A triangle has vertices at (−4,−6),(3,3),(7,2). Rounded to two decimal places, which of the following is the closest approximation of the perimeter of the triangle?

19.34

29.12

12.36

15.52

[tex]\log_2(2\sin x)+\log_2(\cos x)=-1[/tex]

Condense the logarithms on the left side:

[tex]\log_2(2\sin x\cos x)=-1[/tex]

Recall the double angle identity for sine, [tex]\sin(2x)=2\sin x\cos x[/tex]:

[tex]\log_2(\sin(2x))=-1[/tex]

Write both sides as powers of 2:

[tex]2^{\log_2(\sin(2x))}=2^{-1}[/tex]

Simplify this:

[tex]\sin(2x)=\dfrac12[/tex]

Solve for [tex]2x[/tex]:

[tex]2x=\sin^{-1}\left(\dfrac12\right)+2n\pi\text{ OR }2x=\pi-\sin^{-1}\left(\dfrac12\right)+2n\pi[/tex]

(where [tex]n[/tex] is any integer)

Recall that [tex]\sin^{-1}\left(\frac12\right)=\frac\pi6[/tex]:

[tex]2x=\dfrac\pi6+2n\pi\text{ OR }2x=\dfrac{5\pi}6+2n\pi[/tex]

Solve for [tex]x[/tex]:

[tex]x=\dfrac\pi{12}+n\pi\text{ OR }x=\dfrac{5\pi}{12}+n\pi[/tex]

We get solutions in the interval [tex]2\pi <x<\frac{5\pi}2[/tex] when [tex]n=2[/tex], giving

[tex]\boxed{x=\dfrac{25\pi}{12}}\text{ OR }\boxed{x=\dfrac{29\pi}{12}}[/tex]

Answer:

[tex]-|9|[/tex]   and [tex]-|-9|[/tex]

Step-by-step explanation:

Given

Expression = -9

Required

Select all equivalent expressions

To do that, we examine each of the options.

First Option

[tex]-|9|[/tex]

This can be rewritten as

[tex]-|9| = -1 * |9|[/tex]

[tex]|9| = 9[/tex]

So, the expression becomes

[tex]-|9| = -1 * 9[/tex]

[tex]-|9| = -9[/tex]

Hence, |-9| is equivalent to -9

Second Option

[tex]|-9|[/tex]

[tex]|-9| = 9[/tex]

So, the expression is not equivalent to -9

Third Option

[tex]-|-9|[/tex]

This can be rewritten as

[tex]-|-9| = -1 * |-9|[/tex]

[tex]|-9| = 9[/tex]

So, the expression becomes

[tex]-|-9| = -1 * 9[/tex]

[tex]-|-9| = 9[/tex]

Hence, -|-9| is equivalent to -9

Fourth Option

[tex]-(-9)[/tex]

This can be rewritten as

[tex]-(-9) = -1 *(-9)[/tex]

[tex]-(-9) = -1 *-9[/tex]

[tex]-(-9) = 9[/tex]

Hence, -(-9) is not equivalent to -9

Fifth Option

The distance from 0 to 9

This is calculated as thus;

[tex]Difference = 9 - 0[/tex]

[tex]Difference = 9[/tex]

Hence, the difference from 0 to 9 is not equivalent to -9

Answer:

the plant is 79 cm

Step-by-step explanation:

every month = 1 cm

months = 22 months

57 cm + 22 cm

= 79cm

Answer:

It will take them 42 hours

Step-by-step explanation:

Brianna rate = one bed for 2 hours

But for one hour = 0.5 bed per hour

Wesley rate = two bed for 3 hours

For one hour= 2/3 bed per hour

So their total rate for one hour

= 1/2 +2/3

= 7/6 bed per hour

If they received an order of 49 beds

It will take them x hours

Rate= bed/hour

7/6= 49/x

X= 49/(7/6)

X= 49 * 6/7

X= 7*6

X= 42 hours

Answer:

608 squared centimeters.

Step-by-step explanation:

To find the surface area of the net of the triangular prism, we can find the area of each individual shape and add them up.

For the net, we have 3 rectangles and 2 (congruent) triangles.

The left rectangle has a length of 16 and a width of 10. Thus, it's area is A=16(10)=160 squared centimeters.

The middle rectangle has a length of 16 and a width of 12. Thus, it's area is A=16(12)=192 squared centimeters.

