find the midpoint of the line segment with end coordinates of (2,1) and (4,5)​

Answer:

[tex]\rm\displaystyle \theta = \frac{\pi}{4} [/tex]

Step-by-step explanation:

we want to find the acute angle θ (as θ is between (0,π/2)) formed by the line y=-2x+4 and y=3x-3 to do so we can consider the following formula:

[tex] \displaystyle\tan( \theta) = \bigg| \frac{ m_{2} - m_{1} }{1 + m_{1} m_{2} } \bigg | [/tex]

[tex] \rm \displaystyle \implies\theta = \arctan \left( \bigg | \frac{ m_{2} - m_{1} }{1 + m_{1} m_{2} } \bigg | \right)[/tex]

From the first equation we obtain that [tex]m_1[/tex] is -2 and from the second that [tex]m_2[/tex] is 3 therefore substitute:

[tex] \rm\displaystyle \theta = \arctan \left( \bigg | \frac{ 3 - ( - 2) }{1 + ( - 2) (3)} \bigg | \right)[/tex]

simplify multiplication:

[tex] \rm\displaystyle \theta = \arctan \left( \bigg | \frac{ 3 - ( - 2) }{1 + ( - 6)} \bigg | \right)[/tex]

simplify Parentheses:

[tex] \rm\displaystyle \theta = \arctan \left( \bigg | \frac{ 3 + 2 }{1 + ( - 6)} \bigg | \right)[/tex]

simplify addition:

[tex] \rm\displaystyle \theta = \arctan \left( \bigg | \frac{ 5 }{ - 5} \bigg | \right)[/tex]

simplify division:

[tex] \rm\displaystyle \theta = \arctan \left( | - 1| \right)[/tex]

calculate the absolute of -1:

[tex]\rm\displaystyle \theta = \arctan \left( 1\right)[/tex]

calculate the inverse function:

[tex]\rm\displaystyle \theta = \frac{\pi}{4} [/tex]

hence,

the angle θ which is formed by the line y = -2x+4 and y = 3x-3 is π/4

(for more info about the formula refer the attachment thank you!)

Consider two vector-valued functions,

[tex]\vec r(t) = \left\langle t, -2t+4\right\rangle \text{ and } \vec s(t) = \left\langle t, 3t-3\right\rangle[/tex]

Differentiate both to get the corresponding tangent/direction vectors:

[tex]\dfrac{\mathrm d\vec r(t)}{\mathrm dt} = \left\langle1,-2\right\rangle \text{ and } \dfrac{\mathrm d\vec s(t)}{\mathrm dt} = \left\langle1,3\right\rangle[/tex]

Recall the dot product identity: for two vectors [tex]\vec a[/tex] and [tex]\vec b[/tex], we have

[tex]\vec a \cdot \vec b = \|\vec a\| \|\vec b\| \cos(\theta)[/tex]

where [tex]\theta[/tex] is the angle between them.

We have

[tex]\langle1,-2\rangle \cdot \langle1,3\rangle = \|\langle1,-2\rangle\| \|\langle1,3\rangle\| \cos(\theta) \\\\ 1\times1 + (-2)\times3 = \sqrt{1^2 + (-2)^2} \times \sqrt{1^2+3^2} \cos(\theta) \\\\ \cos(\theta) = \dfrac{-5}{\sqrt5\times\sqrt{10}} = -\dfrac1{\sqrt2} \\\\ \implies \theta = \cos^{-1}\left(-\dfrac1{\sqrt2}\right) = \dfrac{3\pi}4[/tex]

Then the acute angle between the lines is π/4.

