What are the legs of a right isosceles triangle?

The two distinct real number zeros of f(x) left parenthesis, x, right parenthesis are -2 + sqrt(7) and -2 - sqrt(7).

The discriminant of a polynomial is a quantity that determines the number of distinct real roots of a polynomial equation. For a polynomial of degree n, the discriminant is calculated as the square of the coefficient of the nth degree term multiplied by the product of the coefficients of the remaining terms, minus the square of the product of the coefficients of the lower degree terms. In the case of [tex]f(x)=3x^2+12x+5[/tex], the discriminant is given by[tex]D=12^2 - 4*3*5 = 144 - 60 = 84.[/tex]

Since the discriminant is greater than 0, it follows that the polynomial equation has two distinct real number zeros. To find the zeros of the equation, we must solve the equation by setting it equal to zero and factorising it. Doing so, we find that the zeros are given by [tex]x = (-12 +/- sqrt(84))/6 = -2 +/- sqrt(7)[/tex]. Hence, the two distinct real number zeros of f(x) are -2 + sqrt(7) and -2 - sqrt(7).

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We're given the equation,

[tex]\longrightarrow\sin\theta+\csc\theta+2=0[/tex]

Take [tex]\sin\theta=x,[/tex] then [tex]\csc\theta=\frac{1}{x}.[/tex] Then the equation becomes,

[tex]\longrightarrow x+\dfrac{1}{x}+2=0[/tex]

Multiply both sides by [tex]x,[/tex] assuming [tex]x\neq 0.[/tex] Then we get,

[tex]\longrightarrow x^2+2x+1=0[/tex]

[tex]\longrightarrow (x+1)^2=0[/tex]

[tex]\Longrightarrow x=-1[/tex]

We're asked to find the value of,

[tex]\longrightarrow y=\sin^{2019}\theta+\csc^{2020}\theta[/tex]

As we've taken [tex]\sin\theta=x,[/tex] this expression becomes,

[tex]\longrightarrow y=x^{2019}+\dfrac{1}{x^{2020}}[/tex]

Now putting [tex]x=-1,[/tex]

[tex]\longrightarrow y=(-1)^{2019}+\dfrac{1}{(-1)^{2020}}[/tex]

[tex]\longrightarrow y=-1+\dfrac{1}{1}[/tex]

[tex]\longrightarrow y=-1+1[/tex]

[tex]\longrightarrow\underline{\underline{y=0}}[/tex]

Hence 0 is the answer.

Answer:

The value of 9^(x) is 3.

Step-by-step explanation:

To find the value of 9^(x), we can use the equation:

(a^x) = (b^y)

Where a and b are different bases, and x and y are different exponents.

We can rewrite this equation as:

x = y * log(a) / log(b)

Plugging in the values given in the question, we get:

x = log(8) / log(4)

Solving for x, we get:

x = 3

Therefore, the value of 9^(x) is 3.

The median for the next batch of observations is 5.

The median is the value in the middle of a data set, which indicates that 50% of the data points have a value less than or equal to the median, and 50% have a value greater than or equal to the median. The median is the number in the middle of a set of numbers. The median represents the 50th percentile of the set of numbers. In other words, the median is the value in the center of a set of numbers where half are less than the median and half are more than the median.

Here,

given set of observation,

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

n=7

median=(7+1)/2

=4th term

=5

The median for following set of observation is 5.

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The age of the son and father obtained after solving the equation will be equal to 6 years and 34 years.

Mathematical expressions with two algebraic symbols on either side of the equal (=) sign are called equations.

This relationship is illustrated by the left and right expressions being equal. The left-hand side equals the right-hand side is a basic, straightforward equation.

As per the given information in the question,

Let the age of the father is f and the age of the son be s.

As per the given statement in the question,

The sum of the ages of a father and his son is 40 years.

f + s = 40   (i)    and,

f + s + 2(f -s) = 96    

3f - s = 96       (ii)

Add equations (i) and (ii)

4f = 136

f = 136/4

f = 34 years

Put f = 34 in equation (i)

f + s = 40

34 + s = 40

s = 40 - 34

s = 6 years.

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Answer:120

Step-by-step explanation:

1,440 divided by 12 equals 120 so 120 shelves are needed

Answer:

15 storage towers

Step-by-step explanation:

first you need to find out how many video games can fit into one storage tower.

8 times 12 is 96,meaning 96 per tower.

Then you would one to get the total number of video games and divide it by 96 to find out how many towers are needed.

If done correctly you will end up with 15.

360° is all the way around a circle. The length of the intercepted arc is equal to the circumference of the circle.

Arc Length of a Circle:

Length. A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement: measure of angle in degrees/360° = L/circumference.

Does arc length have to be in radians?

