A guy wire runs from the top of a cell tower to a metal stake in the ground. Sophia places a 11-foot tall pole to support the guy wire. After placing the pole, sophia measures the distance from the stake to the pole to be 4 ft. She then measures the distance from the pole to the tower to be 32 ft. Find the length of the guy wire, to the nearest foot.

The value of k where the roots of the quadratic equation are real and equal is k = 9.

Option B is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

3x + 6 = 9 is an equation.

We have,

3x² + 2kx + 27 = 0

This is in the form of ax² + bx + c.

a = 3

b = 2k

c = 27

Now,

The roots are real and equal.

D = 0

D = b² - 4ac

Now,

b² - 4ac = 0

(2k)² - 4 x 3 x 27 = 0

4k² - 324 = 0

4k² = 324

k² = 324/4

k² = 81

k = √81

k = ±9

Thus,

The value of k is 9 0r -9

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As per the Maclaurin series, the value of the function is written as 2.8521

The term Maclaurin series is defined as  a function that has expansion series that gives the sum of derivatives of that function.

Here we have given that the function is [tex]\sum\frac{(-1)^n\pi^{2n}}{4^{2n}(2n)!}[/tex]

And we need to find the sum of series by using the Maclaurin series as

Now, the sum of series is calculated as,

=>f(n) =  [tex]\sum\frac{(-1)^n\pi^{2n}}{4^{2n}(2n)!}[/tex]

When we put the value on n = 0, then we get,

=> f(0) = [1 x 3.14]/1] =  3.14

Now, we have to put the value of the function, the value of x as 1,

Then we get,

=> f(1) = [-1 x 9.8596] / [16 x 2]

=> f(1) = -0.308

And then we have to put the value of function for the value of x as 2,

Then we get,

=> f(2) = [1 x 30.959] / [64 x 24]

=> f(2) = 0.0201

Then the sum of the series is

=> 3.14 - 0.308 + 0.0201

=> 2.8521

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Prakash's total score in math and science is (x+90)marks

This word problem will need as to write what we have been given and analyze and come up with an equation to solve it

We are given that Prakash's score in science is = x marks

and that Prakash's score in mathematics = 90 marks

Therefore to get the total score in both math and science will we have to add the score Prakash scored in math and the score Prakash scored in science:

This will translate to (x) +(90)

We will get the total as x+90

hence the total score will be x+90 marks

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Since the provided expression [tex]2x^{2} +8x-x^{2} +16[/tex]  is quadratic, the solution of x are "-4" and "-4" .

Any equation in algebra that can be written in standard form as where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation.

An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles. An expression in mathematics is made up of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) You may think of expressions as being comparable to phrases.

Expression:    [tex]2x^{2} +8x-x^{2} +16[/tex]  

This means that you need to add the two x² terms and the two x terms, which gives you [tex]x^{2} +8x+16[/tex].

Next, you can simplify the equation by taking out a common factor of x from the consecutive two terms, which gives you  [tex]x(x + 4) +4(x+4).[/tex]

Then the equation becomes [tex](x + 4)(x+4).[/tex] which on solving i.e [tex](x+4)=0[/tex]

you will get x= -4 and -4.

Therefore the solution to this equation is x = -4 or x = -4.

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The linear function rule to model the distance d in feet of the car traveled after t seconds is d = 83t.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

The distance traveled after 9 seconds = 747 ft.

The function rule to model the distance d after t seconds.

d = 83t

When t = 9,

d = 83 x 9 = 747 ft

Thus,

The linear function is d = 83t.

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Answer:

The greatest common monomial factor of [tex]3x^2 - 6x[/tex] is 3x.

Step-by-step explanation:

To understand what the problem is asking us, let's define the terms.

⭐What does greatest common factor mean?

⭐What does monomial mean?

Thus, we need to find the greatest term that is a factor of [tex]3x^2 -6x[/tex]. This term must be a monomial.

First, divide both terms of the polynomial [tex]3x^2 -6x[/tex] by the largest term that is common between the terms.

[tex]3x[/tex] is the largest term.

[tex]3x[/tex] is also a monomial.

Therefore, [tex]3x[/tex] is the greatest common monomial factor.

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Answer: x=-3, y=2

Step-by-step explanation: to solve using substitution, we need either x or y by itself in one equation. let's use x from the 2nd equation since it doesn't have a constant.

so the first equation stays the same at 2x+3y=0

for the second equation, rearrange to solve for x by moving the 2y to the right -> x=2y-1

then, we can plug in this x value to x in the first equation to get,

2(2y-1)+3y=0

then solve for y,

4y-2+3y=0

4y-3y=2

y=2

then to solve for x, plug in y in either of the initial equations. lets choose the second one.

so you'd have,

x+2(2)=1

x+4=1

x=-3

Answer: x-intercepts

Step-by-step explanation:

basic math

Answer: X intercept

Step-by-step explanation:

The x-intercept is the point at which a graph crosses the x-axis. The y-intercept is the point at which the graph crosses the y-axis. To find the x-intercept of a line, substitute 0 for y into the equation and solve for x.

