Solve for x using logarithms.

[tex]3^{2-3x} = 4^{2x+1}[/tex]

180 - 115 = 65

4k + 5 + 6k + 10 + 65 = 180

10k + 80 = 180

10k = 100

k = 10

Answer: 1.365 × 10-6

Step-by-step explanation:

All the numbers in scientific notation written in the form a × 10b, where b is an integer and the coefficient a is a non-zero real number between 1 and 10 in absolute value.

To convert 0.000001361 into scientific notation also known as standard form, follow these steps:

Move the decimal 6 times to right in the number so that the resulting number, m = 1.361, is greater than or equal to 1 but less than 10

Since we moved the decimal to the right the exponent n is negative

n = -6

Write in the scientific notation form, m × 10n

= 1.361 × 10-6

Therefore, 0.000001361 in scientific notation is 1.361 × 10-6 and It has 4 significant figures. In scientific e-notation it is written as 1.361e-6.

By applying the proportional relationship formula, the height of the barn is 45 ft.

In a proportional relationship, each pair of data points is related in the same way, typically by multiplying them together. A proportionate relationship can be seen in a set of numbers, an equation, or a graphical representation. In this case, we are given that :

The height of the farmer (X1)  = 6 ft

The height of the farmer’s shadow (Y1) = 10 ft

The height of the barn with farmer’s shadow (Y2) = 75 ft

To calculate the actual height of the barn (X2), we can use this following formula:

X1 : Y1 = X2 : Y2

6 : 10 = X2 : 75

X2 = (6 : 10) x 75

X2 = 45

Thus, the height of the barn is 45 ft

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The correct answer is Between 157,500 and 968,187 phones.

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Answer:

Step-by-step explanation:

Answer: B. x + 52

Step-by-step explanation:

The unknown number is represented as x.

The phrase "52 more than" means adding 52 to something

From this we can say that the expression is B. x + 52

The equation Wendy must have solve is  14+2x = 54 - 6x Option D

A linear equation is an equation whose highest power of the variable is 1.

recall that equation is a mathematical statement showing that two opposite sides are equal

the equation can be solved as follows

14+2x = 54 - 6x

Collecting the like terms from the equation we have

2x+6x = 54 -14

This implies that

8x = 40

Making x the subject of the relation

Therefore x=40/8 = 5

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Answer: 40 cm

Step-by-step explanation: 2/5 x 100 =

200/5 = 40

Answer: Since 2005 the amount of money spent on restaurants in a certain country has increased at a rate of 4% each year. In 2005, about 670 billion was spent on restaurants.

To find the amount spent on restaurants in 2013, we can use the formula:

A = P(1 + r)^t

Where:

A = the final amount

P = the initial amount (670 billion)

r = the rate of increase (0.04 or 4%)

t = the number of years since the initial amount (8 years)

Plugging in the values, we get:

A = 670 billion (1 + 0.04)^8

A = 670 billion (1.04)^8

A = 670 billion * 1.4304

A = 951.8 billion

So, if the trend continues, about 951.8 billion will be spent on restaurants in 2013.

Step-by-step explanation:

The height of the cylinder will be 6 meters.

One of the most fundamental curvilinear geometric shapes, a cylinder has traditionally been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. In some modern branches of geometry, a cylinder may also be defined as an infinite curvilinear surface.

Given that the surface area of a cylinder is 24π m². The distance around the base of the cylinder is 3π meters and the diameter of the base of the cylinder is 4 meters.

The height of the cylinder will be calculated as:-

Surface area = 2πrh

24π = 2πrh

r x h = 12

The diameter is 4 meters so the radius is 2 cm.

r x h =- 12

h = 12 / r

h = 12 / 2 = 6 meters

The height of the cylinder will be 6 meters.

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Their ages are given below:

The sum of their ages would be 57 years.

The current age of Nico is 27 - 7 = 20 years.

Therefore, Maria and Luis's current ages would be 8 years older than when they were 10 and 11 respectively,

so Maria is currently 18 years old and Luis is currently 19 years old.

Therefore, the sum of their current ages is 18 + 19 + 20 = 57 years.

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8 years ago, Maria was 10 years old and Luis was 11 years old. In 7 years Nico will be 27 years old, what is the sum of the current ages of Maria, Luis and Nico?

The Pythagorean identity is proved to be 1 + cot2 = cosec2, which results in sin2 + cos2 = 1.

