Kazuko says the expressions 5x and 6 - x are equivalent expressions, because you
can substitute 1 for x in both expressions and get the same result. Is Kazuko's
reasoning correct? Explain.

The solid S is a triangular prism whose base is a triangle with vertices (0, 0), (1,0), and (0, 1). The cross-sections perpendicular to the x-axis are equilateral triangles of the same height.

The solid S is a three-dimensional shape with a triangular base. The base is a triangle with vertices (0, 0), (1,0), and (0, 1). This triangle has sides of length 1, and the angles at each vertex are equal. The equilateral triangles are the cross-sections of the solid perpendicular to the x-axis. This means that each side of each triangle is the same length, and all the angles are equal. Each cross-section has the same height, determined by the height of the solid. The shape of the solid is a triangular prism, with triangular sides that connect the three vertices. This shape has a three-dimensional volume, and the volume can be calculated by multiplying the area of the base by the height of the solid.

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According to the given  expression There will be 910 women among the 1,560 runners.

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself.

Using mathematical operations like addition, subtraction, multiplication, and division, a combination of numbers and integers can be expressed numerically in mathematics.

I'm looking for 7/12 of 1560. I really do mean, "Divide 1560 among 12 groups and count how many there are in each category overall."

1560 divided into 12 groups.

1560/12

=130.

There are 130 people in each group. We multiply to determine how many there are in the seven groupings.

130/7

=910

Among the 1,560 runners, 910 of them will be female.

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The error in writing the polynomial 3m²-5m⁵+m⁴ in standard form

as -5m³+3m²+m⁴ is, Power of  -5m is 5 not 3 and powers should be in descending order

The polynomials are the algebraic expressions, which have multiple powers. Polynomials are always written in a form of descending powers.

And based on powers of polynomials, we can categorize polynomials into following categories-

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

The polynomial-        3m²-5m⁵+m⁴

The standard form-    -5m³+3m²+m⁴

the errors are in standard form of given polynomial,

 3m²-5m⁵+m⁴

For converting it into standard form,

So, it can be arranged in descending powers

-5m⁵+m⁴+ 3m²

By comparing it with given standard form

It can be seen that there are two errors in given form

1.   Power of -5m is 5 not

2.  Variables should be arranged in descending powers

Hence, the correction is power -5 and descending order of powers

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The roots of the given quadratic equation [tex]\frac{x^{2} }{3} - \frac{5x}{3} - 6= 0[/tex]   after solving are  [tex]\alpha = \frac{5 + 47i }{2}[/tex]  and  [tex]\beta = \frac{5 - 47i }{2}[/tex] .

Given,

          Quadratic Equation is   [tex]\frac{x^{2} }{3} - \frac{5x}{3} - 6= 0[/tex]

Multiply L.H.S. and R.H.S. by 3, and we get:

            [tex]x^{2} -5x - 18 = 0[/tex]

Here,  [tex]a= 1, b= -5 ,c= -18[/tex]

Let's find out the determinant,

         [tex]D = b^{2} - 4ac[/tex]

      [tex]D= (-5)^{2} - 4(1)(-18)\\D = 25 - 72\\D = -47\\[/tex]

Therefore, the roots of the quadratic equation will be :

       [tex]\alpha = \frac{-b + \sqrt{D} }{2a}[/tex]     and    [tex]\beta= \frac{-b - \sqrt{D} }{2a}[/tex]

Now, Substitute the value of [tex]a,b,c[/tex]

      [tex]\alpha = \frac{5 + 47i }{2}[/tex]     and    [tex]\beta = \frac{5 - 47i }{2}[/tex]

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The maximum area of rectangle is 8450 m².

Given that there is fence only on 3 sides and one length side have river flowing trough it

so the perimeter of the plot is given by:

l + w + w

=> 2w+l

where w is width and l is length of the plot

given , perimeter = 260

=> 2w+l = 260  ---- (i)

area of the plot is = l*w

putting the value of w from equation (i) we get

(260-2w)*w

=> 260w-2w²

so A = 260w-2w²

To maximise A we have dA/dw=0

=> d(260w-2w²)/dw

=> 260-4w

so 260 - 4w=0

=> 4w = 260

=> w = 65

l = 260 -2 w

=> 260 -2 *65

=> 260-130

=> 130

so Maximum area of the rectangle = 130*65 = 8450

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Type 540,000,000 as a floating-point literal using scientific notation with a single digit before and after the decimal point is 5.4e8

Type 540,000,000 as a floating-point literal using scientific notation with a single digit before and after the decimal point is 5.4e8.

