this question: 7_4√3+1÷7_4√3=​

Answer:

1. w = [tex]\bold{e^{7+6x}}[/tex]

2. [tex]\bold{t = 2(log_a(7) - log_a(3))}[/tex]

Step-by-step explanation:

1.

a. The exponential form of a logarithmic equation is given by:

[tex]\bold{log_a(b) }[/tex]= c is equivalent to [tex]a^c = b[/tex]

Using this rule, we can rewrite the equation:

ln w = 7 + 6x as w = [tex]\bold{e^{7+6x}}[/tex]

Therefore, the exponential form of the given equation is w = [tex]\bold{e^{7+6x}}[/tex]

2.

a. To solve for t using a logarithm with base a, we can take the logarithm of both sides of the equation:

[tex]\bold{3a^\frac{t}{2} = 7 }\\\bold{log_a3a^\frac{t}{2}= log_a(7)}[/tex]

Using the rule of logarithms that log_a(b^n) = n*log_a(b), we can simplify the left side:

[tex]\bold{log_a(3) +\frac{t}{2} log_a(a) = log_a(7)}[/tex]

Since log_a(a) = 1 for any base a, we can simplify the expression further:

[tex]\bold{log_a(3) + \frac{t}{2}*1= log_a(7)}[/tex]

Now we can solve for t by isolating it on one side of the equation:

[tex]\bold{ \frac{t}{7} = (log_a(7) - log_a(3))} \\ \bold{t = 2(log_a(7) - log_a(3))}[/tex]

Therefore, the solution for t using a logarithm with base a is[tex]\bold{t = 2(log_a(7) - log_a(3))}[/tex]

Answer:

[tex]\textsf{1.} \quad w = e^{7 + 6x}[/tex]

[tex]\textsf{2.} \quad t= \log_a\left(\dfrac{49}{9}\right)[/tex]

Step-by-step explanation:

Exponential form is a way to represent a number using an exponent, where the base is raised to a power.

The natural logarithm function is the inverse of the exponential function:

[tex]\boxed{\ln x=y \iff x=e^y}[/tex]

Therefore we can use this definition to change the given equation to exponential form:

[tex]\begin{aligned}\ln w & = 7 + 6x\\\\e^{\ln w} & = e^{7+6x}\\\\w&=e^{7+6x}\end{aligned}[/tex]

Therefore, the exponential form of the equation is:

[tex]w = e^{7 + 6x}[/tex]

[tex]\hrulefill[/tex]

To solve for t using a logarithm with base a, begin by taking the logarithm of both sides of the equation with base a:

[tex]\log_a (3a^{\frac{t}{2}})=\log_a(7)[/tex]

[tex]\textsf{Apply the log product law:} \quad \log_axy=\log_ax + \log_ay[/tex]

[tex]\log_a (3)+\log_a(a^{\frac{t}{2}})=\log_a(7)[/tex]

Subtract logₐ(3) from both sides of the equation:

[tex]\log_a (3)+\log_a(a^{\frac{t}{2}})-\log_a (3)=\log_a(7)-\log_a (3)[/tex]

[tex]\log_a(a^{\frac{t}{2}})=\log_a(7)-\log_a (3)[/tex]

[tex]\textsf{Apply the log quotient law:} \quad \log_ax - \log_ay=\log_a \left(\dfrac{x}{y}\right)[/tex]

[tex]\log_a(a^{\frac{t}{2}})=\log_a\left(\dfrac{7}{3}\right)[/tex]

[tex]\textsf{Apply the log power law:} \quad \log_ax^n=n\log_ax[/tex]

[tex]\dfrac{t}{2}\log_a(a)=\log_a\left(\dfrac{7}{3}\right)[/tex]

Apply the log law:  logₐ(a) = 1

[tex]\dfrac{t}{2}(1)=\log_a\left(\dfrac{7}{3}\right)[/tex]

[tex]\dfrac{t}{2}=\log_a\left(\dfrac{7}{3}\right)[/tex]

Multiply both sides of the equation by 2:

[tex]2 \cdot \dfrac{t}{2}=2 \cdot \log_a\left(\dfrac{7}{3}\right)[/tex]

[tex]t=2 \log_a\left(\dfrac{7}{3}\right)[/tex]

Finally, apply the log power law:

[tex]t= \log_a\left(\dfrac{7}{3}\right)^2[/tex]

[tex]t= \log_a\left(\dfrac{7^2}{3^2}\right)[/tex]

[tex]t= \log_a\left(\dfrac{49}{9}\right)[/tex]

Therefore, the solution for t in terms of logarithm with base a is:

[tex]\boxed{t= \log_a\left(\dfrac{49}{9}\right)}[/tex]

Answer:

Step-by-step explanation:

The total surface area of the resulting solid is [tex]54 cm^2 + 9 cm^2 = 63 cm^2.[/tex]

What is area of solid?