With the value of sine and the quadrant in which the angle is placed, we can use the Pythagorean identity to get the angle of cosine.

If we are given the cosine value and the quadrant in which the angle is placed, we may similarly compute the angle of sine.

Cosec2 = 1 + cot2

Therefore,

cosec2 = 1/sin2

Tan2 = cos2 / sin2 = cot2 = 1

Hence,

1/sin2 = 1 + cos2 / sin2.

multiplied both sides by sin2 as a result.

1(sin2 ) + (cos2 / sin2 ) = (sin2 / sin2 )

cos2 + sin2 = 1.

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Answer:

The new printer prints 100 pages about 4 minutes faster than the old printer

The new printer prints 22 move pages per minute than the old printer

Step-by-step explanation:

New printer's equation

y = 37x

Old printer's equation

y = 15x  

Since the graph is proportional you can take any ordered pair (except (0,0) and put it the form y/x to find the slope. For example, The point (2,30) 30/2 = 15 or the point (4, 60) 60/4 = 15 and so on.

The difference between the rates is 22 pages a minute (37 - 15).

Substituted in 100 for y and solve for x (minutes)

New:

y = 37x

100 = 37x  Divide both sides by 37

100/37 =37x/37

3 = x  This is rounded

This means that it takes about 3 minutes to print 100 pages.

Old:

y = 15x

100 = 15x  Divide both sides by 15

100/15 = 15x/15

7 = x   This is rounded

This means that it takes about 7 minutes to print 100 pages.

The new printer can print the same amount of page (100) in 4 less minutes (7-3)

Answer:

An exponential function has the form y = ab^x, where a is the initial value, b is the base of the exponential function (in this case, 1 + the growth rate as a decimal) and x is the number of years.

Given that the initial population is 430 and the annual rate of increase is 9%, we can write the exponential function as:

y = 430 (1 + 0.09)^x

Where y is the quail population after x years, and the base (1 + 0.09) represents the growth rate of 9%. This function gives the population of the quail for any number of years x.

It's worth noting that the function is in the form of y = a*(1+r)^x where a is the initial value, r is the annual growth rate, and x is the number of years

The prediction that Lucas can make about the next shipment of ornaments is that H. A delivery of 6 boxes will contain 6 more broken ornaments than a delivery of 3 boxes.

It should be noted that a word problem in mathematics simply refers to a question that is written as a sentence or in some cases more than one sentence which requires an individual to use his or her mathematics knowledge to solve the real life scenario given.

In this case, since the department store receives a shipment of 3 boxes of glass ornaments and there are 25 ornaments in each box.

Lucas then opens the boxes and finds that 6 of the ornaments are broken. This implies that there are 2 broken for each box.

In this case, a delivery of 6 boxes will contain 6 more broken ornaments than a delivery of 3 boxes.

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The expression of (²/₃)⁻¹ without an exponent is given as;

3/2

The different laws of exponents are;

1) Power Law of Exponents which is expressed as; (x^m)ⁿ = x^(mn)

2) Quotient Law of Exponents is expressed as; x^m ÷ x^n = x^(m - n)

3) Product Law of Exponents is expressed as; x^(m) * x^(n) = x^(m + n)

4) Law of Reciprocal is expressed as; a⁻ⁿ = 1/aⁿ

Now, we want to rewrite the expression (²/₃)⁻¹ without using an exponent.

From the law of reciprocal of exponents we can say that;

(²/₃)⁻¹ = 1/(²/₃)

= 3/2

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for any value of n, the nth term of the equation will be:

Aₙ = -3* 2ⁿ⁻¹

A geometric sequence is a sequence of numbers that follows a pattern where the next term is found by multiplying by a constant called the common ratio, r.

From the general formula of the nth term of the geometric sequence:

Aₙ = A₁* rⁿ-1

Given a geometric sequence:

-3, -6, -12, 24,.....

From the general formula of the nth term of the geometric sequence:

Aₙ = A₁* rⁿ⁻¹

in our case

a₁ = -3

ratio(r) = -6/-3 = 2

Thus for any value of n, the nth term of the equation will be

Aₙ = -3* 2ⁿ⁻¹

For instance,

8th term of the given sequence:

A₈ = -3* 2⁸⁻¹

A₈ = -3* 2⁷

A₈ = -384

Therefore, for any value of n, the nth term of the equation will be

Aₙ = -3* 2ⁿ⁻¹

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Answer:

6 = D

7 = F

8 = A

9 = B

10 = C

Have a nice day ;)

The slope of a line in the xy plane b = 13/3

The slope of a line in the xy plane can be calculated by using the equation m = (y2-y1)/(x2-x1). In this equation, m is the slope, x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point.

