89263

Approximate

P(Y ≤ 9) ≈ 0.893

Therefore, the probability that Y is less than or equal to 9 is approximately 0.893

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Complete question

If Y has a binomial distribution with n trials and probability of success p, the moment-generating function for Y is M(t) = (pe⁺ + q)ⁿ, where q = 1 − p.

If Y has moment-generating function M(t) = (0.8e⁺ + 0.2)¹⁰, what is P(Y ≤ 9)?

The accurate definition of a type II error is failing to reject a true null hypothesis.

Type II error is known as a statistical term that happens when a null hypothesis is not rejected when it should have been rejected. Type II error can occur in a study when the researcher has failed to detect a real difference between the research subject group and the comparison group. It's often called the "false negative" because it incorrectly concludes that there is no difference when there actually is a difference.

Type I Error - It is known as a type I error when a researcher rejects a null hypothesis when it is true. Type I errors are often called "false positives."

Type II Error - Type II error is known as a statistical term that happens when a null hypothesis is not rejected when it should have been rejected. Type II error can occur in a study when the researcher has failed to detect a real difference between the research subject group and the comparison group. It's often called the "false negative" because it incorrectly concludes that there is no difference when there actually is a difference.

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Using structural induction, the length of all elements x of B is an integer power of 2, for all n∈N, sumeven(n) = n(n+1), for all elements u of D, u starts and ends with the same character.

a) Basis Step: For the string 0, the length is 1 which is equal to 2^0, and for the string 1, the length is also 1 which is equal to 2^0. Hence, the property holds for the basis step.

Recursive Step: Assume that the length of the binary string x is an integer power of 2. Then, the length of xx is twice the length of x which is also an integer power of 2. Therefore, the length of xx is an integer power of 2. Hence, the property holds for the recursive step.

Therefore, by structural induction, the length of all elements x of B is an integer power of 2.

(b) Basis Step: For n = 0, sumeven(n) = 0 and n(n+1) = 0(0+1) = 0. Hence, the property holds for the basis step.

Recursive Step: Assume that for some n∈N, sumeven(n) = n(n+1). We need to show that sumeven(n+1) = (n+1)(n+2).

Using the recursive step of the definition of sumeven, we have:

sumeven(n+1) = sumeven(n) + (2(n+1)+2)

= n(n+1) + 2n + 4

= n^2 + 3n + 2

= (n+1)(n+2)

Hence, the property holds for the recursive step.

Therefore, by structural induction, for all n∈N, sumeven(n) = n(n+1).

(c) Basis Step: For the strings 0 and 1, the property holds since they start and end with the same character.

Recursive Step: Assume that for some strings u, v, and w, u and w start and end with the same character. We need to show that the string wxw also starts and ends with the same character.

Since u and w start and end with the same character, we can write u = axa and w = byb for some characters a, b, x, and y. Then, the string wxw can be written as axaybybaxa which starts and ends with the same character a.

Similarly, we can write u = axa and w = byb for some characters a, b, x, and y such that a ≠ b. Then, the string wxw can be written as axaybybaxa which starts and ends with the same character a.

Therefore, by structural induction, for all elements u of D, u starts and ends with the same character.

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Answer:

8

Step-by-step explanation:

Not sure this is correct

substitute y in the second equation

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

X-2/4x=4

4/4x-2/4x=2/4x

2/4x=4

4 /2/4

4•4/2

16/2

8

The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.

What is Volume ?

Volume is a measure of the amount of space occupied by an object or substance in three-dimensional space. It is the amount of space that a solid, liquid, or gas occupies.

Liquid volume is the amount of space occupied by a liquid.

The units of liquid volume are typically liters, milliliters, gallons, or fluid ounces.

Liquid volume can be measured using a graduated cylinder or other measuring tools.

The volume of a liquid can be affected by changes in temperature and pressure.

The volume of a liquid can be calculated by multiplying its height, width, and length.

The density of a liquid can also affect its volume, as denser liquids will occupy less space than less dense liquids.

The volume of a liquid can be converted to other units of measurement using conversion factors.

Therefore, The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.

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Values of 12 and 13 are 76° and 63° respectively.

A polygon with three sides and three angles is triangle.It is the simplest polygon and can be classified based on its sides and angles. Triangles are used in various fields, including mathematics, engineering, architecture, and art. They are also used to represent stability, strength, and balance in symbols and logos.