Answer:

1190 m^3

Step-by-step explanation:

l = 1700 cm = 17 m

b = 14 m

h = 1000 cm = 10 m

Total volume = l × b × h

= 17 × 14× 10

= 2380 m^3

since it is half filled ,

Volume is half , so,

volume of water in pond = 2380 ÷ 2

= 1190 m^3

Part one:

[tex]x^2-8x=-26[/tex]

Rewrite in the form [tex](x+a)^{2} =b[/tex]

[tex]\left(x-4\right)^2=-10[/tex]

[tex]\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}[/tex]

Solve [tex]x-4=\sqrt{-10} : x=\sqrt{10} i+4[/tex]

Solve [tex]x-4=\sqrt{-10} : x=-\sqrt{10} i+4[/tex]

[tex]x=\sqrt{10}i+4,\:x=-\sqrt{10}i+4[/tex]

Part two:

[tex]x=\frac{-\left(-8\right)\pm \sqrt{\left(-8\right)^2-4\cdot \:1\cdot \:26}}{2\cdot \:1}[/tex]

Simplify [tex]\sqrt{\left(-8\right)^2-4\cdot \:1\cdot \:26}}: 2\sqrt{10} i[/tex]

[tex]=\frac{-\left(-8\right)\pm \:2\sqrt{10}i}{2\cdot \:1}[/tex]

Separate solutions

[tex]x_1=\frac{-\left(-8\right)+2\sqrt{10}i}{2\cdot \:1},\:x_2=\frac{-\left(-8\right)-2\sqrt{10}i}{2\cdot \:1}[/tex]

[tex]\frac{-(-8)+2\sqrt{10}i }{2*1} :4+\sqrt{10}i[/tex]

[tex]\frac{-(-8)+2\sqrt{10}i }{2*1} :4-\sqrt{10}i[/tex]

[tex]x=4+\sqrt{10}i,\:x=4-\sqrt{10}i[/tex]

The solutions are:-

[tex]x=4+\sqrt{10i}\\\\x=4-\sqrt{10i}[/tex]

What is the equation?

The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

Here given equation is

[tex]x^2- 8x + 26 = 0\\\\x^2-8x=-26\\\\(x-4)^2=-10\\\\[/tex]

[tex](x-4)=[/tex]±[tex]\sqrt{-10}[/tex]

[tex]x=\sqrt{10}i+4\\\\x=-\sqrt{10}i+4[/tex]

So,

[tex]x=\frac{-b+-\sqrt{b^2-4ac}}{2a}\\\\x=\frac{8+-\sqrt{(-8)^2-4(1)(26)}}{2(1)}\\\\x=\frac{8+-\sqrt{(-8)^2-4(1)(26)}}{2(1)}\\\\x=\frac{8+-\sqrt{64-104}}{2}\\\\x=\frac{8+-2\sqrt{10i}}{2}\\\\x=\frac{2(4+-\sqrt{10i})}{2}\\\\x=4+\sqrt{10i}\\\\x=4-\sqrt{10i}[/tex]

Hence, the solutions are:-

[tex]x=4+\sqrt{10i}\\\\x=4-\sqrt{10i}[/tex]

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[tex]\longrightarrow{\green{ (a + b)(a - b) }}[/tex] 

[tex]\sf \bf {\boxed {\mathbb {STEP-BY-STEP\:EXPLANATION:}}}[/tex]

[tex] ={a}^{2} - {b}^{2} [/tex]

[tex] = (a + b)(a - b)[/tex]

[tex]\bold{ \green{ \star{ \orange{Mystique35}}}}⋆[/tex]

Answer:

A

Step-by-step explanation:

Answer:

A and B, every even number is divisible by two, C is odd

Answer:

C

Step-by-step explanation:

C

The output of  [tex]R(x) = x^{2} - 3x- 1[/tex] when x = -2 is 9.

PEMDAS exists as an acronym for the terms parenthesis, exponents, multiplication, division, addition, and subtraction.

To estimate the value of R(-2)

Substitute the value of x = -2, then

[tex]R(-2) = (-2)^{2} - 3(-2) - 1[/tex]

By using the PEMDAS order of operations

Calculate exponents, [tex](-2)^2 = 4[/tex]

= 4 - 3(-2) - 1

Multiply and divide (left to right), 3(-2) = -6

= 4 - (-6) - 1

Add and subtract (left to right),

4 - (-6) - 1 = 9

Therefore, the value of R(-2) = 9.

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Answer:

Kayleigh will have a total of $4608.