Using this formula, we can solve for b.

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

8/3 = (b - (-1)) / (2 - (1/2))

24/3 = (b + 1) / (5/2)

24/3 * (5/2) = (b + 1)

40/3 = b + 1

40/3 - 1 = b

b = 13/3

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On solving the linear equations x + 3y = 5 and -3x - 2y = 20, the value for x is x = -10 and the value of y is y = 5. When solved independently the value is obtained as -2x + y = 25.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The first linear equation is - x + 3y = 5

The second linear equation is - -3x - 2y = 20

On evaluating individually the equation obtained is -

-3x - 2y + x + 3y = 20 + 5

-2x + y = 25

Multiply equation (1) with 3 -

3(x + 3y) = 5 × 3

3x + 9y = 15.....(3)

Adding equation (3) and (2) -

-3x - 2y + (3x + 9y) = 20 + 15

-3x - 2y + 3x + 9y = 35

-2y + 9y = 35

7y = 35

y = 35/7

y = 5

Substituting the value of y in equation (1) -

x + 3y = 5

x + 3(5) = 5

x + 15 = 5

x = 5 - 15

x = -10

Therefore, the value of x and y is -10 and 5 respectively.

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George is not correct because we can find the least common multiple of 8 and 6 which is 24. This allows us to solve the problem with simple numbers and ease our calculations to get the answer in the simplest form.

The equation solved by George is 3/8 - 1/6 = 18/48 - 8/48

To solve the given fractions in the simplest form, we will find the common denominator which would be the least common multiple of both denominators.

we can see that the numbers in the denominator are 8 and 6.

Multiples of 8: 8, 16, 24, 32, 40,...

Multiples of 6: 6, 12, 18, 24, 30,...

Hence, LCM of 8 and 6 is 24.

thus we can solve the equation as follows -

3/8 - 1/6

= 3*3/8*3 - 1*4/6*4

= 9/24 - 4/24

= 5/24

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The functions that show a positive rate of change include the following:

A. y = 2x/3 - 2.

B. y = 7 - (-3x).

In Mathematics, the slope-intercept form of a line can be represented or modeled by using this linear equation:

y = mx + c

Where:

Generally speaking, a function with a positive rate of change represents an increasing function because its numerical value is increasing over time such as in the following increasing function:

y = 2x/3 - 2.

Slope, m = 2/3 (positive slope).

y = 7 - (-3x)

y = 7 + 3x

Slope, m = 3 (positive slope).

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The functions that show a positive rate of change are y = 2/3x - 2 and y = 7 - (-3x).

A positive function is a function that has the values of x that are greater or higher than zero on the graph. It means that they are always increasing on the positive y-axis i.e. f(x) > 0.

The function's outputs known as the y-values, however, must be larger than zero even though the domain values i.e. (x-values) might be negative. A positive function, then, has values that are positive for all of its domain's parameters.

To determine the positivity of the function on the graph, the slope must also be positive. Using the linear function form y = mx + b.

where;

From the given information, the functions that demonstrated a positive slope are:

Therefore, we can conclude that the functions that show a positive rate of change are y = 2/3x - 2 and y = 7 - (-3x).

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Four sides and one angle, three sides and two included angles, or two neighboring sides and three angles, to construct a unique quadrilateral.

If the two diagonals and three sides of a quadrilateral are known, it is possible to create it in a singular way. If the two neighboring sides and three angles of a quadrilateral are known, it can be built in an original fashion.

A polygon called a quadrilateral has four sides, four angles, and four vertices. If the size of a quadrilateral's sides and diagonal are known, it is possible to create it in a singular way.

They must have the same number of sides as angles because they are polygons and have four angles. Every quadrilateral has inner angles that sum to 360 degrees.

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Using the Midsegment Theorem, the value of x is: 6.

The Midsegment Theorem states that in a triangle, the segment that connects the midpoints of two sides of the triangle is parallel to the third side and is half as long as the third side. In other words, the midsegment of a triangle is parallel to the third side and has half the length of the third side.