The right rectangle has a length of 16 and a width of 10. Thus, it's area is identical to the left rectangle. It's area is 160 squared centimeters.

Now, for the two triangles, it's important to note they are the congruent since they belong to one triangular prism. Thus, we only need to figure out the area of one and then multiply by 2 to get the area of both of them.

One triangle has a base length of 12 and a height of 8. In other words, its area is (1/2)(12)(8)=48. Two of them will be 48(2) or 96.

Now, let's add all the areas together to find the surface area of the triangular prism. Thus, the area is:

96+160+160+192=608 squared centimeters.

Answer:

608 cm^2

Step-by-step explanation:

Find the area of the large rectangle

length is 10+12+10 = 32 and width is 16

A = 32*16 =512 cm^2

Then we have 2 triangle

The area of 1 is

A = 1/2 bh where b = 12 and h = 8

A = 1/2 ( 12 *8)

 = 1/2 (96)

But we have 2 triangles

2 * 1/2 (96)

96 cm^2

Add the area of the large rectangle and the two triangles

512+96 =608 cm^2

Answer:

Step-by-step explanation:

Since the figure above is a right angled triangle we can use trigonometric ratios to find x

To find x we use tan

tan∅ = opposite/ adjacent

From the question

x is the opposite

10 is the adjacent

Substitute the values into the above formula and solve for x

That's

[tex] \tan(20) = \frac{x}{10} [/tex]

x = 10 tan 20

x = 3.63970

We have the final answer as

Hope this helps you

Answer:

[tex] \boxed{ \bold{ \huge{ \boxed{ \sf{3 {i}^{2} + 6i}}}}}[/tex]

Step-by-step explanation:

[tex] \sf{(2 + i)(i + 2i)}[/tex]

Use the distribute property to multiply each term of the first binomial by each term of the second binomial.

⇒[tex] \sf{2(i + 2i) + i(i + 2i)}[/tex]

⇒[tex] \sf{2i + 4i + {i}^{2} + 2 {i}^{2} }[/tex]

Collect like terms

⇒[tex] \sf{3 {i}^{2} + 6i}[/tex]

Hope I helped!

Best regards!

A function assigns the values. The equation for each reflected graph of f(x)=x²-4 can be written as shown below.

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

A.) Reflect across the y-axis, to find the equation replace x with -x,

f(-x) = (-x)² - 4

       = x² - 4

No, Change because the function is symmetrical about the y-axis

B.) Reflect across the x-axis, to find the equation replace y with -y,

y = f(x) = y

-y = -f(x)

   = -(x² - 4)

   = -x² + 4

Hence, The equation for the reflection of the function can be done as shown below.

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Answer:

wire for square to maximize total area = 23 m

Wire to minimize total area = 2.019 m

Step-by-step explanation:

For the square, let's say the total length of the square is "y" m.

Thus, length of one side is = y/4

And area of the square = (y/4) = y²/16

Since the wire is 23 m, then total length of equilateral triangle is; 23 - y.

Thus, length of one side of equilateral triangle = (23 - y)/3

Using trigonometric ratio, we can find the height of the triangle and thus area.

Area of triangle = (√3)/4) × ((23 - y)/3)²

Area of triangle = (√3)/36)(23 - y)²

So, total area of square and triangle is;

A_total = (y²/16) + (√3)/36)(23 - y)²

Now, extremizing this function by derivatives, we have;

dA/dy = (y/8) - (√3)/18)(23 - y)

d²A/dy² = ⅛ + (√3)/18)

So, d²A/dy² > 0

Now,let's find the maximum or minimum of the function.

So, we equate dA/dy to zero.

Thus;

(y/8) - (√3)/18)(23 - y) = 0

y/8 = (√3)/18)(23 - y)

(y/8) + (√3)/18)y = 23((√3)/18)

Multiply through by 8 to give;

y + 0.0962y = 2.2132

1.0962y = 2.2132

y = 2.2132/1.0962

y = 2.019 m

2.019 will be a minimum because d²A/dy² > 0

The maximum will occur at a boundary of the allowed values. Thus, the absolute maximum is for y = 23.

Note that a square has more area than a triangle and as such it is normal for the square to get the largest area over the triangle and therefore we will have to use all of the wire to construct the square.

Answer:

10 times.