Step-by-step explanation:

First, you use the formula, I=PRT (Interest=Principal, Rate, Time), then you distribute the numbers: (I=4500x0.8%x3) when you multiply them all, you get $108, then you lastly add 108 to 4500, and you get your final answer of $4608.

Kayleigh will earn $108 in interest over a period of 3 years.

Interest is the additional amount of money that is charged or earned on a principal amount of money. It is typically expressed as a percentage of the principal and is either charged when borrowing money or earned when investing or saving money.

To calculate the interest Kayleigh will earn in 3 years, we can use the formula for simple interest:

Interest = Principal x  Rate x Time

Given:

Principal = $4500

Rate = 0.8% = 0.008 (decimal form)

Time = 3 years

Plugging the values into the formula, we have:

Interest = $4500 x 0.008 x 3

Calculating the expression, we find:

Interest = $108

Therefore, Kayleigh will earn $108 in interest over a period of 3 years.

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Answer:

y = square root x2 - 5

Step-by-step explanation:

y=[tex]\sqrt{x} -5[/tex]

(the top option)

Add all the coins to get total coins:

6 + 8 + 12 + 3 = 29 total coins

Subtract dimes to find total of the coins that are not dimes:

29 -6 = 23

Probability of not picking a dime is the number of coins that aren’t dimes over total coins:

23/29

Answer:

The probability is 1/12.

Step-by-step explanation:

Number of elements in sample space is 6 . Even numbers are 2, 4 and 6 so the there are 3 three even numbers.

So, the probability of getting 3 on the first chance and then an even number in the second chance is

[tex]P = \frac{1}{6}\times \frac{3}{6}\\\\P = \frac{1}{12}[/tex]

Answer:

∠ C = ∠BCD = 30°

Step-by-step explanation:

∠ EDF = ∠GFC  = 110°                [ corresponding angles ]

Now consider triangle BCF

∠FBC  + ∠ABC = 180°                [ straight line angle ]

∠FBC  + 100° = 180°

∠FBC  = 180 - 100 = 80°

∠BFC + ∠GFC = 180°                  [ straight line angle ]

∠BFC + 110° = 180°

∠BFC = 180 - 110 = 70°

Sum of angles of triangle is 180°

Therefore , in triangle BCF

That is ,

       ∠F + ∠B + ∠ C  = 180°

          70° + 80° + ∠C = 180°

           ∠C = 180 - 150 = 30°            

Answer:

B

Step-by-step explanation:

we rule out all other 3 by the simple fact that the side is 9

Answer:

x = 4

Step-by-step explanation:

Corresponding angles are congruent

4x + 44 = 6x + 36

44 = 2x + 36

8 = 2x

x = 4

Answer:

=0=−6

Step-by-step explanation:

Answer:

D. f(t) = 192,345(1.04)^t

Step-by-step explanation:

I took the test and it was right.

Also that is the original price and when you look at exponential functions, the starting point or original price is always first the then rate of increase. The table just shows how it increased in year 1 and 2.

Hope this helps. :)

A function assigns the values. The function that best represents the value of the real estate after t years is f(t) = 192,345(1.04)^t. Thus, the correct option is D.

A function assigns the value of each element of one set to the other specific element of another set.

The initial cost of industrial real estate is $192,345, while the cost after one year is $200,038.80. Therefore, the rate of appreciation is,

[tex]\$200,038.80 = \$192,345(1+R)^t\\\\\$200,038.80 = \$192,345(1+R)^1\\\\\dfrac{\$200,038.80}{\$192,345}=(1+R)^t\\\\1.04 = 1 + R\\\\R = 0.04[/tex]

Hence, the function that best represents the value of the real estate after t years is f(t) = 192,345(1.04)^t. Thus, the correct option is D.

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Answer:

Step-by-step explanation:

It's not the correct choice. If a root is given, that is the same thing as the solution. We go backwards from a solution to a factor, which is what you need to do here.

If x = 2i is the solution, then the factor is

(x - 2i). Likewise, with the other one.

