Match the following reasons with the statements to complete the following proof.

By the application of algebraic procedures and logarithm properties, the roots of the logarithmic equation [tex]\log \sqrt{x} + \log \sqrt{x-1} = 1[/tex] are x = 2 and x = -1.

In this question we need to solve a logarithmic equation by means of algebraic procedures and logarithm properties:

[tex]\log \sqrt{x} + \log \sqrt{x-1} = 1[/tex]

[tex]\log \sqrt{x\cdot (x-1)} = 1[/tex]

[tex]\log \sqrt{x^{2}-x} = 1[/tex]

[tex]\frac{1}{2}\cdot \log (x^{2}-x) = 1[/tex]

[tex]\log (x^{2}-x) = 2[/tex]

[tex]x^{2}-x = 2[/tex]

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

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

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

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

[tex]x^{2} - 2 = x[/tex]

[tex]x^{2}-x-2 = 0[/tex]

[tex](x-2)\cdot (x + 1) = 0[/tex]

[tex]x = 2 \,\lor \,x= -1[/tex]

By the application of algebraic procedures and logarithm properties, the roots of the logarithmic equation [tex]\log \sqrt{x} + \log \sqrt{x-1} = 1[/tex] are x = 2 and x = -1.

To learn more on logarithms: https://brainly.com/question/20785664

#SPJ1

The two options are R₀ 270 , R₀ 90 , Option C and D .

The complete question is attached in the image.

There are two types of transformations namely , Rigid Transformation and Non- Rigid Transformation

The rigid transformations do not change the shape and size of the preimage .

The non Rigid transformation do not change the shape but change the size of the preimage.

The rule rotating the origin , 270 degree is

(x,y) ----> (y , -x)

as (x,y) is (0,5)

Therefore after 270 degree transformation

(y,-x) is (5 , 0)

So, the clockwise rotation is 90 degree about origin

(x,y) ----> (y , -x)

as (x,y) is (0,5)

Therefore after 90 degree transformation

(y,-x) is (5 , 0)

Therefore the two options are R₀ 270 , R₀ 90 , Option C and D .

To know more about Types of Transformations

https://brainly.com/question/5020733

#SPJ1

Mixtures D and E have a ratio of 5:12 and will create the same shade of pink. Options D and E are correct.

A ratio is an ordered pair of integers a and b expressed as a/b, with b never equaling 0. A percentage is a mathematical expression in which two ratios are specified to be equal.

Given data;

red wax  = 12 ml

white wax = 5 ml

A red wax: white wax = 5:12

B. red wax: white wax=4:1

C red wax white wax = 13: 6

D. red wax: white wax=12:5

E. red wax white wax=12:5

Hence, mixtures D and E have a ratio of 5:12 and will create the same shade of pink.

To learn more about the ratio, refer to the link;

https://brainly.com/question/14335762

#SPJ1

Answer:

y = 10x + 422

Step-by-step explanation:

A cylinder is a three-dimensional structure formed by two parallel circular bases connected by a curving surface. The radius of the cylindrical tin is 7 cm.

A cylinder is a three-dimensional structure formed by two parallel circular bases connected by a curving surface. The circular bases' centers overlap each other to form a right cylinder.

The height of the cylindrical tin can is 7 cm, while the total surface area is 462 cm². Therefore, the total surface area of the tin will be,

The total surface area = area of base + Lateral surface area

462 cm² = πR² + 2πRH

462 = π(R² + 2R7)

147 = R² + 14R

R² + 14R - 147 = 0

R = 7, -21

Since the radius can not be negative.

Hence, the radius of the cylindrical tin is 7 cm.

Learn more about Cylinder:

https://brainly.com/question/12248187

#SPJ1

Answer:

Step-by-step explanation:

If the function is f(x) = √(7x – 21). Then the inverse function of the function will be f⁻¹(x) = (1/7) x² + 3.

Let the function will be

f: X → Y

Then the inverse function will be

f⁻¹: Y → X

The function is given below.

f(x) = √(7x – 21)

Then the inverse function will be

x = √[7{f⁻¹(x)} – 21]

x² = 7f⁻¹(x) – 21

7f⁻¹(x) = x² + 21

f⁻¹(x) = (1/7) x² + 3

Learn more about inverse function here:

https://brainly.com/question/19425567

#SPJ1

i love robloox

               n n n nn

Answer:

m = 3

Step-by-step explanation:

Gradient of a line is given by the formula

m= (y2-y1) /(x2-x1) which is change y divided by change in x

m = (-4-2)/(-2-0)

m = -6/-2

m = 3

Answer: the gradient is 3

Step-by-step explanation:

for a full explanation refer to the document and the ones written colorfully are the explanation. So, if you want just the workings they are the ones written in black.

and I drew an eye! Have a nice day!

Volume is a three-dimensional scalar quantity. The volume of the triangular prism is 66cm².

A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.