Given:-∠F = 104°

Let we assume ∠12 = x

therefore,

∠12 + ∠F = 180°

x + 104 = 180°

x = 180 - 104

x = 76°

therefore , ∠12 is 76°.

now ,

∠D = 41°

Sum of all angles of triangle is 180°

so,

∠D + ∠12 + ∠13= 180°

41 + 76 + ∠E = 180

117 + ∠E = 180

∠E = 180 - 117

∠E = 63°

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The new coordinates after the scale factor comes into the scenario is:

1. (-12,3), (-6,9), (0, -6), (-15, -6)

2. (1, -4), (6,4), (7, -2)

3. (4,3), (12,12), (12,8), (4,8)

4. (3, -6), (9, -3), (15, -6), (9, -9)

5. (-14, -8), (-8, -6), (-6, -12), (-12, -14)

6. (-1,3), (2,2), (2,1), (-1,0)

During the process of dilatation, an object must be reduced in size or changed. It is a transformation that uses the given scale factor to shrink or expand the objects. The image is the new figure that forms as a result of dilatation, whereas the pre-image is the original figure. There are two kinds of dilation:

A rise in an object's size is referred to as expansion.

Contraction is the term for a reduction in size.

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Answer:

Using Newton's Method, the points of intersection between f(x) = x^6 and g(x) = cos(x) are approximately (0.8241, f(0.8241)) and (2.3111, f(2.3111)), where f(x) = x^6.

To find the points of intersection of the graphs of f(x) = x^6 and g(x) = cos(x), we can solve the equation h(x) = f(x) - g(x) = x^6 - cos(x) = 0.

Explanation:

We will use Newton's Method to approximate the x-value(s) of intersection. The formula for Newton's Method is:

x_n+1 = x_n - f(x_n)/f'(x_n)

where x_n is the nth approximation of the root, f(x_n) is the function evaluated at x_n, and f'(x_n) is the derivative of the function evaluated at x_n.

Let h(x) = x^6 - cos(x), then

h'(x) = 6x^5 + sin(x)

Now we need to choose a starting value for x. By graphing the two functions, we can see that there are two points of intersection in the interval [0,1]. Let's choose x = 0.5 as our starting value.

x_0 = 0.5

x_1 = x_0 - h(x_0)/h'(x_0) = 0.5352

x_2 = x_1 - h(x_1)/h'(x_1) = 0.8656

x_3 = x_2 - h(x_2)/h'(x_2) = 0.8249

x_4 = x_3 - h(x_3)/h'(x_3) = 0.8241

Thus, the approximate value of the first intersection point is x = 0.8241.

Now we need to find the second intersection point. By graphing the two functions, we can see that there is another intersection point in the interval [2,3]. Let's choose x = 2.5 as our starting value.

x_0 = 2.5

x_1 = x_0 - h(x_0)/h'(x_0) = 2.3214

x_2 = x_1 - h(x_1)/h'(x_1) = 2.3111

x_3 = x_2 - h(x_2)/h'(x_2) = 2.3111

Thus, the approximate value of the second intersection point is x = 2.3111.

Therefore, the points of intersection of the two graphs are approximately (0.8241, f(0.8241)) and (2.3111, f(2.3111)), where f(x) = x^6.

Hope this helps you in some way! I'm sorry if it doesn't. If you need more help, ask me! :]

According to the formula,  the area of rhombus JKI M is approximately 315.3 square centimeters.

What is area of rhombus formula?

The formula for the area of a rhombus is half the product of its diagonals. That is,

Area of rhombus = (diagonal 1 x diagonal 2)/2

where diagonal 1 and diagonal 2 are the lengths of the two diagonals of the rhombus.

Let D be the intersection of diagonals JK and IM.