Therefore the ratio of DE to AB would be: 1/2

Ratio of EC to AC would be: x/(x + 6)

Thus:

x/(x + 6) = 1/2

Cross multiply

2x = x + 6

2x - x = 6

x = 6

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The area of the shaded part of the rectangle is 152 cm².

The area of the shaded part is the difference between the area of the larger rectangle and the smaller square.

Area of the shaded part = area of the larger rectangle - area of the square

Area of the larger rectangle = length x width

12cm x 18cm = 216 cm²

Area of the square = length²

= 8² = 64 cm²

Area of the shaded part = 216 - 64 = 152 cm²

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24 cookies were on the plate before anyone started feeling hungry.

Let X be the number of cookies on the plate before anyone started feeling hungry.

After Mrs. Baxter ate 1/6 of the cookies, there were,

X - (X * 1/6) = 5/6X cookies left.

After Mr. Baxter ate 1/5 of the remaining cookies, there were,

5/6X - (5/6X * 1/5) = 5/6X- 1/6X = 4/6X cookies left.

After Anna ate 1/4 of the remaining cookies, there were,

4/6X - (4/6X * 1/4) = 4/6X- 1/6X = 3/6X cookies left.

After Tyler ate 1/3 of the remaining cookies, there were

3/6X - (3/6X * 1/3) = 3/6X- 1/6X = 2/6X cookies left.

After Adam ate 1/2 of the remaining cookies, there were,

2/6X - (2/6X * 1/2) = 2/6X- 1/6X = 1/6 cookies left.

We know, after Adam eats, 4 cookies are left, therefore,

1/6X = 4

So X = 4 *6 = 24

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The Probability that the player will win against each of the four opponents at least once is 0.7942.

The player has a P(W) = 0.80 winning chance.

The likelihood that the player will lose is hence P(L)= 1 - P(W) = 0.20.

The player is up against 4 different opponents in the video game.

It is stipulated that the game finishes when a player is defeated by an opponent.

The player can win in any of the following ways: L, WL, WWL, WWWL, and WWWW.

The results from each of the four opponents are distinct, meaning that the outcome of a game played against one opponent has no bearing on the outcome of a game against another.

P (Player defeats all 4 opponents) = 0.4096 is the probability that a player will win a game against all four opponents.

Thus, the probability that the player defeats all four opponents in a game is 0.4096.

(b) The likelihood that a player will win a game against at least two opponents is,

P = 1 (Player defeats at least 2) P = 0.64 when the player loses the first and second games.

Therefore, there is a 0.64 percent chance that the player will win at least two games against opponents.

Let X equal the number of times the player triumphs over all four opponents.

The likelihood that a player will win a game against all four opponents is,

P(WWWW) = 0.4096

The probability that a player will triumph over each of the four opponents at least once is,

P (X ≥ 1) = 1 - P (X < 1)

= 1 - P (X = 0)\s = 0.7942

Therefore, there is a 0.7942 probability that the player will win against all four opponents at least once.

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The equation which represents Lisa's earnings, e, relative to the number of hours, h, that she works assume a proportional relationship is e = 7.5h

Amount Lisa makes for babysitting = $30

Number of hours Lisa babysit = 4 hours

Total earnings = e

Number of hours = h

If the relationship is proportional, then

k = constant of proportionality

e = k × h

30 = k × 4

30 = 4k

divide both sides by 4

k = 30/4

k = 7.5

So therefore,

e = k × h

e = 7.5 × h

e = 7.5h

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The vertical stretch of f(x) will be -2f(x), 5.5f(x) - 1, 7f(x)

As is well known, the vertical stretch of f(x) occurs when the value of the parameter is changed in the following general form:

When a constant c with a value larger than one is scaled across the board of a function, it results in a vertical stretch. It is written as the vertical stretch from the original function f(x) to the new, stretched function g(x)=cf(x) where c>1 g ( x ) = c f ( x ) where c > 1.

f(k(x-d)) = y = a+ c

when |a| exceeds 1 since The function is stretched vertically by a dilation factor of |a| when |a| > 1 (when a > 1).

Thus, we select F, D, and A.

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The full question:

Consider the function f(x). Select all of the following that includes a vertical stretch of f(x)

A. -2f(x)

B. 0.25f(x)

C. -f(x) + 6

D. 5.5f(x) - 1

E. f(x) + 9

F. 7f(x)