help me with this geometry question with proofs

Answer:

see explanation

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Calculate m using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (2, 7) and (x₂, y₂ ) = (4, 6) ← 2 points on the line

m = [tex]\frac{6-7}{4-2}[/tex] = - [tex]\frac{1}{2}[/tex]

Note the line crosses the y- axis at (0, 8) ⇒ c = 8

y = - [tex]\frac{1}{2}[/tex] x + 8 ← equation of line

Answer : The slope of the line and equation of the line is, [tex]\frac{-1}{2}[/tex]  and [tex]y=\frac{-1}{2}x+8[/tex] respectively.

Step-by-step explanation :

The general form for the formation of a linear equation is:

[tex](y-y_1)=m\times (x-x_1)[/tex] .............(1)

where,

x and y are the coordinates of x-axis and y-axis respectively.

m is slope of line.

First we have to calculate the slope of line.

Formula used :

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

Here,

[tex](x_1,y_1)=(2,7)[/tex] and [tex](x_2,y_2)=(4,6)[/tex]

[tex]m=\frac{(6-7)}{4-2)}[/tex]

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

Now put the value of slope in equation 1, we get the linear equation.

[tex](y-y_1)=m\times (x-x_1)[/tex]

[tex](y-7)=\frac{-1}{2}\times (x-2)[/tex]

[tex]y-7=\frac{-1}{2}x+1[/tex]

[tex]y=\frac{-1}{2}x+1+7[/tex]

[tex]y=\frac{-1}{2}x+8[/tex]

Thus, the slope of the line and equation of the line is, [tex]\frac{-1}{2}[/tex]  and [tex]y=\frac{-1}{2}x+8[/tex] respectively.

Answer:

The student is expected to spend 15.4 hours doing homework

Step-by-step explanation:

The scattered plot shows there is a close correlation between the variables. A line of best fit will go through the 'center' of the points. Since we are not required to find an exact line, we'll draw it in red color as shown below

To know the equation of that line, we must take two clear points of it from the graph. We'll pick (28,4) and (4,25)

The equation of a line, given two points (a,b) and (c,d) is

[tex]\displaystyle y-b=\frac{d-b}{c-a}(x-a)[/tex]

Using the selected points

[tex]\displaystyle y-4=\frac{25-4}{4-28}(x-28)[/tex]

Simplifying and computing results, the equation is

[tex]\displaystyle y=-\frac{7}{8}x+\frac{57}{2}[/tex]

Using that equation, we can predict how many hours the students will spend doing homework if they spend 15 hours watching TV

[tex]\displaystyle y=-\frac{7}{8}(15)+\frac{57}{2}[/tex]=15.4 hours

So the student is expected to spend 15.4 hours doing homework

Answer:

The number is -84.

Step-by-step explanation:

-x/7-2=10

-x/7=10+2

-x/7=12

-x=12*7

-x=84

x=-84

Answer:

a(1) = 8, a(n + 1) = a(n) + 5

Checking this:

a(2) = a(1) + 5 = 8 + 5 = 13

a(3) = a(2) + 5 = 13 + 5 = 18

a(4) = a(3) + 5 = 18 + 5 = 23

a(5) = a(4) + 5 = 23 + 5 = 28

Answer:

Cut it in half so that the two pieces are each 7 in by 4 in.

Answer:

The proof is given below.

Step-by-step explanation:

Given: ( the correct question and ans is as follow )

sin (x + y) = 3sin ( x - y)

To Prove

tan x = 2 tan y

Proof:

[tex]\sin (x+y) = 3\sin (x-y)[/tex]

Using the identities

[tex]\sin (A+B) = \sin A.\cos B + \cos A.\sin B\\and\\\sin (A-B) = \sin A.\cos B - \cos A.\sin B\\[/tex]

we get

[tex]\sin x.\cos y + \cos x.\sin y = 3(\sin x.\cos y - \cos x.\sin y)\\\sin x.\cos y + \cos x.\sin y = 3\sin x.\cos y - 3\cos x.\sin y[/tex]

Now we will take sin x . cos y to the left hand side and cos x . sin y to the right hand side,we get

[tex]3\sin x.\cos y - \sin x.\cos y = 3\cos x.\sin y +\cos x.\sin y\\2\sin x.\cos y =4\cos x.\sin y\\\sin x.\cos y =2\cos x.\sin y \ \textrm{4 divide by 2}\\ \frac{\sin x}{\cos x} =2\times \frac{\sin y}{\cos y} \\\textrm{we know identity }\\\frac{\sin x}{\cos x} = \tan x\\\frac{\sin y}{\cos y} = \tan y\\\therefore \tan x = 2\tan y\ ...............{PROVED}[/tex]

Answer: 71 hours

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

We have: [tex]$ \frac{18}{4} \times \frac{20}{7} $[/tex]

Simplifying we get: [tex]$ 18 \times \frac{5}{7} $[/tex]

⇒ [tex]$ \frac{90}{17} $[/tex]

This is equal to 5.2 ≅ 5.