Answer:

Step 2

Step-by-step explanation:

3x-24=5x

he did -3x on each side but added 3x to 5x instead of subtracting

Answer: 1.5 and 20

Step-by-step explanation:

Answer:

Step-by-step explanation:

28/4 = 7

40/4 = 10

Then:

28:40 in simplest terms is:

7:10

Answer:

[tex] \sf \: 7 : 10 [/tex]

Step-by-step explanation:

Now we have to,

→ simplify ratio into simplest form.

The given ratio,

→ 28 : 40

Let's simplify the given ratio,

→ 28 : 40

→ (28) ÷ 4 : (40) ÷ 4

→ (7) : (10)

→ 7 : 10

Therefore, the ratio is 7 : 10.

Answer: Angle C

Step-by-step explanation:

The length of C-T and T-B would be longer then C-B and push C-B out making angle C the biggest.

To find potential rational roots, we make use of the rational root theorem

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

Task: How the potential rational root is calculated

To make use of the potential rational root, we use the rational root theorem using the following steps

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Answer:

----------------------

Given options:

A) f(x) = 2x² + 5x − 12,

B) f(x) = 2x + 5,

C) f(x) = 2(x + 5)² − 12,

D) f(x) = 2(x + 5)(x − 12),

Answer:

∠MGN = 24°

∠NGO = 24°

Step-by-step explanation:

⭐What is an angle bisector?

Thus, ∠MGN ≅ ∠NGO.

To solve for ∠MGN and ∠NGO, we need to:

1. Create an equation about their relationship:

If ∠MGN ≅ ∠NGO, then...

[tex]5x-6 = 11x -42[/tex]

2. Solve for "x":

[tex]5x - 6 = 11x -42[/tex]

[tex]42 - 6 = 11x - 5x[/tex]

[tex]36 = 6x[/tex]

[tex]6 = x[/tex]

3. Substitute "x" back into the equations to find the angle measures:

[tex]< MGN = 5x - 6[/tex]

[tex]< MGN = 5(6) - 6[/tex]

[tex]< MGN = 30 - 6[/tex]

[tex]< MGN = 24[/tex]

❗❗❗At this point, you could write that ∠MGN and ∠NGO = 24°, as they are congruent, but you should still substitute "x" into  ∠NGO to make sure "x" is correct.❗❗❗

[tex]< NGO = 11x - 42[/tex]

[tex]< NGO = 11(6) - 42[/tex]

[tex]< NGO = 66 - 42[/tex]

[tex]< NGO = 24[/tex]

∴ ∠MGN = 24°

∴ ∠NGO = 24°

Answer:

The solution on a graph is the point where two lines interesect

Step-by-step explanation:

The slope of equation 3x-4y = 5 is 3/4.

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

Slope-intercept form:

y = mx+b,

where m is the slope and b is the y-intercept.

Given equation:

3x-4y = 5

To find the slope:

First, we simplify the equation.

-4y = 5 -3x

Divide by -4.

(-4/-4)y = 5/-4 + (-3/-4)x

y = (3/4)x -5/4

Now, we know the slope-intercept form:

y = mx+b,

where m is the slope and b is the y-intercept.

Comparing, both the equation,

we get,

m =3/4.

So, the slope is 3/4.

Therefore, the slope is 5/3.

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Yes, 3x²A5y is a polynomial.

Polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables

we need to check whether 3x²A5y is a polynomial.

An expression is combination of variables, numbers and operators.

For an expression to be a polynomial term, it must contain no square roots of variables, no fractional or negative powers on the variables, and no variables in the denominators of any fractions.

3x²A5y is a polynomial.

Hence,  3x²A5y is a polynomial.

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Answer:

8/15

Step-by-step explanation:

There are 15 equal-sized sectors:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

The odd ones are:

1, 3, 5, 7, 9, 11, 13, 15

There are 8 odd numbered sectors.

p(event) = (number of desired outcomes)/(total number of possible outcomes)

p(odd) = 8/15

The greatest product that can be obtained by multiplying two of these 36 integers is; 306

We are told that when 36 consecutive integers are added, the product is -18.

Now, the sequence for these integers would be;

x, (x + 1), (x + 2), (x + 3),......(x + 35)

Now, since the addition of these integers gives a negative number, then it suggests to us that the sequence starts with a negative integer and then ends with a positive integer.