If x = 3i is the solution, then the factor is

(x - 3i). The leading coefficient of 1 just sits outside the first set of parenthesis, and because multiplication is commutative, it doesn't matter which set you put first. Thus, the one you want looks like this:

f(x) = (x - 2i)(x - 3i) or

f(x) = (x - 3i)(x - 2i).

Answer:

4/3

Step-by-step explanation:

The formula is (y1-y2) /(x1-x2)

Answer:

slope = [tex]\frac{4}{3}[/tex]

Step-by-step explanation:

Calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (2, 2) and (x₂, y₂ ) = (- 1, - 2)

m = [tex]\frac{-2-2}{-1-2}[/tex] = [tex]\frac{-4}{-3}[/tex] = [tex]\frac{4}{3}[/tex]

Answer:

100

Step-by-step explanation:

Based on the histogram given :

The vertical axis gives the number of student in the school :

For the range :

0 - 5 = 40

5 - 10 = 30

10 - 15 = 20

15 - 20 = 5

20 - 25 = 3

25 - 30 = 2

Taking the sum :

(40 + 30 + 20 + 5 + 3 + 2) = 100

1 hour is 60 minutes

6 hours x 60 = 360 total minutes

Divide total time by number of lessons:

360/8 = 45

Each lesson was 45 minutes

Answer:

[tex]a = 18x[/tex]

Step-by-step explanation:

Given

[tex]\frac{1}{2}(a + d) = 10x[/tex]

[tex]\frac{1}{2}(b + c) = 8x[/tex]

[tex]\frac{1}{3}(b + c+d) = 6x[/tex]

Required

The value of (a)

We have:

[tex]\frac{1}{2}(a + d) = 10x[/tex] --- multiply by 2

[tex]\frac{1}{2}(b + c) = 8x[/tex] --- multiply by 2

[tex]\frac{1}{3}(b + c+d) = 6x[/tex] --- multiply by 3

So, we have:

[tex]\frac{1}{2}(a + d) = 10x[/tex]

[tex]a + d = 20x[/tex]

[tex]\frac{1}{2}(b + c) = 8x[/tex]

[tex]b + c = 16x[/tex]

[tex]\frac{1}{3}(b + c+d) = 6x[/tex]

[tex]b + c + d= 18x[/tex]

Substitute [tex]b + c = 16x[/tex] in [tex]b + c + d= 18x[/tex]

[tex]16x + d = 18x[/tex]

Solve for d

[tex]d = 18x - 16x[/tex]

[tex]d = 2x[/tex]

Substitute [tex]d = 2x[/tex] in [tex]a + d = 20x[/tex]

[tex]a + 2x = 20x[/tex]

Solve for (a)

[tex]a = 20x - 2x[/tex]

[tex]a = 18x[/tex]

Answer:

The streamer's speed in still water is 90.23 km/h while the stream's speed is 33.33 km/h

Step-by-step explanation:

Let v = streamer's speed in still water and v' = stream's speed. His speed upstream is V = v + v' and his speed downstream is V' = v - v'.

Since he travels 36 km upstream, the time taken is t = 36/V = 36/(v + v').

he travels 32 km downstream, the time taken is t' = 32/V' = 32/(v - v')

The total time is thus t + t' = 36/(v + v') + 32/(v - v')

Since the whole trip takes 6,5 hours,

36/(v + v') + 32/(v - v') = 6.5  (1)

Multiplying each term by (v + v')(v - v'), we have

(v + v')(v - v')36/(v + v') + (v + v')(v - v')32/(v - v') = 6.5(v + v')(v - v')  (1)

(v - v')36 + (v + v')32 = 6.5(v + v')(v - v')  (1)

36v - 36v' + 32v + 32v' = 6.5(v² + v'²)

68v - 4v' = 6.5(v² + v'²)        (2)

Also he travels 4 km upstream, the time taken is t" = 4/V = 4/(v + v').

he travels 40 km downstream, the time taken is t'" = 40/V' = 40/(v - v')

The total time is thus t" + t'" = 4/(v + v') + 40/(v - v')