So, Option C - 5 is the answer.

Answer:

Look the square root up on the internet.

Step-by-step explanation:

Answer:

[tex]y = \frac{1}{3}x + 3[/tex]

Step-by-step explanation:

The slope of two parallel straight lines are the same.

Now, the slope of the line [tex]y = \frac{1}{2} x + 8[/tex] is [tex]\frac{1}{2}[/tex] as the straight line equation is in slope-intercept form.

Now, the equation of the straight line in slope-intercept form which is parallel to the above line is [tex]y = \frac{1}{2} x + c[/tex] ............. (1) , where c is any constant.

Now, point (-2,2) is on the line (1) {Given}

So, this point will satisfy the equation.

Hence, [tex]2 = \frac{1}{2} (- 2) + c[/tex]

⇒ c = 3.

Therefore, the equation of the required straight line is [tex]y = \frac{1}{3}x + 3[/tex] (Answer)

Answer:

123

Step-by-step explanation:

135 - 12 = 123

The concession stand made $123

Profit and loss are the terms used to identify whether a deal is profitable or not. We use these terms very often in our daily lives. If the selling price is greater than the cost price, then the difference between the selling price and cost price is called profit. If the selling price is less than the cost price, then the difference between the cost price and the selling price is called loss.

When, in a transaction, the selling price is greater than the cost price, it means we earn a profit.

When, in a transaction, the cost price is greater than the selling price, it means we incur a loss.

Given:

profit on saturday = $135

loss on sunday = $12

So,

the total profit concession stand made= $135 + (-$12)

                                                                = $135 - $12

                                                                = $123

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z = 37°

Step-by-step explanation:

The exterior angle theorem says that the sum of two internal angles is equal to the exterior angle

In this case the sum of z and z+4 will be equal to z+41

Writing mathematically

[tex]z + z+ 4 = z + 41\\2z+4 = z + 41\\2z -z = 41 -4\\z = 37[/tex]

Hence

z = 37°

Keywords: Angles, Triangles

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Answer:

Equation: 3x + 2 = 50

# of friends: 16

Step-by-step explanation:

3 chocolates each or 3x because what "each" means is not known yet-

3x

Note: Always use x for an unknown value

2 remaining chocolates-

3x + 2

50 total chocolates-

3x + 2 = 50

Remove two to keep the x alone-

3x = 48

Remove the 3 or the "Coefficient" by dividing-

48/3 = 16

x = 16

Equation: 3x + 2 = 50

# of friends: 16

Answer:

The correct answer is 149.00.

Step-by-step explanation:

Part = Percent • Whole

Adding a 65% markup to the original price of 105 is simply multiplying the price by the decimal form of the markup, then adding that to the original price. Next, you would find 20% of your sum by multiplying the sum by 0.20. Then you would find 7.5% of that and add this to the price.

Solve:

105 • 0.65 = 68.25                      Find 65% of 105

105 + 68.25 = 173.25                  Add the markup to the price

173.25 • 0.20 = 34.65                 Find the discount of the new price

173.25 - 34.65 = 138.60              Subtract discount from price

138.60 • 0.075 = 10.395             Find sales tax of new price (Or round to 10.40)

138.60 + 10.395 = 148.995         Add sales tax to new price (Or 149)

Because money only goes to the hundredths place, we can either start rounding to that place in the second to last step - rounding 10.395 to 10.40 - or we can round our answer - 148.995 rounds to 149. Either way, we will still get an answer of $149.00.

Hope this helps.