The area of a solid refers to the total amount of surface space that the solid occupies. This can be measured in square units, such as square centimeters or square meters, depending on the unit of measurement being used.

To calculate the surface area of the resulting solid, we need to consider the surface area of each of the six faces of the cube, as well as the surface area of the three internal hole walls.

The surface area of each face of the cube is 3 cm x 3 cm = [tex]9 cm^2[/tex]. Since there are six faces, the total surface area of the cube is 6 x 9 [tex]cm^2[/tex] = 54 [tex]cm^2[/tex].

Each hole has a length of 3 cm, so the surface area of the internal walls of each hole is 3 cm x 1 cm = 3 [tex]cm^2[/tex]. Since there are three holes, the total surface area of the internal walls is 3 x 3 [tex]cm^2[/tex] = 9 [tex]cm^2[/tex].

Therefore, the total surface area of the resulting solid is 54 [tex]cm^2[/tex] + 9 [tex]cm^2[/tex] = 63 [tex]cm^2[/tex].

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The result of multiplication of complex numbers is ¹/₈sin5π/3 -√3/8 cos5π/3 + i¹/₈(cos5π/3 - √3sin5π/3).

The result of the multiplication of the complex number is determined as follows;

-√3 + i  x ¹/₈ (cos 5π/3  + i sin 5π/3)

Multiply the bracket by grouping them as shown below;

= -√3 x ¹/₈ (cos 5π/3  + i sin 5π/3) + i x ¹/₈ (cos 5π/3  + i sin 5π/3)

Multiply out the bracket;

= -√3/8 cos5π/3 -  i√3/8 sin5π/3 + i¹/₈cos5π/3 + ¹/₈sin5π/3

note: i x i = 1

Collect similar terms together and add them if any;

= ¹/₈sin5π/3 -√3/8 cos5π/3 + i¹/₈(cos5π/3 - √3sin5π/3)

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Answer:

C. Chord

Step-by-step explanation:

Chord: Any segment whose endpoints lie on the circle is a chord.

Answer:

Step-by-step explanation:

A. To find the annual interest rate, compounded continuously, we can use the formula:

A = Pe^(rt)

where A is the final amount, P is the principal amount (initial investment), e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate, and t is the time in years.

We are given that P = $50,000, A = $132,558.36, and t = 15 years. We need to solve for r.

Substituting the given values into the formula, we get:

$132,558.36 = $50,000 e^(15r)

Dividing both sides by $50,000 and taking the natural logarithm of both sides, we get:

ln(2.6511672) = 15r

Solving for r, we get:

r = ln(2.6511672) / 15

r ≈ 0.045 (rounded to three decimal places)

Therefore, the annual interest rate, compounded continuously, that produced this return is approximately 0.045 or 4.5%.

B. To determine the approximate number of years it will take for the account to grow to $300,000, we can again use the formula:

A = Pe^(rt)

where A is the final amount we want to reach, P is the initial investment ($50,000), e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate we just found (0.045), and t is the time in years we want to solve for.

Substituting the given values into the formula, we get:

$300,000 = $50,000 e^(0.045t)

Dividing both sides by $50,000 and taking the natural logarithm of both sides, we get:

ln(6) = 0.045t

Solving for t, we get:

t = ln(6) / 0.045

t ≈ 27.6 years (rounded to the nearest tenth)

Therefore, it will take approximately 27.6 years from the original investment for the account to grow to $300,000.

There are 168 different area codes possible with this plan by using the multiplication principle of counting.

What is multiplication principle ?

The multiplication principle of counting, also known as the rule of product, is a counting principle in combinatorics that states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.

There are 7 possible choices for the first digit (0 through 6), 3 possible choices for the second digit (2, 3, or 4), and 8 possible choices for the third digit (any digit except 0 or 3). To find the total number of possible area codes, we can use the multiplication principle of counting:

Total number of area codes = Number of choices for first digit x Number of choices for second digit x Number of choices for third digit

Total number of area codes = 7 x 3 x 8

Total number of area codes = 168

Therefore, there are 168 different area codes possible with this plan.