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The label of the triangles based on the center are;

A) Incenter

B) Orthocenter

C) Circumcenter

D) Centroid

1) The centroid of a triangle is defined as the point in which the three medians of the triangle intersect.

2) The orthocenter of a triangle is defined as the point where all the three altitudes of the triangle cut or intersect each other.

3) The incenter of a triangle is defined as the intersection point of all the three interior angle bisectors of the triangle.

4) The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.

Looking at the given triangles, we can conclude that triangle A shows the incenter, triangle B shows the orthocenter, triangle C shows the circumcenter while triangle D shows the centroid,

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Answer:

Step-by-step explanation:

(1+1/a)^2-(1-1/a)^2

= (1+1/a)^2 - (1-1/a)(1+1/a)

= (1+1/a)^2 - 1 + 1/a^2

= (1+1/a+1/a^2+1/a^3) - 1 +1/a^2

= 1/a^3+2/a^2+2/a+1

So the final answer is 1/a^3+2/a^2+2/a+1

a. The point slope form of the equation is y-4=-3(x+5)/10.

b. The y-intercept of the line is 2.5.

c. The line crosses the y-axis halfway between the given points so the y-intercept is halfway between the y-coordinate of the point on the left, and that of the point on the right. The number halfway between those values is 2.5.

d. The slope intercept form is y = (-3/10)x + (5/2).

The solution has been obtained using linear equations.

What is linear equation?

A linear equation is one that has the highest degree of 1 possible. This indicates that there are no variables in a linear equation with exponents greater than 1. Such an equation on the graph, forms a straight line.

a. We know that general form of point- slope form of equation is

y-y₁=m(x-x₁)

Here, y₁= 4 and x₁= -5, y₂= 1 and x₂=5

Now, m = (y₂ - y₁) / (x₂ - x₁)

⇒m= (1-4)/(5-(-5))

⇒m= -3/10

Thus, we get the equation as y-4=-3(x+5)/10

b. At the y-intercept, the x coordinate is 0.

So, putting x coordinate as 0 in the equation, we get

⇒y-4=-3(0+5)/10

⇒y-4=-15/10

⇒y=-1.5 + 4

⇒y= 2.5

Thus, the y-intercept of the line is 2.5.

c. The line crosses the y-axis halfway between the given points so the y-intercept is halfway between the y-coordinate of the point on the left, and that of the point on the right. The number halfway between those values is 2.5.

d. The general equation for slope intercept form is y=mx+b

m= -3/10

b = y₁ - x₁(y₂ - y₁)/(x₂ - x₁)

⇒ b = 4 - (-5)(1 - 4)/(5 - (-5))

⇒ b = 4 - 15/10

⇒ b = 25/10

⇒ b = 5/2

So, we get y = (-3/10)x + (5/2).

Hence, the required solutions have been obtained.

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Answer:

The lowest interest rate you would need to earn in order to have 15,000 become 20,000 in 5 years is 6.5%. This is based on the compound interest formula, where you would need to multiply the original amount by the interest rate (15,000 x 0.065 = 975) and add that amount to the original amount to equal your desired total (15,000 + 975 = 20,000).

The volume and surface area of the cube is 343 yd³ and 294 yd² respectively

An equation is an expression showing the relationship between numbers and variables.

a) The volume of the cube is the amount of space that it occupies. It is given that:

Volume = length³

But length = 7 yd, hence:

Volume = 7³ = 343 yd³

b) The surface area of the cube is the sum of the individual surface area. It is given that:

Surface area = 6 * length²

But length = 7 yd, hence:

surface area = 6 * 7² = 294 yd²

The surface area is 294 yd²

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The total capacity of Harold's computer hard drive is 750 GB.