♥A.W.E.S.W.A.N.♥

Answer:

B $149.00

Step-by-step explanation:

105 * 1.65 =   173.25      (100%+ 65%)    Price with the markup            

173.25 *0.80  138.60  (Price after 20% discount .The price will be the 80%)    

 138.60 * 1.075 =   148.995  (price with 7.5  sales tax.)  

a) 5000 represents the initial amount that Sue invested in a money market account

b) The annual percentage rate is 7%

c) Sue will have $7012.76 after 5 years

Step-by-step explanation:

The exponential growth function is [tex]f(t)=P(1+r)^{t}[/tex] , where

Sue invests in a money market account. The balance of the account in dollars after t years can be represented by the function [tex]5000(1.07)^{t}=B[/tex]

∵ The function is [tex]5000(1.07)^{t}=B[/tex]

∵ The form of the exponential growth function is tex]f(t)=P(1+r)^{t}[/tex]

- By comparing the two functions

∴ P = 5000

∵ P is the initial amount

∴ The initial amount is 5000

a) 5000 represents the initial amount that Sue invested in a money market account

∵ [tex](1+r)^{t}=(1.07)^{t}[/tex]

∴ 1 + r = 1.07

- Subtract 1 from both sides

∴ r = 0.07

∵ r is the annual rate in decimal

∵ 0.07 × 100% = 7%

∴ The annual rate is 7%

b) The annual percentage rate is 7%

∵ t = 5

- Substitute the value of t in the function

∴ [tex]5000(1.07)^{5}=B[/tex]

∴ B = 7012.76

c) Sue will have $7012.76 after 5 years

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Answer:

-20/21

Step-by-step explanation:

Answer:

$4,500

Step-by-step explanation:

4,200e^3(.023)

d/dx cos^2(5x^3)

= d/dx [cos(5x^3)]^2

= 2[cos(5x^3)]

= - 2[cos(5x^3)] * sin(5x^3)

= - 2[cos(5x^3)] * sin(5x^3) * 15x^2

= - 30[cos(5x^3)] * sin(5x^3) * x^2

Explanation:

d/dx x^n = nx^(n - 1)

d/dx cos x = - sin x

Chain rule:

d/dx f(g(...w(x))) = f’(g(...w(x))) * g’(...w(x)) * ... * w’(x)

Step-by-step explanation:

f(x) = cos²(5x³)

f(x) = (cos(5x³))²

Use chain rule to find the derivative:

f'(x) = 2 cos(5x³) × -sin(5x³) × 15x²

f'(x) = -30x² sin(5x³) cos(5x³)

If desired, use double angle formula to simplify:

f'(x) = -15x² sin(10x³)

Answer:

$ 929

Step-by-step explanation:

Given: Original price of stay in hotel= $240 per night

           Discount= 25%

           Total taxes and fee = 29%

Now, finding the price of stay for four night.

Using unitary method to solve:

As we know original price of hotel stay per night is $240

∴ Price of four night stay at original price is [tex]240\times 4 = \$ 960[/tex]

Next, 25% discount given on the price of four night stay.

∴ [tex]960 - \frac{25}{100} \times 960 = 960 - 240[/tex]

= [tex]\$ 720[/tex]

∴Price of four night hotel stay after discount is [tex]\$ 720[/tex]

Now, 29% taxes and fees been added to the price.

Price of hotel stay after discount + Taxes paid.

[tex]720 + ( \frac{29}{100} \times 720) = 720 + 209[/tex]

= [tex]\$ 929[/tex]

∴ [tex]\$ 929[/tex] is the price to be paid for four night stay at hotel after discount and taxes.

Answer:

Ariel

Step-by-step explanation:

Ariel is diving which is below the surface of the water

Answer:

ariel

Step-by-step explanation:

Answer:

2 transformations

Step-by-step explanation:

1)Reflection

2) Rotation 180 degrees clockwise

Answer:

(1, -2).

Step-by-step explanation:

y=-4x+2

y=6x-8

Substitute y = 6x - 8 in the first equation

6x - 8 = -4x + 2

10x = 10

x = 1

So y = 6x - 8

= 6(1) - 8

= -2.

The pair should be (1, -2).

The equation is y=-4x+2 and y=6x-8

here we Substitute y = 6x - 8 in the first equation

So,

6x - 8 = -4x + 2

10x = 10

x = 1

Now

y = 6x - 8

= 6(1) - 8

= -2.

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You will pay 5% sales tax.

Step-by-step explanation:

Given,

Cost of item = $40

Sales tax = $2

Percent of sales tax = [tex]\frac{Sales\ tax}{Cost\ of\ item}*100[/tex]

[tex]Percent\ of\ sales\ tax=\frac{2}{40}*100\\Percent\ of\ sales\ tax=\frac{200}{40}\\Percent\ of\ sales\ tax=5\%[/tex]

You will pay 5% sales tax.

Keywords: sales tax, division

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Answer:

A. (2 × 6) + (3 × 8) + 6²

72 sq. inches.