Since JK and IM are perpendicular bisectors of each other, D is the midpoint of both diagonals. Let AD = x and BD = y. Then, we have:

[tex]$$\begin{aligned} x + y &= \frac{1}{2} JM = \frac{1}{2}(KL + KM) = \frac{1}{2}(2 \cdot 12.7 + 25.1) = 25.25 \ y - x &= \frac{1}{2} KL = \frac{1}{2} \cdot 12.7 = 6.35 \end{aligned}$$[/tex]

Solving for x and y, we get:

x = [tex]\frac{25.25 - 6.35}{2}[/tex]= 9.95cm

y = [tex]\frac{25.25 + 6.35}{2}[/tex] = 15.8cm

Therefore, the diagonals of rhombus JKI M have lengths 2x = 19.9 cm and 2y = 31.6 cm, respectively. The area of the rhombus is half the product of the diagonals, so we have:

[tex]$$\begin{aligned} A &= \frac{1}{2} \cdot 19.9 \cdot 31.6 \ &= 315.32 , \text{cm}^2 \end{aligned}$$[/tex]

Rounding to the nearest tenth, we get:

[tex]$$A \approx 315.3 , \text{cm}^2$$[/tex]

Therefore, the area of rhombus JKI M is approximately 315.3 square centimeters.

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Answer:

If a = -1/2 is a root of the quadratic equation 8x² - bx - 3, then we know that when x = -1/2, the equation is equal to 0. We can use this information to solve for b.

Substituting x = -1/2 into the equation, we get:

8(-1/2)² - b(-1/2) - 3 = 0

Simplifying and solving for b, we get:

2 - (b/2) - 3 = 0

b/2 = -1

b = -2

Therefore, b = -2 is the value we are looking for.

To find the other root, we can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient. In this case, the constant term is -3 and the leading coefficient is 8. Therefore, the product of the roots is:

(-1/2) times the other root = -3/8

Solving for the other root, we get:

(-1/2) times the other root = -3/8

other root = (-3/8) / (-1/2)

other root = (3/8) * 2

other root = 3/4

Therefore, the other root is 3/4.

Finally, to find (1/a - 1/b)², we can substitute a = -1/2 and b = -2 into the expression:

(1/a - 1/b)² = (1/(-1/2) - 1/(-2))²

= (-2 - 1/2)²

= (-5/2)²

= 25/4

Therefore, (1/a - 1/b)² is equal to 25/4.

Answer:

[tex]b=2[/tex]

[tex]\textsf{Other root} = \dfrac{3}{4}[/tex]

[tex]\left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2=\dfrac{25}{4}[/tex]

Step-by-step explanation:

Roots are also called x-intercepts or zeros.  They are the x-values of the points at which the function crosses the x-axis, so the values of x when f(x) = 0.

If x = α is a root of a polynomial f(x), then f(α) = 0.

Therefore, given that a = -1/2 is a root of the quadratic equation 8x² - bx - 3, substitute x = -1/2 into the equation and set it to zero:

[tex]\implies 8\left(-\dfrac{1}{2}\right)^2-b\left(-\dfrac{1}{2}\right)-3=0[/tex]

Solve for b:

[tex]\implies 8\left(\dfrac{1}{4}\right)+\dfrac{1}{2}b-3=0[/tex]

[tex]\implies \dfrac{8}{4}+\dfrac{1}{2}b-3=0[/tex]

[tex]\implies 2+\dfrac{1}{2}b-3=0[/tex]

[tex]\implies \dfrac{1}{2}b-1=0[/tex]

[tex]\implies \dfrac{1}{2}b=1[/tex]

[tex]\implies b=2[/tex]

Therefore, the quadratic equation is:

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

The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.  

The constant term of the quadratic equation is -3 and the leading coefficient is 8.  Let the other root be "r". Therefore:

[tex]\implies a \cdot r=\dfrac{-3}{8}[/tex]

Substitute the known value of a = -1/2 and solve for r:

[tex]\implies -\dfrac{1}{2} \cdot r=\dfrac{-3}{8}[/tex]

[tex]\implies r=\dfrac{3}{4}[/tex]

Therefore, the other root of the quadratic equation is 3/4.

To find the value of (1/a - 1/b)²​, substitute the given value of a and the found value of b into the equation and solve:

[tex]\implies \left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2[/tex]

[tex]\implies \left(\dfrac{1}{-\frac{1}{2}}-\dfrac{1}{2}\right)^2[/tex]

[tex]\implies \left(-2-\dfrac{1}{2}\right)^2[/tex]

[tex]\implies \left(-\dfrac{5}{2}\right)^2[/tex]

[tex]\implies \dfrac{25}{4}[/tex]

Answer:

Step-by-step explanation:

93+23 then u takeaway the whole number to 93

Answer:

The answer is 70.