Since the whole trip takes 180 minutes = 3 hours,

4/(v + v') + 40/(v - v') = 3  (3)

Multiplying each term by (v + v')(v - v'), we have

(v + v')(v - v')4/(v + v') + (v + v')(v - v')40/(v - v') = 3(v + v')(v - v')  (1)

(v - v')4 + (v + v')40 = 3(v + v')(v - v')  (1)

4v - 4v' + 40v + 40v' = 3(v² + v'²)

44v - 36v' = 3(v² + v'²)      (4)

Dividing (2) by (4), we have

(68v - 4v')/(44v - 36v') = 6.5(v² + v'²)/3(v² + v'²)      

(68v - 4v')/(44v - 36v') = 6.5/3

3(68v - 4v') = 6.5(44v - 36v')

204v - 12v' = 286v - 234v'

204v - 286v = 12v' - 234v'

-82v = -222v'

v = -222v'/82

v = 111v'/41

Substituting v into (2), we have

68v - 4v' = 6.5(v² + v'²)      

68(111v'/41) - 4v' = 6.5[(111v'/41)² + v'²]        

[68(111/41) - 4]v' = 6.5[(111/41)² + 1]v'²    

[68(111/41) - 4]v' = 6.5[(111/41)² + 1]v'²

[7548/41 - 4]v' = 6.5[12321/1681 + 1]v'²

[(7548 - 164)/41]v' = 6.5[(12321 + 1681)/1681]v'²

[7384/41]v' = 6.5[14002/1681]v'²

[7384/41]v' = [91013/1681]v'²

[91013/1681]v'² - [7384/41]v' = 0

([91013/1681]v' - [7384/41])v' = 0

⇒ v' = 0 or ([91013/1681]v' - 7384/41) = 0

⇒ v' = 0 or [91013/1681]v' - 7384/41) = 0

⇒ v' = 0 or v' =  7384/41 × 16841/91013

⇒ v' = 0 or v' =  180.097 × 0.185

⇒ v' = 0 or v' =  33.33 km/h

Since v' ≠ 0, v' = 33.33 km/h

Substituting v' into v = 111v'/41 = 111(33.33 km/h)/41 = 3699.63 km/h ÷ 41 = 90.23 km/h

So, the streamer's speed in still water is 90.23 km/h while the stream's speed is 33.33 km/h

Answer:

its true trust biggie on this one fam

Step-by-step explanation:

its true broski

Answer:

12

Step-by-step explanation:

using pythagoras theorem

here 15 is hypotenuse since it is opposite 0f 90 degree

9 and x are the other smaller sides of a triangle

according to pythagoras thorem the sum of square of two smaller sides of a triangle is equal to the square of hypotenuse. So,

9^2 + x^2 = 15^2

81 + x^2 = 225

x^2 = 225 - 81

x^2 = 144

x = [tex]\sqrt{144[/tex]

x = 12

Answer:

ok so we can simpley this is 1:106

so we multiply

20*106=2120 this is cm so we make it meters...

21.2meters!!!

Hope This Helps!!!

Answer:

2 and -1

Step-by-step explanation:

sum = 2 + (-1)

= 1

product = 2 × ( -1)

= -2

Answer:

first

2/5m + 8/5

second

22/5

third

11

Answer:

[tex]MK\approx 87.7[/tex]

Step-by-step explanation:

By definition, similar polygons have corresponding sides in a constant proportion. Therefore, we can set up the following proportion (ratio of corresponding sides) to solve for [tex]MK[/tex]:

[tex]\frac{20}{13}=\frac{MK}{57},\\MK=\frac{57\cdot 20}{13}\approx \boxed{87.7}[/tex]

Answer:

87.7

Step-by-step explanation:

Like stated previously similar triangle have side lengths with common ratios

*Create a proportionality to solve for MK*

57/13 = MK / 20

Now solve for MK

Multiply each side by 20

57/13 * 20 = 87.7 ( rounded )

MK/20 * 20 = MK

We're left with MK = 87.7