Step-by-step explanation:

We need to find that the value of 3 in 546300 is how many times more than 3456.

The place value of 3 in 546300 is 300 and the place value of 3 in 3456 is 3000.

Now dividing 3000 by 300 as follows :

[tex]\dfrac{3000}{300}=10[/tex]

It means that the value of 3 in 546300 is 10 times more than the value of 3 in 3456.

Answer: Hi!

121 ÷ 3 = 40.3

987 ÷ 9 = 109.7

675 ÷ 5 = 135

432 ÷ 7 = 61.7

Hope this helps you!

Answer:

40 1/3

109 2/3

135

61.7

Step-by-step explanation:

Hey there!

Well lets divide all given expressions,

121 ÷ 3 = 40 1/3

987 ÷ 9 = 109 2/3

675 ÷ 5 = 135

432 ÷ 7 = 61.7 rounded to the nearest tenth

Hope this helps :)

Let's take as x and y the age of the man and of his son

We will do a system, so we can satisfy all the request:

1. sum of ages = 2 (difference)

2. product = 675

[tex]\left \{ {x + y = 2 (x-y)} \atop {x*y=675}} \right.[/tex]

semplify the first equation

x + y = 2x -2y

now let's choose one of the incognite (x) and we solve for it

x - 2x = - 2y - y

- x = - 3y

x = 3y

Let's substitute this solution in the second equation

[tex]\left \{ {x=3y} \atop {(3y)*y=675}} \right.[/tex]

note: x = 3y, so in the second equation x * y = 3y * y

Now let's solve the second equation

3y * y = 675

3y² = 675

y² = 675 / 3 =

y² = 225

y = 15

Son's age is 15

Man's age is 15 * 3 = 45 (See the first equation [x = 3y])

Full Question:

The data show systolic and diastolic blood pressure of certain people. Find the regression equation, letting the first variable be the independent (x) variable. Find the best predicted diastolic pressure for a person with a systolic reading of 113. use a significance level of 0.05.

Systolic| 150 129 142 112 134 122 126 120

Diastolic| 88 96 106 80 98 63 95 64

a. What is the regression equation?

^y = __ + __x (Round to two decimal places as needed.)

b. What is the best predicted value?

^y is about __ (Round to one decimal place as needed.)

Answer:

A. yhat = a + bx = -10.64 + 0.75x

B. 74.0

Step-by-step explanation:

A. To find the regression equation here, we apply the formulas and then apply it to find the value of y given value of x:

calculate xbar and ybar which is the average of the variables:

Where n(number of values in x or y)=8

xbar = sum of x/n = 129.375

ybar = sum of y/n = 86.25

to calculate b

b= [Sum x^2 * Sum y - Sum x * Sum x*y] / [N*Sum x^2 - (Sum x)^2]

b = 0.74891

To calculate a

a = ybar - b * xbar = -10.64023

Regression equation:

y=mx+b= -10.64 + 0.75x

B. given x = 113,

y = -10.64023 + 0.74891 * 113

y= 74.0

Answer: 93 and 4/5

Step-by-step explanation:

6 7/10 divided by 1/14

= 67/10 divided by 1/14

= 67/10 x 14/1

= 67/5 x 7/1

= 469/5

Answer:

Step-by-step explanation:

If a mass of 2.5 grams costs a penny, we want to know the amount of mass that will cost $75.  Before we get the mass, we need to convert 75 dollars to penny. Using the conversion:

$1 = 100pennies

$75 = (75*100)pennies

$75 = 7,500 pennies

Using the law of equivalence

1 penny = 2.5 grams

7,500pennies = x grams

Cross multiply

1 * x = 2.5 * 7500

x = 18,750 grams

Hence 18,759 grams will cost $75

Converting grams to kilograms

If 1000 grams = 1kg

18,750grams -= y

cross multiply

1000*y = 18750

Divide both sides by 1000

1000y = 18750/1000

y = 18.750kg

Hence the amount in kilogram of $75 coin in pennies is 18.750kg

Answer:

x = 18

Step-by-step explanation:

ΔABC ≅ ΔDEC

∠ B ≅ ∠E

40 = 2x + 4

-2x = 4 - 40

-2x = -36

2x = 36

x = 36/2

x = 18

Answer: 18 rows.

Step-by-step explanation:

The theatre has 4 chairs in a row.