Step-by-step explanation:

If you subtract 23 from 93, you will get 70.

Answer: x = 1 ± √ 13 / 6

Step-by-step explanation:

For ax^2 + bx + c= 0, the values of x which are the solutions to the equation are given by: x = − b ± √ b^2 − (4ac) / 2 ⋅ a

Substituting:

3 for a

−1 for b

−1 for c gives: x = − ( −1 ) ± √ ( − 1 )^2 − (4 ⋅ 3 ⋅ − 1) / 2 ⋅ 3

x = 1 ± √ 1 − ( − 12 ) / 6

x = 1 ± √ 1 + 12 / 6

x = 1 ± √ 13 / 6

Hope this helps!

Therefore , the solution of the given problem of triangle comes out to be   m∠B = 29.5 degrees , m∠C = 132.25 degrees and m∠D = 18.25 degrees.

Because a triangle has two or so more extra parts, it is a polygon. It has a straightforward rectangular shape. Only two of a triangle's three sides—A and B—can differentiate it from a regular triangle. Euclidean geometry produces a single area rather than a cube when boundaries are still not perfectly collinear. Triangles are defined by their three sides and three angles. Angles are formed when a quadrilateral's three sides meet. There are 180 degrees of sides on a triangle.

Here,

Angles B and D are congruent because triangle BCD is isosceles with basis BD. As a result, we can equalise their measurements and find x:

=> m∠B = m∠D

=> (5x + 4) = (x + 15)

=> 4x = 11

=> x = 11/4

Knowing x allows us to determine the size of each angle.

=> m∠B = 5x + 4 = 5(11/4) + 4 = 29.5 degrees

=> m∠D = x + 15 = (11/4) + 15 = 18.25 degrees

Angles B and D being congruent, we can determine what mC is as follows:

=> m∠C = 180 - m∠B - m∠D = 180 - 29.5 - 18.25 = 132.25 degrees

As a result, the triangle's angles are each measured in degrees as follows:

=> m∠B = 29.5 degrees

=>  m∠C = 132.25 degrees

=>  m∠D = 18.25 degrees

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Answer:

A. area of larger rectangle: 18x^2

area of smaller rectangle: 8x^2

B. area of shaded region: 10x^2

Step-by-step explanation:

Larger rectangle: A = lw = (6x)(3x) = 18x^2

Smaller rectangle: A = lw = (4x)(2x) = 8x^2

The shaded region = larger rectangle - smaller rectangle

=> 18x^2 - 8x^2 = 10x^2

Answer: Based on the description of the figure, the first three parts of the rectangle are shaded dark to represent the fraction 3/8, and the next three parts are shaded light to complete the fraction 3/4. The last two parts are not shaded.

To find the quotient of 3/4 divided by 3/8, we can use the rule that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 3/8 is 8/3, so we have:

3/4 ÷ 3/8 = 3/4 × 8/3

To simplify this expression, we can cancel out a factor of 4 from the numerator and denominator of 3/4, and a factor of 3 from the numerator and denominator of 8/3. This gives us:

3/4 × 8/3 = (3 × 2)/(1 × 1) = 6

Therefore, the quotient of 3/4 divided by 3/8 is 6.

Step-by-step explanation:

The mean of the given frequency distribution is 587.12.

What is frequency distribution?

In frequency tables or charts, frequency distributions are displayed. The actual number of observations that fall into each range can be seen in frequency distributions, as well as the proportion of observations that do.

We are given a frequency distribution table.

We know that the mean is the average of sum of all the values.

So, we first get the values as :

⇒ 610 * 12 = 7320

⇒ 537 * 6 = 3222

⇒ 597 * 10 = 5970

⇒ 572 * 14 = 8008

⇒ 590 * 9 = 5310

⇒ 606 * 6 = 3636

Now, on adding all the values, we get

⇒ 7320 + 3222 + 5970 + 8008 + 5310 + 3636

⇒ 33466

So,

⇒ Mean =  33466 ÷ 57

⇒ Mean = 587.12

Hence, the mean of the given frequency distribution is 587.12.