Arc length is a measurement of distance, so it cannot be in radians. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians, however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius.

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The first ten even natural numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. The mean of these numbers is the sum of the numbers divided by the number of numbers in the set. The sum of these numbers is 110 and there are 10 numbers in the set, so the mean is 11.

The mean of the numbers 9, 10, and 11 is 10. The mean of the numbers 12, 13, and 14 is 13. So the mean of the numbers 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18 is (10+13)/2 = 11.5.

Standard Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 are the numbers we use to count or arrange things in numerical order. Detailed numbers the set of integers that includes zero and the natural numbers. neither a fraction nor a decimal.

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Answer: see below

Step-by-step explanation:

The table is based on the rule y=-2x+5

let's use the last numbers as an example. x=10 and y= -15

since we know the values of x and y, we can substitute them into the rule equation

-15=-2(10)+5, see how I replaced y with 15 and x with 10?

Now we solve and see if they match.

-15=-20+5

-15=-15

since they match, they are correct.

Basically, replace x in the rule with the x values in the table, and if the y value is the same as the y value in the table, it is correct.

Answer:

Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think of them is that, in a proportional relationship, one variable is always a constant value for the other. That constant is known as the "constant of proportionality."

Step-by-step explanation:

Answer:

A whole number is a number without fractions or intergers

Step-by-step explanation:

An example of this wou;d be 0, 16, etc too find one you have to look at the qualities and see if it fits

The general case when G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ) is that the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

G⊕H is cyclic

That is G⊕H = <(g, h)>

Let us choose an arbitrary element g₁ ∈ G, then as G⊕H is cyclic

(g₁, h₁) ∈ G⊕H = <( g, h)>

This means that there exists an integer x such that

(g₁, h₁) = (g, h)ˣ

(g₁, h₁) = (gˣ, hˣ)

that is g₁ = gˣ

implies g₁ ∈ <g>

that is G ⊆ <g> as g₁ is an arbitrary element belonging to G

But we know <g> ⊆ G

Hence G = <g>

Thus G is a cyclic group generated by g.

Similarly we can prove that H group is cyclic generated by h, that is H = <h>

So the general case is that G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ), then the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

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The quadratic equation, which is written in the form [tex]ax^{2} +bx+c=0[/tex] has two solutions because it is a second-order polynomial equation.

What is a quadratic equation?

Any equation of the form  [tex]ax^{2} + bx + c=0[/tex] where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.

As we know, the formula for the roots of the quadratic equation is given by :

[tex]x=[/tex][tex]\frac{-b+/-\sqrt{b^{2}-4ac } }{2a}[/tex]

where a, b, and c are the coefficients of the quadratic equation, and [tex]\sqrt{b^{2}-4ac }[/tex] represents the square root function. The plus-minus symbol (±) indicates that there are two solutions, one for each value of the symbol. These two solutions represent the roots of the equation, which are the values of x that make the equation true.

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There are commonly two answers to quadratic equations because the answers are where the parabola crosses the x -axis.

Quadratic equations are equations in two variables whose graph is a U-shaped parabola that is symmetrical about a vertical line, called the axis of symmetry. The standard form of a quadratic equation is [tex]y = ax^2 + bx + c[/tex], where [tex]a,b[/tex] and [tex]c[/tex] are real numbers, and x and y represent all (x,y) pairs that satisfy the equation. The solution(s) to a quadratic equation is the point(s) where the parabola crosses the x-axis. A quadratic expression can be written as the product of two linear factors and each factor can be equated to zero, So there exist two solution.

There are commonly two answers to quadratic equations because the answers are where the parabola crosses the x -axis.

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Step-by-step explanation:

∠1 + 49° + 49° = 180°

∠1 = 180-98

Value of  [tex]1^{3} + 10^{3} + 11^{3} +12^{3} +......+30^{3}[/tex] is "216225".therefore the correct option is C.

Natural Numbers are the counting numbers that start from 1 and go on till infinity. To find the sum of cube of first n natural numbers means to add the cube of a specific number of natural numbers starting from 1 and get the answer.

There are a lot of patterns in mathematics. One analogous pattern of numbers is the sum of cube of n natural numbers.

Given:  Equation  = [tex]1^{3} + 10^{3} + 11^{3} +12^{3} +......+30^{3}[/tex]

            Here  N=30  , We have given

solution:

                                              [tex]S=\frac{{n^{2} (n+1)^{2}}}{4} \\=\frac{30^{2} (30+1)^{2}}{4}\\=\frac{900 (31)^{2}}{4}\\=\frac{900 (961)}{4}\\=\frac{864900}{4}\\=216225[/tex]

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The area of the shaded region of the figure composed of a trapezoid and triangle is 26.25 m².