There are 5 rows.

If we consider rows as unit, then we observed that

Length = 4 rows

Width = 5 rows

Perimeter of Rectangle = 2(length +breadth)

= 2(5+4)

= 2(9)

=18 rows

Hence, the perimeter is 18 rows.

Answer: c(x) = y = $0.132*x + $27.16

Step-by-step explanation:

The chargers are a fixed price plus a charge for every copy, then we have a linear relationship.

A linear relationship can be written as:

c(x) = y = a*x + b

where a is the slope and b is the y-axis intercept.

For a line that passes through the points (x1, y1) and (x2, y2), the slope can be written as:

a = (y2 - y1)/(x2 - x1).

In this case, c(x) = y represents the cost in dollars,  x is the number of copies bought, and b is the fixed cost.

We know two points in this line:

(80, $37) and (209, $54)

Whit those two points we can find the slope:

a = ($54 - $37)/(209 - 80) = $17/129 = $0.132 per copy.

Then our equation is:

c(x) = y = $0.132*x + b

To find the value of b, we know that 80 copies cost $37, we can replace those values in the equation:

$37 = $0.132*80 + b

$37 = $9.84 + b

($37 - $9.84) = $27.16 = b.

Then the equation is:

c(x) = y = $0.132*x + $27.16

Answer:

Step-by-step explanation:

The distance between two points can be found by using the formula

where

(x1 , y1) and (x2 , y2) are the points

From the question the points are

(0,5) and (8,0)

So the distance between them is

d = 9.4339811

We have the final answer as

Hope this helps you

Answer:

y = 1

Step-by-step explanation:

5·x

x=2

5·2=10

10+2y

y=1

2·1=2

10+2=12

Answer:1

Step-by-step explanation:5(x)+2(y)=12

5 x 2 =10

2 x 1=2

10 + 2=12

Answer:

C. 2/25

Step-by-step explanation:

P(A ∩ B) = P(A) P(A|B)

P(A ∩ B) = 2/5 * 1/5 = 2/25

Answer option is:

C. 2/25

Answer:

The distance d = 1

Step-by-step explanation:

The objective is to compute the distance from y to the line through u and the origin.

Given that :

[tex]y = \left[\begin{array}{cc}5\\5\end{array}\right][/tex]  and [tex]u = \left[\begin{array}{cc}6\\8\end{array}\right][/tex]

Recall that:

from the  slope - intercept on the graph, the equation of line can be expressed as :

y = mx + b

where;

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

b = y - intercept

Similarly, we are being informed that the line  passed through [tex]u = \left[\begin{array}{cc}6\\8\end{array}\right][/tex] and origin, so ;

[tex]x_1 = 0 , y_1 = 0 \\ \\ x_2 =6 , y_2 = 8[/tex]

the slope m = [tex]\dfrac{y_2-y_1}{x_2-x_1}[/tex]

= [tex]=\dfrac{8 - 0}{6-0}[/tex]

[tex]= \dfrac{4}{3}[/tex]

Also, since the line pass through the origin:

Then

y = mx + b

0 = m(0) + b

b = 0

From y = mx + b

y = mx + (0)

y = mx

[tex]y = \dfrac{4}{3}x[/tex]

3y = 4x

3y - 4x = 0

4x - 3y =0

The distance of a point (x,y) from a line ax +by + c = 0 can be represented with the equation:

[tex]d = \dfrac{|ax+by +c|}{\sqrt{a^2 +b^2}}[/tex]

∴ the distance from [tex]y = \left[\begin{array}{cc}5\\5\end{array}\right][/tex]   to the line 4x - 3y = 0  is

[tex]d = \dfrac{|4x-3y +0|}{\sqrt{4^2 +3^2}}[/tex]

[tex]d = \dfrac{|4(5)-3(5) +0|}{\sqrt{16+9}}[/tex]

[tex]d = \dfrac{20-15 }{\sqrt{25}}[/tex]

[tex]d = \dfrac{5}{5}[/tex]

The distance d = 1

The distance from y(5,5) to the line through u(6,8) and the origin(0,0) is √2 units.