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The length of arc of the sector is 52.2 cm and the area of the sector is 260.8 cm²

Arc length is defined as the distance between the two points placed on the circumference of the circle and measured along the circumference. Arc length is the curved distance along the circumference of the circle.

area of an arc = tetha/360 × πr²

l = 299/360 × 3.14 × 10²

l = 93886/360

l = 260.8 cm² ( 1 dp)

The length of arc of the sector

=( tetha)/360 × 2πr

= 299/360 × 2 × 3.14 × 10

= 18777.2/360

= 52.2 cm

therefore the area of the sector is 260.8cm² and the length of the arc is 52.2 cm

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Answer:

13.2

Step-by-step explanation:

using Pythagorean theorem, create the equation for the unknown side, x.

x^2+9^2=16^2

subtract 9^2

x^2=16^2-9^2

Use difference of squares.

x^2=(16-9)*(16+9)

Solve

x^2=7*25

x^2=175

Take the square root of both sides

x=sqrt175

x=13.2

The correct step in the solution of the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 is option C: [tex]\sqrt[4]{}[/tex](2m-1) = 1.

An equation is a mathematical statement that indicates that two expressions are equal. It consists of two sides separated by an equal sign (=). The expressions on either side of the equal sign can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value of the variable that makes the equation true. Equations are used in many areas of mathematics, as well as in physics, engineering, and other sciences, to model and solve problems.

We can start solving the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 by simplifying the left side of the equation first. We have:

[tex]\sqrt[4]{}[/tex](2m+1-2) = 1

[tex]\sqrt[4]{}[/tex](2m-1) = 1

²(2√(2m-1)) = 1 (using the fact that 4 = 2²)

2sqrt(2m-1) = 0 (taking the square root of both sides)

At this point, we can see that the equation simplifies to 2*√(2m-1) = 0, which means that √(2m-1) = 0 (since 2 ≠ 0). Therefore, we can solve for m by squaring both sides:

√(2m-1) = 0

(√(2m-1))² = 0²

2m-1 = 0

2m = 1

m = 1/2

Therefore, the correct step in the solution of the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 is option C: [tex]\sqrt[4]{}[/tex](2m-1) = 1.

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The value of sin θ is -[tex]\frac{7\sqrt{130} }{130 }[/tex] and cos θ is [tex]\frac{9\sqrt{130} }{130 }[/tex] . The solution has been obtained by using trigonometry.

What is trigonometry?

The study of right-angled triangles, including their sides, angles, and connections, is referred to as trigonometry.

We are given that tan θ is -7/9. The minus sign is there because it lies in the fourth quadrant.

This means that the perpendicular is 7 and the base is 9.

Let the hypotenuse be x.

Now, by using Pythagoras theorem, we get

⇒ [tex]7^{2}[/tex] + [tex]9^{2}[/tex] = [tex]x^{2}[/tex]

⇒ 49 + 81 = [tex]x^{2}[/tex]

⇒ [tex]x^{2}[/tex] = 130

⇒ x = √130

By trigonometry,

⇒ Sin θ = -[tex]\frac{7}{\sqrt{130} }[/tex]

⇒ Sin θ = -[tex]\frac{7\sqrt{130} }{130 }[/tex]

Similarly,

⇒ Cos θ = [tex]\frac{9}{\sqrt{130} }[/tex]

⇒ Cos θ = [tex]\frac{9\sqrt{130} }{130 }[/tex]

Hence, the values for sin θ and cos θ have been obtained.

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The sunflower's height is increasing at an average rate of 13.045 inches per week over the third week.

The average rate of change of a function over an interval is the slope of the secant line that passes through the two endpoints of the interval. Mathematically, if we have a function f(x) and an interval [a,b], the average rate of change of f(x) over [a,b] is given by:

average rate of change = (f(b) - f(a))/(b - a)

For our problem, the function is f(t) = 5(1.7)ˣ, and we need to find the average rate of change over different time intervals. Let's consider each interval separately:

The average rate of change over the [0,1] interval is:

average rate of change = (f(1) - f(0))/(1 - 0) = (5(1.7)¹ - 5(1.7)⁰)/(1 - 0) = 4.5

Therefore, the sunflower's height is increasing at an average rate of 4.5 inches per week over the first week.

The average rate of change over the [1,2] interval is:

average rate of change = (f(2) - f(1))/(2 - 1) = (5(1.7)² - 5(1.7)¹)/(2 - 1) = 7.65

Therefore, the sunflower's height is increasing at an average rate of 7.65 inches per week over the second week.