Given the length of the triangle is 4cm, while the width of the triangle is 11cm, therefore, the area of the base of the triangle will be,

Area of triangle = 0.5 × 4cm × 11cm

Area of triangle = 22 cm²

Now, the volume of the prism with a base area of 22cm² and a height of 3 cm is,

Volume of the triangular prism = 22cm² × 3cm = 66cm³

Hence, the volume of the triangular prism is 66cm².

Learn more about Volume:

https://brainly.com/question/13338592

#SPJ1

Answer: Below is the correct data.

Fraction: [tex]\frac{23}{100}[/tex]

Decimal: 0.23

Percent: 23%

I hope this helps! Pls mark brainliest!! :)

Answer:

7:24pm

Step-by-step explanation:

At 50 mph, it will take 195/50 = 3.9 hours to reach London

3.9 hours = 3 .9 * 60 = 3 hours and 54 minutes(54 = 0.9 * 60 )

3:30 + 3:34 ==> 7:24

(one way to do this is to add up the hours and minutes separately so we get 6:84) Since 84 minutes = 1 hour and 24 minutes, just add 1:24 to 6 to get 7:24

Another way to do this is to see that if the distance was 200 miles, then it would take exactly 4 hours (200/50) to reach at 7:30pm But it is short by 5 miles. At the speed of 50mph, the time taken to travel 5 miles = 60/50 * 5 = 6 minutes so deduct 6 minutes from 7:30pm to get 7:24pm

If the train is slower average speed, then the arrival time is delayed beyond 7:24pm

Answer:

Basically find the unit rate by doing 3.99/24 = $0.16625/1oz

1 quart = 32 ounces so 4.8/32 = $0.15/1oz

1 gallon = 153.722 ounces so 11/153.722 = 0.07

49.6/768.61 = 0.06

Step-by-step explanation:
D.

Since the length of this rectangle is twice its breadth, the length is equal to 40 cm.

Mathematically, the area of a rectangle can be calculated by using this formula:

Area = Length × Width

Based on the information provided, we have:

Substituting the parameters into the formula, we have:

LB + 75 = (L - 5)(B + 5)

2B² + 75 = (2B - 5)(2B + 5)

2B² + 75 = 2B(B + 5) - 5(B + 5)

2B² + 75 = 2B² + 10B - 5B - 25

5B = 75 + 25

5B = 100

B = 100/20

B = 20 cm.

For the length, we have:

L = 2B

L = 2 × 20

L = 40 cm.

Read more on rectangle here: https://brainly.com/question/25292087

#SPJ1

The point slope form of line is y+4 = -4( x+3)

The point slope form is used to find the equation of the straight line which is inclined at a given angle to the x-axis and passes through a given point.

Given:

m=4

We know the general form is

y - b = -4(x - a)

So, from the graph attached below

b= -4, a= -3

So, y-(-4)= -4(x-(-3))

y+4 = -4( x+3)

Hence,  point slope form of line is y+4 = -4( x+3).

Learn more about tis concept here:

https://brainly.com/question/13436232

#SPJ1

Answer:

$10

Please mark me as brainliest

The equation of the parabola is y = 2(x - 2)^2 + 4

The given parameters are:

A parabola is represented as:

y = a(x - h)^2 + k

Substitute (h,k) = (2,4)

y = a(x - 2)^2 + 4

Substitute (x,y) = (3,6)

6 = a(3 - 2)^2 + 4

Evaluate the difference

6 = a(1)^2 + 4

This gives

6 = a + 4

Subtract 4 from both sides

a = 2

Substitute a = 2 in y = a(x - 2)^2 + 4

y = 2(x - 2)^2 + 4

Hence, the equation of the parabola is y = 2(x - 2)^2 + 4

Read more about parabola at:

https://brainly.com/question/4061870

#SPJ1

Answer:

85° = ∠x {corresponding angles}

∠x = ∠1 {vertically opp. angles}

And, ∠2 + 103° = 180° ( co-interior angles)

∠2 = 180° - 103°

∠2 = 77°

Now, m∠1 + m∠2 = 85° + 77°

= 162°

Hemce, option A. is the right answer.

Answer:

[tex]x=-18, y=-44[/tex]

Step-by-step explanation:

[tex]y=3x+10\\y=2x-8\\\\3x+10=y\\2x-8=y\\3x-y=-10\\2x-y=8\\\\-2(3x-y=-10)\\3(2x-y=8)\\\\-6x+2y=20\\6x-3y=24\\-y=44\\y=-44\\-44=3x+10\\-54=3x\\x=\frac{-54}{3} \\x=-18\\\\x=-18, y=-44\\[/tex]

CHECK

[tex]-44=3(-18)+10\\-44=-54+10\\-44=-44\\\\-44=2(-18)-8\\-44=-36-8\\-44=-44[/tex]

The graph which resembles the graph of f(x)=[tex]2^{x}[/tex] but is reflected over the x-axis is g(x)=[tex]2^{x} +1[/tex]

Given Function f(x)=[tex]2^{x}[/tex].

We have to generate a function whose graph is similiar to the given function and is reflected over the x axis.

If we take first function g(x)=2x-1

put the value of x=-1 then the value of function g(x)=2*(-1)-1=-2-1=-3 means the value of y in this function will be negative means the graph will be under x-axis.

