Write an algebraic expression that models each situation.
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The length of the unknown leg is 8 yd in the given right triangle.

The Pythagoras theorem states that a right-angled triangle's hypotenuse square (90 degrees) equals the sum of the squares of its other two sides.

Let the three legs of the right triangle be a, b and c where a is the altitude, b is the base and c is hypotenuse of the given right angled triangle.

Here, a = [tex]\sqrt{23}[/tex] yd, b = ? and c = [tex]\sqrt{87}[/tex]

According to the Pythagoras theorem,

Hypotenuse² = Base² + Altitude²

c² = b² + a²

[tex](\sqrt{87})^{2}[/tex] = b² + [tex](\sqrt{23})^{2}[/tex]

87 = b² + 23

Subtracting 23 from both sides of the equation.

87 - 23 = b²

b² = 64

Taking square root of the both sides.

b = ± 8

Since the value of length can't be negative so the value of b = - 8 is rejected.

Therefore, the value of b = 8 yd.

As we have calculated above, the length of the unknown leg is 8 yd.

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The set notation will be S ∈ { 14, 16, 18, 20, 22, 24, 26, 28}.

When we have to represent set having defined specific number, we do my writing the number in curly bracket. When we have to represent the set having collection of continuous number between two specific defined number, we do by writing first and last number in small bracket.

As the terms given for the set is 14, 16, 18, 20, 22, 24, 26, 28 which are defined and specific terms so it will be represented by writing it in curly brackets

i.e. S ∈ { 14, 16, 18, 20, 22, 24, 26, 28}

Final answer,  the set will be S ∈ { 14, 16, 18, 20, 22, 24, 26, 28}

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Answer: v= root[3]  {12}=2.2894

Step-by-step explanation:

Answer:Your answer is V=4

Step-by-step explanation:Need brainliest award

The events are independent. The probability of getting 5 each dice after rolling the dice 2 times is 1/36.

As we are rolling two different die simultaneously, the events are independent.

The probability of getting a 5 in the first die = 1/6

The probability of getting a 5 in the second die = 1/6

Hence, as the events are independent, we can write :-

The probability of getting 5 each dice after rolling the dice 2 times =

probability of getting a 5 in the first die * probability of getting a 5 in the second die = (1/6)*(1/6) = 1/36

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Answer: r=-3

Step-by-step explanation:

12-6r=2r+36

12=8r+36

-24=8r

r=-3

What is the value of Σ⁵n=1 (2 n-3) ? 15

             [tex]\sum_{n=1}^{5} (2 n-3)\\\\ (2 (1)-3)+ (2 (2)-3)+ (2 (3)-3)+ (2 (4)-3)+ (2 (5)-3)\\\\ -1+1+3+5+7=15[/tex]

The sum of the given expression is 15

What is summation?

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Answer: 3.162

Step-by-step explanation:

Answer:

√10 should be ~3.16

Answer:

How should this model be completed to represent 820 ÷ 17? Enter your answers by filling in the boxes. I'm in 5the BTW

The equation of to represent the two unknowns y and x is undefined hence it can be said not to be existing.

Equation of a straight line is the equation of the form

y = mx + c

y and x are the unknown variables

m is the slope

c is the intercept in the y axis

The equation given in the problem is a special type which is a vertical line

A vertical line has no intercept on the y axis hence c =0/ This reduces the equation of a straight line to

y = mx + c

at c = 0 we have:

y = mx + 0

y = mx

The slope of the vertical line is undefined. From first principles we have that slope is

m = change in y / change in x

for a line to be vertical the change in x is at the lowest absolute value and the smallest absolute number is zero. note negative number is not inclusive.

Therefore the slope at change in x = 0 represents a vertical line which is

m = change in y / change in x

m = change in y / 0

m = undefined

so y = undefined

check using calculator any number divided by zero gives error which is defined mathematically to be undefined,

We can conclude that the relationship between y and x from the graph is undefined.

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The probability of getting a number greater than 1 is 5/6

What is the formula of Probability?