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Answer:

10 pounds

Step-by-step explanation:

First, we need to determine the total weight of the pencils in ounces:

64 boxes x 2.5 ounces per box = 160 ounces

To convert this weight to pounds, we need to divide by the number of ounces in a pound:

160 ounces / 16 ounces per pound = 10 pounds

Therefore, the case of wooden pencils weighs 10 pounds.

The missing length of the triangle above would be =

A.Using radicals, the shorter leg is exactly 9. The shorter leg, up to three decimal places, is approximately 8.660.

To calculate the missing length of the given triangle, the sine formula is used such as;

a/sinA = c/sinC

where;

a = 15

A = 60°

c= ?

V= 90°

15/sin60° = b/sin90°

Make be the subject of formula;

c = 15× 1/0.866025403

c = 17.32

Using the Pythagorean formula;

c² = a² + b²

17.32² = 15² + b²

b² = 300 - 225

b² = 75

b = √75

b = 8.660.

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Solving the radical expression √(x + 3) + 4 = 5 for x gives x = -2

From the question, we have the following parameters that can be used in our computation:

√(x + 3) + 4 = 5

Subtract 4 from both sides of the equation

So, we have

√(x + 3) = 1

Take the square of both sides

x + 3 = 1

So, we have

x = -2

Hence, the solution for x is -2

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The explicit formula for an, the nth term of the sequence is f(n) = 48 * (3/2)ⁿ ⁻ ¹

From the question, we have the following parameters that can be used in our computation:

48, 72, 108…

The above definitions imply that the sequence is a geometric sequence with the followin features

First term, a = 48

Common ratio, r = 72/48 = 3/2

Using the above as a guide,

So, we have the following representation

f(n) = a * rⁿ ⁻ ¹

So, we have

f(n) = 48 * (3/2)ⁿ ⁻ ¹

Hence, the sequence is f(n) = 48 * (3/2)ⁿ ⁻ ¹

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The correct answer would be option B i.e. The inverse of the function x = -y² + 4 is a function.

What is a horizontal shift?

A horizontal shift is a type of transformation in which a function is shifted left or right along the x-axis without changing its shape or orientation. If a function f(x) is shifted to the right by a distance of c units, the resulting function is denoted by f(x - c), and if it is shifted to the left by a distance of c units, the resulting function is denoted by f(x + c). Horizontal shifts are commonly used in calculus, geometry, and other branches of mathematics to analyze the behavior of functions and to solve problems related to graphs and equations.

By plotting the inverse of the function the following graph is obtained.

We know that the function is x = -y² + 4.

The inverse of the function x = -y² + 4 would be y = , which is a vertical parabola opening upwards. When we use the horizontal line test, we find that any horizontal line would intersect the once, which means that a single input (x) would have single corresponding output (y). Therefore, the inverse of the function x = -y² + 4 is a function.

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The ordered pairs that represent points on the graph of the equation 2x = -1/2y are d) (-1,4) and e) (0,0).

An equation is a mathematical statement that shows the equality of two expressions. It consists of two sides, a left-hand side and a right-hand side, separated by an equal sign. An equation can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

The equation 2x = -1/2y can be rewritten as y = -4x. This is a linear equation in slope-intercept form, with a slope of -4 and y-intercept of 0.

To determine which ordered pairs represent points on the graph of this equation, we can substitute the x and y values into the equation and check if it satisfies the equation.

a) (4,6):

y = -4x

6 = -4(4)

6 = -16

This ordered pair does not satisfy the equation.

b) (-6,-1):

y = -4x

-1 = -4(-6)

-1 = 24

This ordered pair does not satisfy the equation.

c) (0,-4):

y = -4x

-4 = -4(0)

-4 = 0

This ordered pair does not satisfy the equation.

d) (-1,4):

y = -4x

4 = -4(-1)

4 = 4

This ordered pair satisfies the equation.

e) (0,0):

y = -4x

0 = -4(0)

0 = 0

This ordered pair satisfies the equation.

f) (4,-2):

y = -4x

-2 = -4(4)

-2 = -16

This ordered pair does not satisfy the equation.

Therefore, the ordered pairs that represent points on the graph of the equation 2x = -1/2y are d) (-1,4) and e) (0,0).