The average rate of change over the [2,3] interval is:

average rate of change = (f(3) - f(2))/(3 - 2) = (5(1.7)³ - 5(1.7)²)/(3 - 2) = 13.045

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a)  The probability that there will be exactly four droughts in the region over the next decade is approximately 0.193.

b) The probability that it will be a severe drought given that exactly one drought actually occurred in 2 years is approximately 0.164.

c)  The probability that all three droughts that actually occurred in 5 years will be moderate is 0.016.

To determine the probability of there being exactly four droughts in the region over the next decade, the expected value of droughts over a decade must first be calculated. λ, the expected number of droughts over a period, can be calculated using the formula:λ = (number of droughts in the last 60 years)/(60 years)λ = (6+16)/(60)λ = 0.367

Therefore, the expected number of droughts in the region over the next decade is 0.367 x 10 = 3.67.Using the Poisson distribution formula, the probability of there being exactly four droughts in the region over the next decade can be calculated as:P(4) = (e^-3.67)(3.67^4)/(4!)P(4) ≈ 0.193

Therefore, the probability that there will be exactly four droughts in the region over the next decade is approximately 0.193.

Assuming that exactly one drought actually occurred in 2 years, the probability that it will be a severe drought can be calculated using Bayes' theorem:P(severe | 1) = P(1 | severe)P(severe) / P(1)First, P(1) must be calculated:P(1) = P(1 | severe)P(severe) + P(1 | moderate)P(moderate)P(1 | severe) = e^-λ(λ^1) / 1! = e^-0.367(0.367^1) / 1! ≈ 0.312P(1 | moderate) = e^-λ(λ^1) / 1! = e^-0.367(0.367^1) / 1! ≈ 0.592P(moderate) = 16 / 60 = 0.267P(severe) = 6 / 60 = 0.1P(1) ≈ 0.312(0.1) + 0.592(0.267) ≈ 0.279Next, P(severe | 1) can be calculated:P(severe | 1) = P(1 | severe)P(severe) / P(1)P(severe | 1) ≈ (0.312)(0.1) / 0.279 ≈ 0.164

Therefore, the probability that it will be a severe drought given that exactly one drought actually occurred in 2 years is approximately 0.164.

Assuming that exactly three droughts actually occurred in 5 years, the probability that all will be moderate droughts can be calculated using the binomial distribution formula:P(3 moderate) = (n choose k)(p^k)(1-p)^(n-k)where n = 3, k = 3, and p = 16 / 60 = 0.267(n choose k) = (n! / k!(n-k)!) = (3! / 3!(3-3)!) = 1P(3 moderate) = (1)(0.267^3)(1-0.267)^(3-3) = 0.016

Therefore, the probability that all three droughts that actually occurred in 5 years will be moderate is 0.016.

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The area of rectangle DAEF is 40 square units.

Since rectangle ABCD is similar to rectangle DAEF, their corresponding sides are proportional.

Let the length of rectangle DAEF be x.

Then, we have the following ratios:

AB/DA = EF/DA (corresponding sides of similar rectangles are proportional)

10/4 = x/4 (substituting AB=10 and AD=4)

Solving for x, we get:

x = 40/10 = 4

Therefore, the length of rectangle DAEF is 4.

Now, the area of rectangle DAEF is:

Area = length x width

Area = 4 x 10 = 40 square units.

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The standard value that corresponds to a sample mean age of 60 years is 0.

We can use the formula for the z-score (standardized value) to find the answer:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, we have:

x = sample mean age of the 400 trees

μ = population mean age of trees in the large park = 60 years

σ = population standard deviation of tree ages in the large park = 2.2 years

n = sample size = 400

We don't know the value of x, but we do know that the distribution of sample means is approximately normal, with a mean of μ = 60 and a standard deviation of σ / sqrt(n) = 2.2 / sqrt(400) = 0.11.

So, we want to find the standardized value for a sample mean that is 0 standard deviations away from the population mean, which means:

z = (x - 60) / 0.11 = 0

Solving for x, we get:

x - 60 = 0

x = 60

Therefore, the standardized value that corresponds to a sample mean age of 60 years is:

z = (x - μ) / (σ / sqrt(n))

z = (60 - 60) / 0.11

z = 0

So the answer is 0.

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