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Probability of event happen P(E) = Number of favourable outcomes / Total number of outcomes.

The possible outcomes when a die thrown is {1, 2, 3, 4, 5, 6}. Therefore, the number of possible outcomes of a die is 6.

Now, the numbers greater than 1 between 1 to 6 are P{2, 3, 4, 5, 6} then the number of favourable outcome is 5.

Therefore, probability P of getting a number greater than 1 is:

P = 5/6

Hence, the probability of getting a number greater than 1 is 5/6

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[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

By Pythagoras theorem :

[tex]\qquad \sf  \dashrightarrow \: hypotenuse = \sqrt{(perpendicular) {}^{2} + (base) {}^{2} } [/tex]

[tex]\qquad \sf  \dashrightarrow \: h= \sqrt{15 {}^{2} + {8}^{2} } [/tex]

[tex]\qquad \sf  \dashrightarrow \: h = \sqrt{225 + 64} [/tex]

[tex]\qquad \sf  \dashrightarrow \: = \sqrt{289} [/tex]

[tex]\qquad \sf  \dashrightarrow \: h = 17 \: \: units[/tex]

Now, we know ~

[tex]\qquad \sf  \dashrightarrow \: \sin(A) = \dfrac{opposite \: \: side}{hypotenuse} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \sin(A) = \dfrac{8}{17} [/tex]

Answer:

True

Step-by-step explanation:

Its true because the absolute value of integers is either its positive side or negative side. For example -4 turn into 4 and -5 turn into 5. Hope this helps!

Answer:

r=2

Step-by-step explanation:

To find the slope, we take the difference in the y values over the difference in the x values

m = ( y2-y1)/(x2-x1)

2 = ( 9-3)/ (5-r)

2 = (6)/(5-r)

Multiply each side by (5-r)

2 ( 5-r) = 6

Divide each side by 2

5-r = 6/2

5-r = 3

Subtract 5 from each side

5-r-5 = 3-5

-r = -2

r = 2

Answer:

r = 2

Step-by-step explanation:

To solve this problem, we have to use the following formula for slope:

[tex]\boxed{m= \frac{y_2 - y_1}{x_2 - x_1}}[/tex],

where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are the coordinates of two points.

We are given the coordinates [tex](r, 3)[/tex] and [tex](5, 9)[/tex], and told that [tex]m = 2[/tex]. To find the value of r, we have to substitute the given values into the formula and then solve for r :

[tex]2 = \frac{9 - 3}{5-r}[/tex]

⇒ [tex]2 = \frac{6}{5-r}[/tex]

⇒ [tex]2\times (5-r) = \frac{6}{5-r} \times (5 - r)[/tex]         [Multiplying both sides by (5 - r)

⇒ [tex]2(5-r) = 6[/tex]

⇒ [tex]\frac{2}{2}(5-r) = \frac{6}{2}[/tex]                                [Dividing both sides by 2]

⇒ [tex]5- r = 3[/tex]

⇒ [tex]5 - r - 5 = 3 - 5[/tex]                         [Subtracting 5 from both sides]

⇒ [tex]-r = -2[/tex]

⇒ [tex]\frac{-r}{-1}= \frac{-2}{-1}[/tex]                                       [Dividing both sides by -1]

⇒ [tex]r = \bf 2[/tex]

Answer: 95%

Step-by-step explanation:

To find the percentage, we divide the number of students who chose red by the total number of students and then multiply by 100.

[tex]\frac{285}{300} \times 100=95[/tex]

The solution of the given linear equation is y = 80.

According to the given question.

We have an equation.

-y + 13 = -67

Since, we have to solve the above linear equation in variable.

As we know that, the linear equations in one variable is an equation which is expressed in the form of ax+b = 0, where a and b are two integers, and x is a variable and has only one solution.

Thereofre, the solution of the linear equation in one variable -y + 13 = -67 is given by

-y + 13 = -67

⇒ - y = -67 -13 (subtractting 13 from both the sides)

⇒ -y = -80

⇒ y = 80

Hence, the solution of the given linear equation is y = 80.