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Answer:

6.2

Step-by-step explanation:

For this we use a method called Pythagoras theorem.

to find '?' we will have to use this method

[tex]A^{2}[/tex] +[tex]B^{2}[/tex] =[tex]C^{2}[/tex]

since we don't know what A is we will name it x

[tex]x^{2}[/tex]+[tex]5^{2}[/tex]=[tex]8^{2}[/tex]

x+25=64

64-25=39

[tex]x^{2}[/tex]=39

x=[tex]\sqrt{39}[/tex]

x=6.2

Answer:

P=32 yards, A=64 yards^2

Step-by-step explanation:

We know that P = 4 x s, where s is the side length. In this case, we know that s=8. So, the perimeter is 32 yards. We know that A= s^2. Substituting for s=8 makes the area 64 yards^2. Thus, the perimeter of this square is 32 yards, while the area is 64 yards squared.

Answer:

-6x + 15 < 10 - 5x or 11x > 5

Step-by-step explanation:

The area of the shape is equal to 60 square centimeters

The shape is observed to be made up of a triangle and a rectangle, so we shall calculate for each area and sum the results to get the total area of the shape as follows:

area of triangle = 1/2 × 4 cm × 12 cm

area of the triangle = 24 cm²

area of the rectangle = 9 cm × 4 cm

area of the rectangle = 36 cm²

total area of the shape = 24 cm² + 36 cm²

total area of the shape = 60 cm²

Therefore, the area of the shape is equal to 60 square centimeters

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Answer: The volume of the cylindrical pipe is 76pi cubic feet.

Step-by-step explanation: Sincerely, answered by Lizzy ♡ :: as an A+ student, I want to make sure other students can succeed as well, so in my free time: i answer questions like yours on Brainly! if you could click the thanks heart, give me brainliest by clicking the crown, and rate my answer 5 stars, it would be appreciated! have a lovely day! (ᵔᴥᵔ)

Answer:

Step-by-step explanation:

239 cubic feet

i asked my math teacher

Answer: We are given that:

42^a + 2^2 + 2^2 = 2336

Simplifying the expression on the left-hand side, we get:

42^a + 4 = 2336

Subtracting 4 from both sides, we get:

42^a = 2332

Taking the logarithm of both sides with base 42, we get:

a = log₄₂₂₋₃₂ ≈ 1.417

Since a is a positive integer, the closest integer to 1.417 is 1. Therefore, the value of a is 1.

Step-by-step explanation:

[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{0}~,~\stackrel{y_1}{-7})\qquad (\stackrel{x_2}{-8}~,~\stackrel{y_2}{5})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(~~-8 - 0~~)^2 + (~~5 - (-7)~~)^2} \implies d=\sqrt{(-8 +0)^2 + (5 +7)^2} \\\\\\ d=\sqrt{( -8 )^2 + ( 12 )^2} \implies d=\sqrt{ 64 + 144 } \implies d=\sqrt{ 208 }\implies d\approx 14.42[/tex]

Answer:

4√13 or 14.422205101856

Step-by-step explanation:

Distance (d) = √(-8 - 0)2 + (5 - -7)2

= √(-8)2 + (12)2

= √208

= 4√13

= 14.422205101856

In statement c), A and B are sets, and x is an element that belongs to both sets A and B.

What is equation?

An equation is a mathematical statement that shows the equality between two expressions, often denoted by placing an equals sign (=) between the two expressions. The expressions on either side of the equals sign are typically composed of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and so on.

An equation is said to be true if the expressions on both sides of the equals sign represent the same quantity or value. Equations are often used in mathematics and science to model real-world phenomena, to make predictions, and to solve problems.

a) P: It is a rock.

Q: It contains magnesium.

P → Q

b) P: The weather is nice.

Q: I am chilly.

P ∨ Q

c) P: x belongs to set A.

Q: x belongs to set B.

P ∧ Q

d) P: It is raining.

Q: The ground is wet.

P → Q

e) P: The food is hot.

Q: I will blow on it.

P → Q

f) P: The car is red.

Q: It is a convertible.

P ↔ Q

Note: In statement c), A and B are sets, and x is an element that belongs to both sets A and B.

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Step-by-step explanation:

Slope  = rise / run

Using points ( -1/2  - 1/2 ) and (1/2, 0 )  

  rise = 1/2    run = 1

     rise / run = 1/2 /1 = 1/2 = slope

you will need 180 mL of the 55% alcohol mixture to create a 25% alcohol solution.

How to solve the question?

To find out how much of the 55% alcohol mixture is needed to create a 25% alcohol solution, we can use the formula:

(amount of pure alcohol in the 10% mixture + amount of pure alcohol in the 55% mixture) / total volume of solution = desired alcohol concentration

Let x be the amount of the 55% alcohol mixture needed.

First, we need to calculate the amount of pure alcohol in the 10% mixture. We know that the 10% mixture contains 10% alcohol, which means that there are 0.10 x 360 = 36 mL of pure alcohol in the mixture.