Thus, we have;

x + (x + 1) + (x + 2) +(x + 3) + (x + 35) = -18

The formula for sum of an arithmetic sequence is;

Sₙ = (ⁿ/₂)(2a + (n - 1)d)

where a is first term and d is common difference

Thus;

-18 = (36/2)(2x + (36 - 1)1)

-18 = 18(2x + 35)

-18/18 = 2x + 35

2x + 35 = -1

x = -34/2

x = -17

Thus;

The first term = -17

Last term = -17 + 35 = 18

Thus, the greatest product that can be obtained  = 18 * 17 = 306

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Answer:

306

Step-by-step explanation:

The required median of the given data set is 8.

Finding the middle number requires sorting all the data points and choosing the center one (or if there are two middle numbers, taking the mean of those two numbers).

Example: When the numbers are arranged (1, 4, 7), the number 4 lies in the middle, making it the median of 4, 1, and 7.

So, we have the data:

4, 8, 10, 5, 9

Arrange in ascending order as follows:

4, 5, 8, 9, 10

Since 8 is the number in the middle then it is the median.

Therefore, the required median of the given data set is 8.

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The nature of the roots of a quadratic equation is described by the discriminant calculated by seeing the equation and comparing it with the standard equation which is given by -

[tex]ax^{2} + bx+c[/tex] , where a , b , c are real numbers .

The root of the equation is given by -

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

Here, the term (b^2 - 4ac) is called the discriminant of the quadratic equation .

This discriminant is the defining factor of the nature of the roots.

Now, here arise three cases-

(i) If (b^2 - 4ac) > 0, the roots are real and distinct .

(ii) If (b^2 - 4ac) = 0 , the roots are real and equal.

(iii) If (b^2 - 4ac) < 0 , the roots are not real and they are imaginary i.e. they are complex in nature and of the form a+ib.

Hence, it is the discriminant which determines the nature of roots.

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A symbol (often a letter) used in mathematics to represent an unknowable numerical value in quite an equation or algebraic statement.

Example:

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The coefficient of x in the expression 4x+ 3y is 4. This means that for every x there is 4 times as much as for every y. The coefficient of x is the number that is multiplied by x in the expression.

In the expression 4x+ 3y, the coefficient of x is 4. This means that the x is multiplied by 4. The coefficient of a term is the number that is multiplied by the variable. In this expression, the x is multiplied by 4 and the y is multiplied by 3. This means that for every x there is 4 times as much as for every y. In order to find the coefficient of x in the expression, we must look at the number that is multiplied by x. In this case, the number is 4. This means that the coefficient of x is 4. For example, in this expression, the coefficient of x is 4 which means that for every x there is 4 times as much as for every y.

4x + 3y

Coefficient of x = 4

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The value of g'(t) is  ([tex]e^t[/tex], -sin t, cos(t)).

In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.

Given:

f(x, y, z) = xz + yz + xy,

g(t) = ([tex]e^t[/tex], cos(t), sin(t))

Now, differentiating g(t) w.r.t t

g'(t) =  ([tex]e^t[/tex], -sin t, cos(t))

Hence, the g'(t) =  ([tex]e^t[/tex], -sin t, cos(t)).

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x + 2y = 12 and x = 4y represents the given word problem in equations

How to form an equation for a word problem?

Here are some steps to follow:

1. Understand all the words used in stating the problem. Understand what you are asked to find.

2. Translate the problem to an equation.

Assign a variable (or variables) to represent the unknown. Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Use substitution, elimination or graphing method to solve the problem.

According to the given question:

Isaiah scored a total of 12 points. In this game, he made some free throws (worth one point each) and some two point shots.

Let number of free throws be x and number of two point shots be y.

x(1) + y(2) = 12

x + 2y = 12

This equation defines the total number of points scored with free throws and two point shots.

He made 4 times as many free throws as two point shots.

Using the pre defined variables to form the equation as

x = 4y

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Answer:

Step-by-step explanation:

Square's mission statement is simple. Square deliver “Economic Empowerment” to the thousands of business owners who use Square to manage their businesses.

The correct statement regarding the proportional relationships is given as follows:

B. Karleigh ran a greater distance. The slope of the line described by the data in her table increased at a rate of 1/10, compared to 1/12 for Andrew.

To obtain the person who runs faster, we need to look at their rates, as the time and the distance form a proportional relationship.

Andrew ran 1.5 miles in 18 minutes and 2 miles in 24 minutes, hence his rate is given as follows:

k = 1.5/18 = 2/24 = 1/12 = 12 minutes per mile.

Karleigh ran 3 miles in 30 miles and 4 miles in 40 minutes, hence her rate is given as follows:

k = 3/30 = 4/40 = 1/10 = 10 minutes per mile.

Due to the lower number of minutes per mile, Karleigh ran faster, and the correct option is given by option B.

Andrew ran 1.5 miles in 18 minutes and 2 miles in 24 minutes.

Karleigh ran 3 miles in 30 miles and 4 miles in 40 minutes.

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