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The formula for finding any term of the given sequnce is [tex]a_{n}[/tex]  = n^3.

According to the given question.

We have a sequence

1, 8, 27, 64, 125

As we know that, a sequence or number pattern is an ordered set of numbers or diagrams that follow a rule.

The above of given sequence 1, 8, 27, 64, 125 is following a rule i.e each term is the cube of its number of term.

Here,

The first term is 1, so the cube of 1 is 1.

The second term is 8, so the cube of 2 is 8.

So, the nth term of the given sequnce is n^3.

Therefore, the formula for finding any term of the given sequnce is given by

[tex]a_{n}[/tex]  = n^3

Where, n = 1, 2, 3, 4...

So, for n = 1

The first term will be 1.

for n = 2

The second term will be 8

for n= 3

The fourth term will be 64.

Hence, the formula for finding any term of the given sequnce is [tex]a_{n}[/tex]  = n^3.

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The final amount when $10,000 is invested for 3 years at 3% the interest is compounded continuously is  $10,942

What is compound interest ?

A compound interest rate is a rate of interest calculated on both the amount saved and the interest earned on it.

In the given question the expression to calculate amount is given as :

A = Pe^(rt)

where,

P = principle amount that is $10,000

r = rate of interest that is 3%

t = time period that is 3 years

Now, calculating Amount by substituting the above value in the formulae :

A = Pe^(rt)

A = 10000 e^(0.03 x 3)

A = 10000 e^(0.09)

A = 10000 x 1.0942

A = $10,942

Therefore, the final amount when $10,000 is invested for 3 years at 3% the interest is compounded continuously is $10,942

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The probability that the number spun is a 3 given that it was less than 4 is 1/6.

Let A is the event of a number less than 4.

P(A) is the probability of getting a number less than 4.

The total number less than 4 is 3

Here, the total numbers in the game are 6.

By the Formula for the probability of an event E can be observed as:

P(A)= no of favorable cases/total no. of cases  = 3/6 = 1/2

B is the event of getting the number  3

P(B) be the probability of getting 3

The total  possible cases  is 1

Therefore , P(B)= 1/6

Now, P(B/A) is the probability of getting 3 which is an odd number.

P(B/A)= P(A ∩ B) / P(A)

          =1/6 / 1/2

   = 1/3

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Answer:

x = 21, -21

Step-by-step explanation:

Let's solve the problem,

→ |x/7| = 3

→ x/7 = 3 ll -x/7 = 3

→ x = 3 × 7 ll -x = 3 × 7

→ [ x = 21 ] ll [ x = -21 ]

So, the value of x is 21, -21.

Step-by-step explanation:

[tex]x_{1,\:2}=\frac{-6\pm \sqrt{6^2-4\cdot \:1\cdot \left(-16\right)}}{2\cdot \:1}\\\sqrt{6^2-4\cdot \:1\cdot \left(-16\right)}\\\sqrt{6^2+64}\\\sqrt{36+64}\\\sqrt{100}\\\sqrt{10^2}\\=10\\x_1=\frac{-6+10}{2\cdot \:1},\:x_2=\frac{-6-10}{2\cdot \:1}\\\\\bold{x_1:2}\\\frac{-6+10}{2\cdot \:1}\\\frac{4}{2\cdot \:1}\\\frac{4}{2}\\=2\\\\\bold{x_2:-8}\\\frac{-6-10}{2\cdot \:1}\\\frac{-16}{2\cdot \:1}\\\frac{-16}{2}\\-\frac{16}{2}\\=-8[/tex]

Answer:

x₁ = 2

x₂ = -8

The slope of the line is 0 since the y-value doesn't change, meaning it is a horizontal line.

Answer:

Slope ⇒ 0

Step-by-step explanation:

The formula to find the slope of a line is:

[tex]m = \frac{y_1-y_2}{x_1-x_2}[/tex]

We can use the given two coordinates of the line to find the slope.