Next, we need to calculate the amount of pure alcohol in the 55% mixture. We know that the 55% mixture contains 55% alcohol, which means that there are 0.55 x x = 0.55x mL of pure alcohol in the mixture.

The total volume of the solution will be 360 mL + x mL.

Using the formula above, we can set up the equation:

(36 + 0.55x) / (360 + x) = 0.25

Simplifying the equation, we get:

36 + 0.55x = 0.25(360 + x)

36 + 0.55x = 90 + 0.25x

0.3x = 54

x = 180 mL

Therefore, you will need 180 mL of the 55% alcohol mixture to create a 25% alcohol solution.

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The true statement about monthly income is "The family's monthly income must be less than $5,092." (option a).

The table provided shows the median income levels for different family sizes in various states, including Georgia. To determine whether the family of three is eligible for Chapter 7 bankruptcy, we need to compare their monthly income to the median income for a three-person family in Georgia, which is $61,104.

If the family's monthly income is less than the median income for their family size in their state, they are presumed to be eligible for Chapter 7 bankruptcy. In this case, the family's monthly income must be less than $61,104.

Looking at the table, we can see that the median income for a three-person family in Georgia is $61,104, which means that the family's monthly income must be less than $5,092 in order to be eligible for Chapter 7 bankruptcy.

Therefore, the correct answer is (a) - the family's monthly income must be less than $5,092 for them to be eligible to file for Chapter 7 bankruptcy.

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Answer:

x=41

Step-by-step explanation:

If this is a polygon, I'm going to assume that it has 5 sides.

s=(5-2)(180)

s=3(180)

=540

(2x+4)+(3x+6)+(4x-23)+(3x-18)+(2x-3)=540

Now simplify.  Add the x's and the numbers together. Then divide.

14x-34=540

14x=540+34

14x=574

x=574/14

x=41

The expression which has an approximate value be between the numbers 6 and 7 are (a) π + 3 and (b) √39.

In order to determine which expression has an approximate value between the numbers, 6 and 7, we can estimate the value of each expression and compare them to 6 and 7.

Option(a) π + 3 ;

We know that the approximate value of "π" is 3.14,

So, 3.14 + 3 = 6.14 (it is between 6 and 7)

Option(b) : √39 ≈ 6.24 (it is between 6 and 7)

Option (c) : 6π , We substitute, π ≈ 3.14,

We get, ≈ 18.85 (it is not between 6 and 7)

Option (d) : √35 ≈ 5.92 (it is not in between 6 and 7)

Therefore, the correct options are (a) and (b).

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The given question is incomplete, the complete question is

Which expression has an approximate value be between 6 and 7?

Select all that apply

(a) π + 3

(b) √39

(c) 6π

(d) √35

Volume is Length x width x height

1 box = 4.5 x 2 x 4 = 36

Each box is 36 cubic feet. Now divide the closet size by box size

185 cubic feet/36 cubic feet per box = 5.138

Since you can't have .138 of a box, and rounding up to 6 boxes is too much, the closet will hold 5 boxes (with .138 cubic feet left over).

Answer:

The slope is 1/3

Step-by-step explanation:

y+6=1/3(x-4)
-6         -6

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

y = 1/3 * x + 1/3 * (-4) - 6
y = 1/3x -7 1/3

Answer:

[tex]\frac{1}{3}[/tex]

Step-by-step explanation:

First, start by simplifying the right side of the equation, [tex]y+6=\frac{1}{3}(x-4)[/tex].

When we simplify the right side using distributive property, we will get:

[tex]y+6=\frac{1}{3}x-\frac{4}{3}[/tex]

Subtract 6 from both sides to get:

[tex]y=\frac{1}{3}x-\frac{4}{3}-6[/tex]

Simplify further and get:

[tex]y=\frac{1}{3}x-\frac{22}{3}[/tex]

By viewing the equation, we can tell the slope is 1/3.

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The age of oldest is 44 and age of youngest is 18.

We have,

Sum of age of 3 brother = 36

let the age of three brother be x, y and z.

so, x + y + z = 36

x = y + z

After 8 year, (z+ 8) = 2(x+8)

z+ 8 = 2x+ 16

2x - z +8=0

Thus, the age of oldest is 44 and age of youngest is 18.

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The Translation of the given question is:

The sum of the ages of 3 brothers is 36. The age of the eldest is equal to the sum of the two younger brothers, within 8 years the younger will double the age of the older, what is the age of the brothers?