The distance or length of any line on the graph is given by the formula,

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

where,

d = distance/length of the line between points 1 and 2,

(x₁, y₁) = coordinate of point 1,

(x₂, y₂) = coordinate of point 2,

In the given equation, we need to find the distance between the line and the point u (6,8). Now, we try to find the equation of the line, with points y=(5,5) and origin (0,0). Therefore, the slope of the equation can be written as,

[tex]m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{5-0}{5-0} = 1[/tex]

Now, if we substitute the value of the slope and a point in the general equation of the line, we will get,

[tex]y = mx+c\\\\0 = 1(0)+c\\\\c = 0[/tex]

Further, if we draw the line on the graph, the nearest point to point u(6,8) is a(7,7). Therefore, the distance between the two points can be written as,

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\D = \sqrt{(7-6)^2+(7-8)^2}\\\\D = \sqrt2[/tex]

Hence, the distance from y(5,5) to the line through u(6,8) and the origin(0,0) is √2 units.

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Answer:

0

Step-by-step explanation:

First, simplify for both sides

6/15 -> 2/5

2/5 -> 2/5

x+2/5 = 2/5

Subtract 2/5 from each side.

(x+2/5)-2/5-> x

2/5-2/5 -> 0

x = 0

Answer:

The number of 1 point throw is 5, the number of 2 point throw is 8 and the number of 3 point throw is 4

Step-by-step explanation:

Let x represent the number of  1-point free throws, y is the number of 2-point field goals, and z is the number of 3-point field goals. A basketball player scored 33 points during a game by shooting 1-point free throws, 2-point field goals, and 3-point field goals. Each of 1 point free throw is 1 point, each of 2 point free throw is 2 points and each of the 3 point free throw is 3 points. This is represented by the equation:

x + 2y + 3z = 33             (1)

Also, The player scored 17 times, therefore it is represented by:

x + y + z = 17                   (2)

The player scored  3 more 2-point field goals than 1-point free throws, it is represented by:

y = x + 3                      

-x + y = 3                         (3)

Solving equation 1, eqn 2 and eqn 3 simultaneously:

x = 5, y = 8, z = 4.

The number of 1 point throw is 5, the number of 2 point throw is 8 and the number of 3 point throw is 4

Answer:

No. of 1 pt free throws = 5, No. of 2 pt goals = 8, No. of 3 pt goals = 4

Step-by-step explanation:

Equations : x + y + z = 17 [ Total times taken to score ]

1x + 2y + 3z = 33 [ Total Score ]

Also, y = x + 3

Putting the value of 'y' in both equations :

x + (x + 3)+ z = 17 → 2x + 3 + z = 17 → 2x + z = 14  (i)

1x + 2 (x + 3) + 3z = 33 →  x + 2x + 6 + 3z = 33 → 3x + 3z = 27 (ii)

Solving these equations :

From (i), z = 14 - 2x

Putting this value in (ii), 3x + 3(14 - 2x) = 27 → 3x + 42 - 6x = 27

42 - 3x = 27 → 3x = 15 → x = 5

y = x + 3 = 5 + 3 → y = 8

z = 17 - x - y → z = 17 - 5 - 8 = 17 - 13 → z = 4

Answer:

4×(-5)=-20

Step-by-step explanation:

Answer:

[tex]\huge \boxed{-20}[/tex]

Step-by-step explanation:

[tex]4*(-5)[/tex]

[tex]\Rightarrow 4* -5[/tex]

Multiplying.

[tex]\Rightarrow -20[/tex]

Answer:

0.0512

Step-by-step explanation:

We are told in the question that X isa Binomial variable. Hence we solve this question, using the formula for Binomial probability.

The formula for Binomial Probability = P(x) = (n!/(n - x) x!) × p^x × q ^n - x

In the above question,

x = 3

n = 5

p = probability for success = 0.2

q = probability for failures = 1 - 0.2 = 0.8

P(x) = (n!/(n - x) x!) × p^x × q ^n - x

P(3) = (5!/(5 - 3) 3!) × 0.2^3 × 0.8^5-3

P(3) =( 5!/2! × 3!) × 0.2³ × 0.8²

P(3) = 10 × 0.008 × 0.64

P(3) = 0.0512

Answer: $428.00

Step-by-step explanation: First, we need to calculate the person’s total hours, (30 x 2 = 60 hours) then the money per hour (60 x 9) then take away the taxes and account deductions.

Answer:

12 ab²c³

Step-by-step explanation:

√144 a² b⁴ c6

144 would be 12

a² would be a

b⁴ would be b²

c6 would be c³