B. 9 in. by 8 in.

Step-by-step explanation:

The white shape has dimensions of 2 in. by 6 in. and the green shape is a square.

So, the dimension of the green shape is 6 in. by 6 in.

Therefore, the length of the blue shape is (6 + 2) = 8 in.

Now, given that the area of the blue shape is 24 sq. in.

So, the width of the blue shape is [tex]\frac{24}{8} = 3[/tex] in.

Now, if we consider the total figure, then area of the entire figure = area of the  white shape + area of the green shape + area of the blue shape.

= (2 × 6) + (3 × 8) + 6² (Answer)

= 12 + 24 + 36  

= 72 sq, inches. (Answer)

Now, the length of the entire figure is (6 + 3) = 9 in. and width of the entire figure is (2 + 6) = 8 in.

Therefore, the dimensions of the entire figure will be 9 in. by 8 in. (Answer)

Answer:

Elga's age is 5.

Step-by-step explanation:

3x=y

x+y=20

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

x+3x=20

4x=20

x=20/4

x=5

3(5)=15

y=15

x=5, y=15

(a) The recursive rule for the sequence is:

     [tex]a_{1}[/tex] = 5 ;  [tex]a_{n}=a_{n-1}+15[/tex]

(b) The explicit rule for the sequence is: [tex]a_{n}=15n-10[/tex]

Step-by-step explanation:

The form of the recursive rule of an arithmetic sequence sequence is

[tex]a_{1}[/tex] = first term ; [tex]a_{n}=a_{n-1}+d[/tex] , where

The form of the explicit rule of nth term of an arithmetic sequence is

[tex]a_{n}=a+(n-1)d[/tex] , where

∵ The sequence is 5, 20, 35, 50, ….......

∵ 20 - 5 = 15 , 35 - 20 = 15 and 50 - 35 = 15

∴ The sequence is an arithmetic sequence with a common

   difference 15

∵ The form of the recursive rule of the arithmetic sequence is

   [tex]a_{1}[/tex] = first term ; [tex]a_{n}=a_{n-1}+d[/tex]

∵ First term = 5

∵ d = 15

∴ [tex]a_{1}[/tex] = 5 ;  [tex]a_{n}=a_{n-1}+15[/tex]

(a) The recursive rule for the sequence is:

     [tex]a_{1}[/tex] = 5 ;  [tex]a_{n}=a_{n-1}+15[/tex]

∵ The form of the explicit rule of nth term of an arithmetic

   sequence is [tex]a_{n}=a+(n-1)d[/tex]

∵ a = 5 and d = 15

- Substitute these values in the rule

∴ [tex]a_{n}=5+(n-1)15[/tex]

- Simplify the right hand side

∴ [tex]a_{n}=5+15n-15[/tex]

- Add like terms

∴ [tex]a_{n}=15n-10[/tex]

(b) The explicit rule for the sequence is [tex]a_{n}=15n-10[/tex]

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Answer:

Kelan gets 135 Euros.

Step-by-step explanation:

First of all you add up the ratio getting 10, with 10 you divide it from 450 getting 45. Then you multiply 3 (Kelan's Part) by 45 getting your answer. Tell me if this helped, i'd appreciate knowing.

This question is based on simple mathematics. Therefore, the answer is £135.

Given:

Harry, regan and kelan share £450 in the ratio 2 : 5 : 3.

We have to determined the money would kelan get.

According to questions,

First of all, adding  up the ratios.

We get,

2+5+3 = 10

Regan and kelan share £450. So divided £450  by 10, and solve it further.

We get,

[tex]\dfrac{450}{10} = 45[/tex]

Now, multiplying 45 by 3 (Kelan's part).

We get,

[tex]45\times3= 135[/tex]

Therefore, the money that kelan get is £135.

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Answer:

[tex]\theta=54.25\°[/tex]

Step-by-step explanation:

we know that

When the sine of angle X is equal to the cosine of angle Y, then angle X and angle Y are complementary angles

so

[tex]X+Y=90\°[/tex]----> equation A

In this problem

[tex]X=\theta[/tex]

[tex]Y=(\theta-18.5)[/tex]

substitute in the equation A

[tex]\theta\°+(\theta-18.5)\°=90\°[/tex]

[tex]2\theta=90+18.5[/tex]

[tex]2\theta=108.5[/tex]

[tex]\theta=54.25\°[/tex]

Answer:

Step-by-step explanation:

It's perfectly logical that as the number of hours spent watching TV increases, the grades received on the quizzes drop.  There's a negative correlation between the two variables.