( 1, -7 ) ⇒ ( x₁, y₁ )

( -2, -7 ) ⇒ ( x₂, y₂ )

Let us solve this now.

[tex]m = \frac{y_1-y_2}{x_1-x_2}\\\\m = \frac{-7 -(-7)}{1-(-2)}\\\\m = \frac{0}{3}\\\\m=0[/tex]

Answer:

Pencils = 35

Notebooks = 15

Step-by-step explanation:

15 cents = 15/100 = $0.15

85 cents = 85/100 = $0.85

p + n = 50                    Eq. 1

0.15p + 0.85n = 18       Eq. 2

p = number of pencils

n = number of notebooks

From Eq. 1

p = 50 - n               Eq. 3

Replacing Eq. 3 in Eq. 2:

0.15(50-n) + 0.85n = 18

(0.15*50 + 0.15*-n) + 0.85n = 18

(7.5 - 0.15n) + 0.85n = 18

7.5 + 0.7n = 18

0.7n = 18 - 7.5

0.7n = 10.5

n = 10.5 / 0.7

n = 15

From Eq. 3:

p = 50 - n

p = 50 - 15

p = 35

Check:

From Eq. 2

0.15p + 0.85n = 18

0.15*35 + 0.85*15 = 18

5.25 + 12.75 = 18

e

Answer: 8x^2 + 55x + 42

Step-by-step explanation:

(x+6)(8x+7)=

(8x*x)+(7x)+(6*8x)+(6*7)=

8x^2 + 7x + 48x + 42=8x^2 + 55x + 42

The given pattern 5/7, 8/7, 11/7, 2, ....... is in the form of the arithmetic progression AP. The next 3 terms are  17/7, 20/7, 23/7.

An arithmetic progression is a sequence whose terms continue to increase and otherwise decrease by a constant number. The common difference is the set amount through which they either increase or decrease.

The following arithmetic progression formulas are frequently utilized to solve various AP problems for the initial term 'a' of an AP and the common difference 'd'-

Now, as per the stated question;

The AP given as; 5/7, 8/7, 11/7, 2, ...........

The series consists of four given terms.

Consider the initial term be 'a₁' = 5/7.

Then, the second term be 'a₂' = 8/7.

And, the third term be 'a₃' = 11/7.

And, the fourth term is 'a₄' = 2.

The AP have the same common difference. so,

d = a₃ - a₂

Substitute the values.

d = 11/7 - 8/7

d = 3/7

Thus, the common difference is 3/7.

or d = a₄ - a₃ (Put the values)

d = 2 - 11/7

d = 3/7

As, a₃ - a₂ = a₄ - a₃

Thus, we can conclude that the given sequence is in AP.

The fifth term will be; a₅ = a₄ + d = 2 + 3/7

a₅ = 17/7

Now, the 6th term will be; a₆ = a₅ + d = 17/7 + 3/7

a₆ = 20/7

Similarly, the 7th term will be; a₇ = a₆ + d = 20/7 + 3/7

a₇ = 23/7

Therefore, it can be said that the given sequence is in AP next three terms be 17/7, 20/7, 23/7.

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The science and engineering practices involves studying graphs directly in order to develop initial meaning is  c) Analyzing and Interpreting data

Analyzing and interpretation provide answers to the research questions postulated in the study.

Science and engineering practices that involves studying graphs directly in order to develop initial meaning

Analysis means the ordering, manipulating and summarizing of data to obtain answers to research questions. Its purpose is to reduce data to intelligible and interpretable form so that the relations of research problem can be sudden and tested.

Interpretation gives the results of analysis, makes inferences pertinent to the research relations studied, and draws conclusions about these relations.

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Directly analyzing graphs to get initial meaning is a practice used in research and engineering. b) Data analysis and interpretation

The research questions posed in the study are answered via analysis and interpretation.