The figure shown is composed of a triangle within a trapezoid. The area of the shaded portion can be found as follows:

The area of the shaded portion is the subtraction of area of the triangle from the area of the trapezoid.

Hence,

area of the shaded portion = area of the trapezoid - area of the triangle

area of the trapezoid  = 1 / 2 (5 + 9)6

area of the trapezoid  = 1 / 2 (14)6

area of the trapezoid  = 42 m²

area of the triangle = 1 / 2 × 7 × 4.5

area of the triangle = 1 / 2 × 31.5

area of the triangle = 15.75 m²

Therefore,

area of the shaded portion = 42 - 15.75

area of the shaded portion = 26.25 m²

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The solution of the given system of equations 3x + 6y = 2 ,  15x -30y = 10 is equal to x = 2/3 and y = 0.

As given in the question,

Given system of equations are :

3x + 6y = 2      ____(1)

15x -30y = 10   _____(2)

Multiply equation (1) by 5 and add it to equation (2) to eliminate y and get the value of x we have,

15x + 30y = 10

15x - 30y = 10

30x      = 20

⇒x = 20 / 30

⇒ x = 2 / 3

Substitute value of x = 2/3 in equation (1) to get the value of y,

3(2/3) + 6y = 2

⇒ 2 + 6y = 2

⇒ 6y = 0

⇒ y = 0

Therefore, the solution of the given system of equations is equal to x = 2/3 and y =0.

The above question is incomplete, the complete question is:

What is the solution of the system of equations 3x + 6y = 2 ,  15x -30y = 10?

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The integer set that makes the inequality 3p - 10 > 7p + 6 true is given as follows:

{- infinity, ..., -6, -5}.

The inequality in this problem is defined as follows:

3p - 10 > 7p + 6

To solve the inequality, we treat it similarly to an equality, isolating the variable of interest. The difference is that the solution is a set with infinity values, and not a single value.

Hence the solution of the inequality is obtained as follows:

3p - 10 > 7p + 6

-4p > 16

4p < -16 (multiplying by -1 due to the negative coefficient of x, the sign also changes)

p < -16/4

p < -4.

Hence the integer set of the solution of the inequality is given as follows:

{- infinity, ..., -6, -5}.

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The system of linear equations has no solution. Then the number of the solution of the equations will be zero.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

The equations are given below.

y = 0 and y = - 7

The lines are parallel to each other. Then they will never intersect with each other.

The system of linear equations has no solution. Then the number of the solution of the equations will be zero.

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Answer:

(a) similar

Step-by-step explanation:

20/10 = 1/2

8/4 =1/2

22/11 =1/2

The sides are proportional in the same ratio hence they are similar triangles

Answer:

a

Step-by-step explanation:

it is not D because if it was a replica it would be exactly the same. If it was a conjourent than it would be if all three corresponding sides are equal and all the three corresponding angles are equal in measure. so it leaves similar

To identify corresponding parts of congruent triangles, you can use the transformations of translation, rotation, and reflection.

Translation involves moving a figure a certain distance in a certain direction, and all parts of the figure move the same distance in the same direction. If you translate two congruent triangles, their corresponding parts will still be corresponding.

The term "trigon" is occasionally (though not very frequently) used to refer to a triangle, a 3-sided polygon. Three sides and three angles, some of which may be the same, characterize every triangle.

The term "acute triangle" refers to a triangle with all three angles being smaller than. The term "obtuse triangle" refers to a triangle with any one angle greater than. The terms "right triangle" and "right-angled triangle" both refer to triangles with at least one angle of.

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The length of the side of the smaller box is 6 and the length of the side of the larger box is 20. The equation for the difference in volume of the boxes is, y3 - x3 = 7784 .

Let us consider the length of side of small box be x inch & length of large box be y inch.

Length of smallest box is 30% of the length of the side of the largest box 

x=30%*y

   = 30 * y / 100

x = 3y / 10

Volume of the smallest box is x3.

volume of the largest box is y3.

Volume of largest box is 7784 cubic inches more than the volume of smallest box. Therefore,

y3 =  x3 + 7784

     = (3y/ 10 )3 + 7784

     = 27y3 / 1000 + 7784

y3 - 27y3 / 1000 = 7784

973 y3  / 1000 = 7784

973 y3 = 7784 * 1000

y3 = 7784 * 1000 / 973

y3 = 8000

   y = 20

Now find the smaller side.

x = 3y / 10

putting the value of y  this we get,

x = 3 * 20 / 10

   x= 6    

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In mathematics the expression of i² refers to or has a value equal to -1

We are going to define complex numbers, since "i" is a letter that denotes that it is an imaginary number.