Total free space on Harold's computer hard drive = 131.25 GB

Percentage of free space on Harold's computer hard drive = 17.5%

Let

Total capacity of the hard drive = x

17.5% of x = 131.25

17.5/100 × x = 131.25

0.175 × x = 131.25

0.175x = 131.25

divide both sides by 0.175

x = 131.25 / 0.175

x = 750 GB

Hence, Harold's computer hard drive has a total capacity of 750 GB

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Answer:

Step-by-step explanation:

The slope-intercept form is y=mx+b, where m is the slope and b is the y intercept.  y=mx+b

y=4x3−3

Reorder terms.

y=4 3x−3

Find the values of m and b using the form y= mx+b. 4 3b=−3

The slope of the line is the value of

m

, and the y-intercept is the value of

b

.

Slope:

4

3

y-intercept:

(0,−3)

The slope of the line is -3.

Given equation,

⇒ y - 4 + 3(x-3) = 0

⇒ y = 4 - 3(x-3)

⇒ y = 4 - 3x + 3

⇒ y = -3x + 7

Equation of slope-intercept form: y = mx + b

We know the slope is the value multiplied by x,

so the slope of the line is -3.

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The domain and the range of the composite function are given as follows:

a) Domain: {3, 5, 6, 7}.

b) Range: {0, 9}.

The composite function of f(x) and g(x) is given by the following rule:

(f ∘ g)(x) = f(g(x)).

It means that the output of the inside function serves as the input for the outside function.

The points on the composite function are given as follows:

The domain is the set of input values, hence it is given by:

{3, 5, 6, 7}.

The range is the set of output values, hence it is given by:

{0, 9}.

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The square root of 50 is the distance between the points i.e. 5√2.

It is possible to measure a distance between two points or objects quantitatively or sporadically qualitatively. The word "distance" can be used to describe a physical length or an estimate based on other factors in physics or everyday speech.

To do this we know that a² +b²  = x²

for b:

The distance between -1 and -4 is 5. we will use 5²

for a:

The distance between2 and 1 is also 5. we we use 5²

put into the formula

5² +5² = c²

25+25 = c²

50 = c²

so the square root of 50 is the distance between the points i.e. 5√2.

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The cost in dollars and cents per boxed lunch will be $9.40.

It should be noted that an equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.

Since the proportional relationship between the number of boxed lunches ordered, b, and the total cost in dollars and cents, c, can be represented by the equation c=9.4b

The cost will be:

= 9.4 × 1

= 9.40

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Approximately 99.7 % of the students spent between $ 271 and $ 407  on textbooks in a semester.

Then , according to the Empirical rule , approximately 99.7% of the data lies with in the 2 standard deviations from mean.

Given : The distribution of the amount of money spent by students on textbooks in a semester is approximately normal in shape with

Mean = 339     standard deviation =  34.

Then ,  by

Empirical rule , approximately 99.7 % of the students spent between Mean ± 2 (Standard deviation) on textbooks in a semester.

where , Mean ± 2 (Standard deviation) = 339 ±  2(34)

=(339-2(34) , 339+2(31))

=(271 , 407)

Hence ,  Approximately 99.7% of the students spent between $ 271 and $ 407 on textbooks in a semester.

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Derivative of f(x) = 8√(10x⁴+x²-1)⁹ is (1440x³ + 72x)(10x⁴ + x² - [tex]1)^{(7/2)}[/tex].

The derivative is what calculus is all about. The instantaneous pace at which a function changes with regard to one of its variables is known as the derivative. Finding the slope of the tangent line to the function at a certain location is equal to doing this.

Calculus defines a derivative as the rate at which one quantity, y, changes in relation to another, x. The differential coefficient of y with respect to x is another name for it. Finding a function's derivative is the process of differentiation.

The typical representation of a function's derivative is d/dx (f(x)) (or) df/dx (or) Df(x) (or) f' (x). Let's examine the technical definition of a derivative. Two points on a function f(x) curve are (x, f(x)) and ((x + h), f(x + h)), respectively. If so, [f(x + h) - f(x)]/(x + h - x) = [f(x + h) - f(x) / h] will be the slope of the secant line across these locations. The second point overlaps the first point in the image below when the distance between two points is almost equal to 0 (i.e., when h approaches 0), and the secant line turns into the tangent line.

Given that;

f(x) = 8√(10x⁴+x²-1)⁹

So, First derivatives is ;

[tex]\begin{aligned}& \frac{d}{d x}[f(x)]=f^{\prime}(x) \\& \frac{d}{d x}\left[8\left(10 x^4+x^2-1\right)^{\frac{9}{2}}\right] \\& =8 \cdot \frac{d}{d x}\left[\left(10 x^4+x^2-1\right)^{\frac{9}{2}}\right] \\& \left.=8 \cdot \frac{9}{2}\left(10 x^4+x^2-1\right)^{\left(\frac{9}{2}\right.}-1\right) \cdot \frac{d}{d x}\left[10 x^4+x^2-1\right] \\\end{aligned}[/tex]

[tex]\begin{aligned} & =36\left(10 \cdot \frac{d}{d x}\left[x^4\right]+\frac{d}{d x}\left[x^2\right]+\frac{d}{d x}[-1]\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}} \\& =36\left(10 \cdot 4 x^3+2 x+0\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}} \\& \left.=36\left(40 x^3+2 x\right)\left(10 x^4\right)+x^2-1\right)^{\frac{7}{2}} \\& =\left(1440 x^3+72 x\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}}\\\end{aligned}[/tex]

So, the Derivative of f(x) = 8√(10x⁴+x²-1)⁹ is (1440x³ + 72x)(10x⁴ + x² - [tex]1)^{(7/2)}[/tex].

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Answer:  .9072 is the awnswer

This figure's surface measures 2332 square inches.

The area or region that an object’s surface occupies is known as its surface area. Volume, on the other hand, refers to how much room an object has. There are numerous shapes and dimensions in geometry, including spheres, cubes, cuboids, cones, cylinders, etc. Each form has its own volume and surface area.

Given, The figure below is a net for a triangular prism. Side a = 46inches, side b = 12 inches, side c = 32inches, and altitude d = 22inches

From the figure,

The surface area of triangle = 1/2* d * a +  1/2* d * a

The surface area of the rectangle = b *a  +2 * c * b

Thus the complete surface area of Shape =  The surface area of the rectangle + The surface area of a triangle

Total surface area =d*a + b*a + 2*c*b

Total surface area =  22 * 46 + 12 * 46 + 2 * 32 * 12

Total surface area = 2332

Therefore, The surface area of this figure is 2332 square inches.

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Complete question:

Additional information is needed to prove that the triangles are congruent by SAS.

Triangles are said to be congruent if, under a correspondence, two of a triangle's sides and the angle that connects them are equal to two of another triangle's sides and the angle that connects them.

You know the following information about triangle ABC and triangle DEF:

AB = DE

∠A = ∠D

Since The first such theorem is the side-angle-side (SAS) theorem, which states that two sides and an included angle of one triangle are congruent if they are equivalent to two sides and an included angle of another triangle.

Thus to prove that triangles are congruent by the property of SAS.

Two sides of triangles should be equal.

Therefore,

Additional information is needed to prove that the triangles are congruent by SAS.

Either BC = EF

or AC = DF

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The golfer's average performance over the two-year period was [mean] out of 10,with a standard deviation of [standard deviation].

The mean is a measure of central tendency, which refers to a single value that is representative of a series of values. In this case, the mean is the average performance the golfer achieved over the two-year period. To calculate the mean, add up all the performance scores and divide by the total number of scores. The mean can then be rounded to the nearest whole number. The standard deviation is a measure of variability, which indicates how much the scores vary from the mean. To calculate the standard deviation, subtract each performance score from the mean, square the differences, and then add them up. Finally, divide the sum by the number of scores, and then take the square root. The standard deviation can then be reported to the nearest decimal.

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locating a number's component. Doing percentage and decimal calculations proportional relationships.So,  7/6 times taller is Sally than May.

Why is it important to use fractions?

Identifying the portion of a number is one application for fractions. figuring out percentages and decimals. Verhältnis und Proportion.

Learning more complex mathematical concepts requires a strong foundation in fractions. The greatest way to introduce algebra is through fractions because they are a student's initial exposure to mathematical abstraction.

Let's call Jenny's height H. Next, we have this:

Sally's height is equal to 7/8*H.

May's height is equal to 6/8"H.

We only need to divide Sally's height by May's height to compare their heights:

Sally's height divided by May's height equals (7/8)*H / (6/8)*H.

= (7/8) / (6/8) = (7/8) * (8/6) = 7/6

Sally therefore outranks May by 7/6.

Therefore, the answer is 7/6.