Practices in science and engineering that need direct graph analysis to provide initial meaning

Analysis is the process of arranging, modifying, and summarizing data to find solutions to research problems. Its goal is to condense data into an understandable and interpretable form so that the connections between the research problem may be quickly and accurately verified.

The interpretation presents the findings of analysis, draws conclusions about the relations under study in the research, and makes relevant inferences.

10th term of the geometric sequence = 2560

The explicit formula for the geometric sequence is

[tex]a_{n}[/tex] = a[tex]r^{n-1}[/tex]

Where  

r = common ratio

a = first number of the series

n = number of terms in the series

Common ratio (r) = 10/5 = 20/10 = 2

Here we have to find the 10th term of the series

[tex]a_{10}[/tex] = 5× [tex]2^{10-1}[/tex]

     = 5×[tex]2^{9}[/tex]

     = 5 × 512

     = 2560

So the 10the term geometric series is 2560.

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Answer:

[tex]\textsf{1.} \quad x^2-2x-1[/tex]

[tex]\textsf{2.} \quad -3x^4-5x^3+14x^2+20x-8[/tex]

[tex]\textsf{3.} \quad -\dfrac{5}{2}[/tex]

Step-by-step explanation:

Given functions:

[tex]\begin{cases}f(x) = x^2 - 4\\g(x) = x + 2\\h(x) = -3x + 1\end{cases}[/tex]

Question 1

The composite function (f + g - h)(x) means to add functions f(x) and g(x) then subtract function h(x):

[tex]\begin{aligned}(f+g-h)(x) & = f(x)+g(x)-h(x)\\& = (x^2-4)+(x+2)-(-3x+1)\\& = x^2-4+x+2+3x-1\\& = x^2+3x+x-4+2-1\\& = x^2+4x-3\end{aligned}[/tex]

Question 2

The composite function (fgh)(x) means to multiply functions f(x), g(x) and h(x):

[tex]\begin{aligned}(fgh)(x) & = f(x)\cdot g(x) \cdot h(x)\\& = (x^2-4)(x+2)(-3x+1)\\& = (x^2-4)(-3x^2-5x+2)\\& = -3x^4-5x^3+2x^2+12x^2+20x-8\\& = -3x^4-5x^3+14x^2+20x-8\end{aligned}[/tex]

Question 3

The composite function (f/g)(h)(1/2) means to substitute the value of function h(x) when x = 1/2 into function f(x) and function g(x) and divide the former by the latter:

[tex]\begin{aligned}\left(\dfrac{f}{g}\right)(h)\left(\dfrac{1}{2}\right) & = \dfrac{f\left(h\left(\dfrac{1}{2}\right)\right)}{g\left(h\left(\dfrac{1}{2}\right)\right)}\\\\& = \dfrac{f\left(-3\left(\dfrac{1}{2}\right)+1\right)}{g\left(-3\left(\dfrac{1}{2}\right)+1\right)}\\\\& = \dfrac{f\left(-\dfrac{1}{2}\right)}{g\left(-\dfrac{1}{2}\right)}\\\\& = \dfrac{\left(-\dfrac{1}{2}\right)^2-4}{\left(-\dfrac{1}{2}\right)+2}\\\\& = \dfrac{-\dfrac{15}{4}}{\dfrac{3}{2}}\\\\& = -\dfrac{5}{2}\end{aligned}[/tex]

Coordinates of point G is ( 10 , 2) in this equation.

What do mathematical point coordinates mean?

M is the mid points of FG.

coordinates of M is ( 8,5)

coordinates of F is ( 6,8 )

Let coordinates of G point is ( x₂, y₂)

[tex]x = \frac{x_{1} + y_{1} }{2} , y = \frac{x_{2}+ y_{2} }{2}[/tex]

[tex]8 = \frac{6 + x_{2} }{2} , 5 = \frac{8 + y_{2} }{2}[/tex]

[tex]16 = 6 + x_{2} , 10 = 8 + y_{2}[/tex]

x₂ = 10  ,  y₂ = 2

So, coordinates of point G is ( 10 , 2)

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