Complex numbers are known as imaginary numbers and can be defined as an extension of the set of real numbers, since in its structure it has a real number followed by an imaginary number denoted with the letter "i".

The value of "i" starts from the square root of (-1)

√(-1) = i

Then if we want to know the value of i squared we do the following:

√(-1) = i

[√(-1)]² = i²

(-1) = i²

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The value of the given expression ( 402 ) ^ (1/2) using binomial theorem is equal to option a. 20.05.

As given in the question,

Given expression is equal to :

( 402 ) ^(1/2)

Standard form of binomial theorem is:

( a + b )ⁿ = ⁿC₀aⁿb⁰ +  ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² +..............+  ⁿCₙa⁰bⁿ

Rewrite the 402 = 400 + 2 and represent it in binomial theorem form:

( 400 + 2 ) ^(1/2)

Here take out 400 as common factor :

= (400)^(1/2) ( 1 + 2/400)^(1/2)

= 20 ( 1 + 1/200) ^(1/2)

Using one of the result of binomial theorem to get the approximate value if other values are extremely small.

( 1 + x )ⁿ = 1 + nx

Here n = 1/2 and x = 1/ 200

= 20 [ 1 + (1/2) (1/200)]

= 20 ( 1 + 1 /400)

= 20 ( 401/400)

= 401 / 20

= 20.05

Therefore, the value of (402)^(1/2) using binomial theorem is equal to option a. 20.05.

The above question is incomplete , the compete question is:

The value of (402)^(1/2) according to the binomial theorem is

a.  20.05        b. 20.01         c. 20.00          d.  20.2​

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Answer:

80π feet

Step-by-step explanation:

The question is asking us for the circumference of a circle with diameter 80 ft.

The formula for the circumference of a circle is πd. Therefore, the circumference is π * 80 = 80π ft!

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Answer:

The circumference of a circle is equal to the diameter times pi, which is approximately 3.14. Therefore, the distance of one complete loop on a circular path with a diameter of 80 feet is approximately 80 * 3.14 = 251.2 feet.

Step-by-step explanation:

The vertical asymptotes for the function f(x) = (x^2)(x-6)(x+3)^-1 are x=6 and x=-3.

The vertical asymptotes for the function f(x) = (x^2)(x-6)(x+3)^-1 are x=6 and x=-3. This is because the denominator can be factored into two linear expressions: x-6 and x+3. When either of these expressions equals 0, the fraction becomes undefined and a vertical asymptote is created. To find the asymptotes, set each linear expression in the denominator equal to 0 and solve for x. For x-6, x=6; for x+3, x=-3. Therefore, the vertical asymptotes of the function are x=6 and x=-3. When x is close to either of these values, the fraction will become very large, making the graph approach the vertical asymptotes without ever touching them.

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The value of p is 60°.

A triangle is a polygon with three sides and three vertices.

Sum of all the angles of the triangle is 180°.

Given, the triangle ABC;

∠A = p +  49° + ∠CAB = 180°  

∠CAB = 180° - p - 49°

And the sum of all the angles of the triangle is 180°.

∠CAB + ∠CBA + ∠BCA = 180°

180° - p - 49° + p - 15° + p -16° = 180°

p = 60°

Therefore, Evaluation of p is 60°

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An employee earns $14 per hour and a trainer charges $20 for every 30 minutes of training plus a $5 tip are a constant rate per unit interval. Therefore, options A and E are the correct answers.

Given that:

an employee earns $14 per hour.

constant rate of change :

When something has a constant rate of change, one quantity changes in relation to the other. For example, for every half hour the pigeon flies, he can cover a distance of 25 miles. We can write this constant rate as a ratio. For ratios, it's always a good idea to state both units in whole terms.

A). Let the number of hours be x and the total money earned be y

So, y = 14x

B). Let the money earn by an employee each year be a and total earning in a year be b.

So, b=a+3% of a

⇒ b = a + 0.03a

⇒ b = 1.03a

C) Let the number of months be m and the total charge be t

So, t=50+30m

D) Let cost of gym equipment be p, rate of change =8% and time period = 6 months

So,

E) Let the number of hour be x.

Here, 30 minutes = 1/2 hours

Total charge = 20 + 5x/2

An employee earns $14 per hour and a trainer charges $20 for every 30 minutes of training plus a $5 tip are a constant rate per unit interval. Therefore, options A and E are the correct answers.

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Full question :

For which situation does one quantity change at a constant rate per unit interval relative to another quantity? responses.

Select all that apply

A. An employee earns $14 per hour.

B. An employee earns a 3% raise each year.

C. A gym charges a $50 down payment plus $30 per month.

D. A piece of gym equipment decreases 8% in value every 6 months.

E. A trainer charges $20 for every 30 minutes of training plus a $5 tip.