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The expected value of a random variable is the sum of the probabilities of all possible outcomes which is 10.083

In this case, since the random variable is the sum of the three independent rolls of a fair die, the expected value, E[X], is the sum of all the possible outcomes of three die rolls, each multiplied by the probability of 1/6 that each outcome will occur:

E[X] = (1 + 2 + 3 + 4 + 5 + 6) × (1/6)³

= 3.5

The variance of a random variable is the expected value of the squared deviation from the mean. In this case, the variance for X is:

Var(X) = E[X^2] - (E[X])^2

= (1² + 2² + 3² + 4²+ 5² + 6²) × (1/6)³ - (3.5)²

= 10.0833

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Answer:

5/3

Step-by-step explanation:

1 week = 7 days

5 weeks = 5 × 7 days = 35 days

5 weeks to 21 days =

= 35 days to 21 days

= 35/21

= 5/3

11.9% of students have height less than 136.4cm. The solution has been obtained using probability.

What is probability?

Probability is the ability to happen. The subject of this area of mathematics is the occurrence of random events. From 0 to 1 is used to express the value.

Let the height be 'd' cm.

We are given mean as 160 cm and standard deviation of 20 cm.

Probability(x>d) = 0.12

Probability(x<d) = 0.881

Probability(z ≤ d-160/20) = 0.881

⇒d-160/20 = 1.1800  (using table)

⇒d-160 = 23.6

⇒d = 183.6

Therefore, d-mean = -23.6

⇒d-160 = -23.6

⇒d = 160-23.6

⇒d = 136.4cm

Hence, 11.9% of students have height less than 136.4cm.

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The cumulative frequency for the class with a lower class limit of 27 is 4 + 7 + 7 = 18.

What is the cumulative frequency?

The cumulative frequency for a class is the total frequency of that class and all the classes before it.

In order to find the cumulative frequency for the class with a lower class limit of 27, you would need to add the frequency of that class with the frequencies of all the classes before it.

The cumulative frequency for a class with a lower class limit of 27 can be found by adding up the frequencies of all the classes that come before it.

In this case, the classes that come before the class with a lower class limit of 27 are "Ages 15-18", "Ages 19-22", and "Ages 23-26",

Hence, the cumulative frequency for the class with a lower class limit of 27 is 4 + 7 + 7 = 18.

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Complete Question:

Ages 15-18 19-22 23-26 27-30 31-34 35-38 Number of students 4 7 7 2 5 10 Based on the frequency distribution above, find the cumulative frequency for the class with a lower class limit 27?

The probability that the total weight of the three randomly selected grapefruits is more than 3.4 pound will be equal to 0.842.  

Probability is calculated as the proportion of favorable events to all potential scenarios of an event. The proportion of positive results, or x, for an experiment with 'n' outcomes can be expressed.

From the data given in the question,

x > 3.4 lb

The mean is 1 lb.

The difference from the mean is 0.12 lb.

The z-score will be,

z = (3.4 - 1) / 0.12

z = 20

This is equivalent to an 84.20% or 0.842 probability.

the complete question is,

The weights of grapefruits of a certain variety are approximately normally distributed with a mean of 1 pound and a standard deviation of 0.12 pounds. Use scenario 6-9. what is the probability that the total weight of three randomly selected grapefruits is more than 3.4 pounds?

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Answer:

[tex]\mathrm{Flagpole\;Height = 17 \;feet}[/tex]

Step-by-step explanation:

Here we have two similar right triangles

One right triangle has base of 8 ft and Jeremy's height of 68" as the height

The other right triangle has the base (8 + 16 = 24) and the height is the height of the flagpole. This is the large triangle

Since the triangles are similar,

The ratios of the sides must be the same

[tex]\mathrm{\dfrac{Jeremy's \;Height}{Flagpole \; Height}} = \dfrac{8}{24}= \dfrac{1}{3}\\\\\\68" = \dfrac{68}{12} \; feet\\\\\textrm{Dividing both numerator and denominator by 4:}\\\\= \dfrac{17}{3}[/tex]

Therefore

[tex]\mathrm{\dfrac{17/3}{Flagpole\;height} = \dfrac{1}{3}}\\\\[/tex]

Cross -multiplying we get
[tex]\mathrm{\dfrac{17}{3} \times 3 = Flagpole\; Height}\\\\\mathrm{Flagpole\;Height = 17 \;feet}[/tex]