Taking second function g(x)=[tex]2^{x} +1[/tex]

for all the negative values of x it posses a positive value of y so the graph will be over the x axis.

Taking third function g(x)=-[tex]2^{x}[/tex]

In the above function for each value of x the function possess a negative value so the graph will be under x axis.

Taking fourth function g(x)=2x-1

For all values of x less than 1 the function becomes negative which takes graph below the x axis. First graph is for [tex]2^{x}[/tex] and the second is for [tex]2^{x} +1[/tex]

Hence the graph similar to [tex]2^{x}[/tex] and reflected over x axis is g(x)=[tex]2^{x} +1[/tex].

Learn more about graphs here https://brainly.com/question/4025726

#SPJ10

Problem 1

Reason:

24 people total with 3 absent. So 24-3 = 21 are present

21/24 = (7*3)/(8*3) = 7/8

==========================================================

Problem 2

Work Shown:

1 hour = 60 min

(5/6)*(1 hour) = (5/6)*(60 min)

5/6 of an hour = 50 min

1 hr + 30 min = 60 min + 30 min = 90 min

The first show is 90 minutes. Add on two more shows that are 50 min each to get 90+50+50 = 190 minutes used up

4 hours = 4*60 = 240 min

240 - 190 = 50 min

There are 50 minutes left on the tape.

==========================================================

Problem 3

Explanation:

1 1/2 = 1 + 1/2 = 1 + 0.5 = 1.5

6/(1.5) = 4

Each 6 meter board is cut into 4 equal pieces each 1.5 meters long

There are 6 boards this applies to which means so far that's 6*4 = 24 pieces

Now to the other 6 boards.

3/4 = 0.75

6/(0.75) = 8

Each of the other 6 boards are cut into 8 equal pieces of 0.75 meters each

That's another 6*8 = 48 pieces

In total there are 24+48 = 72 pieces

When cutting into wood, there always is a small sliver that is discarded. While small, these slivers do add up. So yes there is some waste. Unfortunately, we don't have enough info to calculate. We would need to know the width of the saw blade.

Using proportions, considering the weight of each bar and the ratio, it is found that the ship is carrying 52 gold bars.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

Each bar weighs 5 kg, and the total weight is of 600 kg, hence the number of bars is given by:

n = 600/5 = 120.

The proportion of gold bars is given by:

p = 13/(13 + 17) = 13/30

Hence, out of 120 bars, the number of gold bars is given by:

nG = 120 x 13/30 = 4 x 13 = 52.

More can be learned about proportions at https://brainly.com/question/24372153

#SPJ1

Answer:

D:  -5 ≤ x < 1

R:  -4 ≤ y < 7

Step-by-step explanation:

The x values go between -5 and 1. The closed circle on -5 means it is included. The open circle on 1 means it is not included.

The y values go between -4 and 7. The closed circle means -4 is included, the open circle means the 7 is not included.

Answer:

A. y < -2x + 6

y≤ x + 2

Step-by-step explanation:

I put it in desmos.

Hope this helps!

If not, I am sorry.

Answer:

the leading coefficent is 16 or 16n (leading term)

Step-by-step explanation:

Step-by-step explanation:

area of shaded area = area of larger circle - area of smaller circle

a = pi * (r+7)^2 - pi * r^2

a = pi * r^2 + 14pi*r + 49* pi - pi * r^2

pi r^2 cancels

a = 14*pi*r + 49 *pi

a = 7pi(2r + 7)

Answer:

14πr + 49π

Step-by-step explanation:

Logically, to find the area of the border of a circle, we need to find both the area of the inner circle [not including border] and the area of the outer circle [including border]

Once we have found these two areas,  we find the difference (difference meaning subtraction) between the two values.

We know that the area of a circle = πr²

We can use this formula to find the area of both circles.

Inner circle:

πr²

(cannot be simplified further)

Outer circle:

π(r + 7)²

(r + 7)² = (r + 7)(r + 7) = r² + 7r +7r +49 {FOIL method}

  outer       -   inner

π(r² + 14r + 49) - πr²

π (r² +14r +49)= πr² + 14πr + 49π

πr² + 14πr + 49π - πr²

πr² - πr² = 0

So, the area of our border is: 14πr + 49π

hope this helps!!

(also, you don't have to, but marking brainliest gives you some of your points back anyway)

The equation 17q - 13- 3q = 14q + 14 have no solution option first "No solution" is correct.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

The question is incomplete.

The complete question is:

How many solutions does this equation have?

17q - 13- 3q = 14q + 14

no solution

one solution

infinitely many solutions

We have a linear equation:

17q - 13- 3q = 14q + 14

After solving:

14q = 14q + 27

0 ≠ 27 (false)

No solution

Thus, the equation 17q - 13- 3q = 14q + 14 have no solution option first "No solution" is correct.

Learn more about the linear equation here:

brainly.com/question/11897796

#SPJ1

Answer:

(49p2–490p) ÷(p–10)

Step-by-step explanation: