Help needed

65. Solve for in the logarithmic equation 3 log x = 2.
O * = 1,000,000
O * = 150
O * = 56.423
O * = 4.642

The variation given in the first question is not a direct variation and there is no constant of variation.

y = k × x

when

y = -40 and x = -0.5

y = k × x

-40 = k × -0.5

-40 = -0.5k

k = -40/-0.5

k = 80

When y = 8 and x = 2.5

y = k × x

8 = k × 2.5

8 = 2.5k

k = 8 / 2.5

k = 3.2

When y = 4 and x = -3

y = k × x

4 = k × -3

4 = -3k

k = -4/3

when y = 8 and x = -6

y = k × x

8 = k × -6

8 = -6k

k = 8/-6

k = -4/3

This is a direct variation and the constant of proportionality is -4/3

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4.440 minutes of the ride is spent higher than 37 meters above the ground.

We have to find the amount of time spent by the ride 37 meters above the ground.

We know that 37 meters above the ground is actually one meter above the center.

We can find the angle above the center as shown below:

arcsin(1/34) = 0.0294 radians

Now we have to find the angle of rotation:

The angle of rotation of the Ferris wheel above 37 meters = π - (2*0.0294)

= 3.1 radians

The angular velocity of the Ferris wheel = Angle covered/Time

= 2π/9 rad/min.

Therefore time spent above 37 meters = 3.1*9/2π

= 3.1*9/6.28319

= 4.440 minutes

Therefore, we have found that 4.440 minutes of the ride is spent higher than 37 meters above the ground.

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Answer:  [tex]f^{-1}(\text{x}) = (\text{x}-9)^2+4[/tex] when [tex]\text{x} \ge 9[/tex]

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

Work Shown:

[tex]f(\text{x}) = 9 + \sqrt{\text{x} - 4}\\\\\text{y} = 9 + \sqrt{\text{x} - 4}\\\\\text{x} = 9 + \sqrt{\text{y} - 4}\\\\\text{x}-9 = \sqrt{\text{y} - 4}\\\\(\text{x}-9)^2 = (\sqrt{\text{y} - 4})^2\\\\(\text{x}-9)^2 = \text{y}-4\\\\(\text{x}-9)^2+4 = \text{y}\\\\f^{-1}(x) = (\text{x} - 9)^2+4[/tex]

Explanation:

I replaced f(x) with y. After that I swapped x and y, then solved for y to get the inverse.

The smallest that [tex]\sqrt{\text{x}-4}[/tex] can get is 0, which means the smallest f(x) can get is 9+0 = 9. The range for f(x) is [tex]\text{y} \ge 9[/tex]

Since x and y swap to determine the inverse, the domain and range swap roles. Therefore, the domain of the inverse [tex]f^{-1}(\text{x})[/tex] is [tex]\text{x} \ge 9[/tex]

So we will only consider the right half portion of the parabola.

The graph is below. The red curve mirrors over the black dashed line to get the blue curve, and vice versa.

Answer:

[tex]f^-^1(x)=(x-9)^2+4[/tex]   OR [tex]x^2-18x+85[/tex] if you simplify it

Step-by-step explanation:

to find the inverse function you have to switch x and y with each other and solve for y.

[tex]y=9+\sqrt{x-4}\\x=9+\sqrt{y-4}[/tex]           step 1: switch x and y with each other

[tex]x-9=\sqrt{y-4}[/tex]           step 2: subtract 9 from both sides

[tex](x-9)^2=\sqrt{y-4}^2[/tex]     step 3: square both sides to get rid of the square root

[tex](x-9)^2=y-4\\(x-9)^2+4=y\\[/tex]         step 4: add 4 to both sides

you could leave the answer like this or you can simplify to get [tex]x^2-18x+85[/tex]

Step-by-step explanation:

Since there is 2 unknown variable ,

so we need 2 equation to find their values.

The researcher is using a within-subjects factorial design.

According to statement

A researcher conducts an experiment to determine whether the type of car and color of the car impact how fast people drive.

This can be done by within-subjects factorial design.

In a within-subjects factorial design, all of the independent variables are manipulated within subjects. This would mean that each participant was tested in all conditions.

Another common example of a within-subjects design is medical testing.

So, The researcher is using a within-subjects factorial design.

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The number the teacher gave Ross is 5127

From the question, we are to determine the number that the teacher gave Ross

From the given information,

Mr. David gave Ross a number and asked him to divide it by 15.

Let the number be x

That is,

x/15

The quotient and remainder obtained by ross are 341 and 12 respectively

That is,

x/15 = 341 + 12/15

Now, we will solve the equation for x

x/15 = 341 + 12/15

Multiply through by 15

x = 15×341 + 12

x = 5115 + 12

x = 5127

Hence, the number the teacher gave Ross is 5127

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The pH value of the apple juice is 3.5, is the correct answer.

The pH value of the ammonia is 8.9,  is the correct answer.

The question seems to be incomplete and complete question is shown below

Tests show that the hydrogen ion concentration of a sample of apple juice is 0.0003 and that of ammonia is 1.3 x 10-9. Find the approximate pH of each liquid using the formula pH = -log (H+), where (H+) is the hydrogen ion concentration The pH value of the apple juice is___ The pH value of ammonia is____

1.pH of apple juice

A. 8.11

B. 1.75

C. 3.5

D. 2.1

2. pH of ammonia

A. 1.1

B. 7.0

C. 5.4

D. 8.9

The pH of a solution is defined as the logarithm of the reciprocal of the hydrogen ion concentration  [H+] of the given solution.

From the formula;

pH = -log[ H⁺ ]

Given the data in the question.

For the Apple juice

hydrogen ion concentration H⁺ = 0.0003

pH of the apple juice pH = ?

pH = -log[ H⁺ ]

pH = -log[ 0.0003 ]

pH = 3.5

The pH value of the apple juice is 3.5

For the ammonia

hydrogen ion concentration H⁺ = 1.3 × 10⁻⁹

pH of the ammonia pH = ?

pH = -log[ H⁺ ]

pH = -log[ 1.3 × 10⁻⁹]

pH = 8.9

The pH value of the ammonia is 8.9

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Answer:

3x +8 = -5x

3x + 5x = -8

8x = -8x

x = -8 /8

x = -1

Hope it helps...

Please contact me for further explanation....

Here is the completed table:

                                                           Plays a sport      Doesn't play a sport

Plays a musical instrument                  0.46                         0.54

Doesn't play a musical instrument       0.73                          0.27

Relative frequency measures how often a value appears relative to the sum of the total values.

                                                    Plays a sport              Doesn't play a sport

Plays a musical instrument              (6/13)  = 0.46                   (7/13)  = 0.54

Doesn't play a musical instrument       (8/11) 0.73                       (3/11) =0.27

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The length and the width of the rectangle are 10 inches and 6 inches respectively.

In the question, we are given that the length of a rectangle exceeds its width by 4 inches, and the area is 60 square inches.

We are asked for the length and the width of the rectangle.

We assume the width (w) of the rectangle to be x inches.

Its length (l), exceeds its width (w) by 4 inches.

Thus, its length (l) = x + 4 inches.

Now, the area can be calculated using the formula, A = l*w, where A is its area, l is its length, and w is its width.

Thus, the area = (x + 4)x = x² + 4x.

But, we are given that the area is 60 square inches.

Putting the value, we get a quadratic equation:

x² + 4x = 60.

or, x² + 4x - 60 = 0,

or, x² + 10x - 6x - 60 = 0,

or, x(x + 10) - 6(x + 10) = 0,

or, (x - 6)(x + 10) = 0.

By the zero-product rule, we get:

Either, x - 6 = 0, or, x = 6,

or, x + 10 = 0, or, x = -10, but this is not possible as the width of a rectangle cannot be negative.

Thus, the width = x = 6 inches.

The length = x + 4 = 10 inches.

Thus, the length and the width of the rectangle are 10 inches and 6 inches respectively.

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Answer:

[tex]\dfrac{991}{40\sqrt{2}}[/tex]

Step-by-step explanation:

Given expression:

[tex]\sqrt{72}-\dfrac{48}{\sqrt{50}}-\dfrac{45}{\sqrt{128}}+2\sqrt{98}[/tex]

Rewrite 72 as 36·2, 50 as 25·2, 128 as 64·2 and 98 as 49·2:

[tex]\implies \sqrt{36 \cdot 2}-\dfrac{48}{\sqrt{25 \cdot 2}}-\dfrac{45}{\sqrt{64 \cdot 2}}+2\sqrt{49 \cdot 2}[/tex]

[tex]\textsf{Apply radical rule} \quad \sqrt{ab}=\sqrt{a}\sqrt{b}:[/tex]

[tex]\implies \sqrt{36}\sqrt{2}-\dfrac{48}{\sqrt{25}\sqrt{2}}-\dfrac{45}{\sqrt{64}\sqrt{ 2}}+2\sqrt{49}\sqrt{2}[/tex]

Rewrite 36 as 6², 25 as 5², 64 as 8² and 49 as 7²:

[tex]\implies \sqrt{6^2}\sqrt{2}-\dfrac{48}{\sqrt{5^2}\sqrt{2}}-\dfrac{45}{\sqrt{8^2}\sqrt{ 2}}+2\sqrt{7^2}\sqrt{2}[/tex]

[tex]\textsf{Apply radical rule} \quad \sqrt{a^2}=a, \quad a \geq 0[/tex]

[tex]\implies 6\sqrt{2}-\dfrac{48}{5\sqrt{2}}-\dfrac{45}{8\sqrt{ 2}}+2\cdot 7\sqrt{2}[/tex]

Simplify:

[tex]\implies 6\sqrt{2}-\dfrac{48}{5\sqrt{2}}-\dfrac{45}{8\sqrt{ 2}}+14\sqrt{2}[/tex]

Combine like terms:

[tex]\implies 20\sqrt{2}-\dfrac{48}{5\sqrt{2}}-\dfrac{45}{8\sqrt{ 2}}[/tex]

Make the denominators of the two fractions the same:

[tex]\implies 20\sqrt{2}-\dfrac{384}{40\sqrt{2}}-\dfrac{225}{40\sqrt{ 2}}[/tex]

Rewrite 20√2 as a fraction with denominator 40√2:

[tex]\implies 20\sqrt{2}\cdot\dfrac{40\sqrt{2}}{40\sqrt{2}}-\dfrac{384}{40\sqrt{2}}-\dfrac{225}{40\sqrt{ 2}}[/tex]

[tex]\implies \dfrac{1600}{40\sqrt{2}}-\dfrac{384}{40\sqrt{2}}-\dfrac{225}{40\sqrt{ 2}}[/tex]

Combine fractions:

[tex]\implies \dfrac{991}{40\sqrt{2}}[/tex]

The difference between two numbers are [tex]3[/tex] and numbers are [tex]x[/tex] and [tex]y[/tex] so the equation is [tex]x-y=3[/tex]

How to find the equation and what is the equation ?

equation, statement of equality between two expressions consisting of variables and/or numbers. In essence, equations are questions, and the development of mathematics has been driven by attempts to find answers to those questions in a systematic way.

if the difference of two numbers are [tex]3[/tex]

And one number is [tex]x[/tex] and another is [tex]y[/tex]

So we write the equation with operation of subtraction

[tex]x-y=3[/tex]

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Answer: 7(8x + 6) = -1: Exact form: x = -43/56, decimal form: x = -0.76785714

Thats all i have because you can't solve for x for the second one!

Answer:

[tex]x=6[/tex]

Step-by-step explanation:

Given equation:

[tex]x-\left(2x-\dfrac{3x-4}{7}\right)=\dfrac{4x-27}{3}-3[/tex]

Expand the left side:

[tex]\implies x-2x+\dfrac{3x-4}{7}=\dfrac{4x-27}{3}-3[/tex]

Let all terms on the left side have the same denominator of 7:

[tex]\implies \dfrac{7x}{7}-\dfrac{14x}{7}+\dfrac{3x-4}{7}=\dfrac{4x-27}{3}-3[/tex]

Join the terms on the left side:

[tex]\implies \dfrac{7x-14x+3x-4}{7}=\dfrac{4x-27}{3}-3[/tex]

[tex]\implies \dfrac{-4x-4}{7}=\dfrac{4x-27}{3}-3[/tex]

Let all the terms on the right side have the same denominator of 3:

[tex]\implies \dfrac{-4x-4}{7}=\dfrac{4x-27}{3}-\dfrac{9}{3}[/tex]

Join the terms on the right side:

[tex]\implies \dfrac{-4x-4}{7}=\dfrac{4x-27-9}{3}[/tex]

[tex]\implies \dfrac{-4x-4}{7}=\dfrac{4x-36}{3}[/tex]

Cross multiply:

[tex]\implies 3(-4x-4)=7(4x-36)[/tex]

Expand:

[tex]\implies -12x-12=28x-252[/tex]

Add 12x to both sides:

[tex]\implies -12=40x-252[/tex]

Add 252 to both sides:

[tex]\implies 240=40x[/tex]

Divide both sides by 40:

[tex]\implies 6=x[/tex]

[tex]\implies x=6[/tex]

Answer:

See my picture below for the answer.

Step-by-step explanation:

If you look at the unit circle and locate 240 degrees.  You will see an ordered pair.  The first number -1/2 is the cos of 240 and the second number is the sin.

Answer:

it’s the square root of 3, divided by 2 (sorry, can’t type any special characters at the moment)

Step-by-step explanation:

-240 = 120 because you measure negative angles clockwise starting at 0.

Using unit circle, value of sin for an angle is the 2nd number

Answer:

  D.  completing the square

Step-by-step explanation:

The process shown rearranges the standard-form quadratic equation to vertex form. It involves factoring out the leading coefficient and distributing the constant so that the variable terms can be written as a perfect square.

A perfect square trinomial is the square of a binomial. It has the form ...

  (x +b)² = x² +2bx +b²

Given two variable terms, x² and 2bx, the perfect square trinomial can be formed by adding b², which is the square of half the x-term coefficient.

When b² is added to the sum (x² +2bx), the square is complete. The sum (x² +2bx +b²) can be written as the square (x +b)².

This process of adding b² to the sum (x² +2bx) will change the polynomial unless a similar quantity is subtracted. That is why the 3rd line of the problem statement show 4 being added and subtracted inside parentheses:

  y = 5(x² +4x +4 -4) -17

When we're applying this transformation, we do not want to change the equation. We simply want to rearrange it.

In the next step, the subtracted term is brought outside the parentheses. This lets us write the quantity in parentheses as a perfect square.

  y = 5(x² +4x +4) +5(-4) -17

  y = 5(x² +4x +4) -20 -17

This process of adding and subtracting a suitable quantity and rearranging the equation so the variable terms make a perfect square is called ...

  "completing the square."

__

Additional comment

The equation in this form is said to be in "vertex form."

  y = a(x -h)² +k

In this form, the constant 'a' is a vertical scale factor; the ordered pair (h, k) identifies the vertex of the quadratic on a graph.

The system has:

A system of two linear equations is given by:

y = a*x + b

y = c*x + d

1) The system has no solutions when both lines are parallel lines, this happens when both lines have the same slope and different y-intercept.

y = a*x + b

y = a*x + d

2) The system has one solution when the lines have different slopes:

y = a*x + b

y = c*x + d

3) The system has infinite solutions when both equations describe the same line:

y = a*x + b

y = a*x + b

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21 % is the answer.

Problem is that the height of this plant 6.3 and 7.8 cm

P (6.3 <x< 7.8)

12 -5 = 7cm

7.8 - 6.3 = 1.5cm

1.5 / 7 = 0.21

=21%

Probability is a field of mathematics that deals with the numerical description of the likelihood that an event will occur or that a statement is true. The probability of an event is a number between 0 and 1, with approximately 0 indicating the impossibility of the event and 1 indicating certainty.

Probability is a measure of the likelihood that an event will occur. Many events cannot be predicted with absolute certainty. You can only predict the probability that an event will occur.

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Answer:

It is stretched taller by a factor of 2.

The value of N is 28.

A mathematical statement that has a "equal to" symbol between two expressions with equal values is called an equation.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation. To determine the value of an unknown variable that represents an unknown quantity, equations can be solved.

According to the question,

a+b+c=60

a-7=N

b+7=N

7c=N

From the above three equations,

a=N+7

b=N-7

c=N/7

So, N+7+N-7+N/7=60

2N+N/7=60

14N+N=420

15N=420

N=28

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[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{14+10y\geq 3(y+14) } \end{gathered}$}[/tex]

[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{Simplify \ both \ sides \ of \ the \ inequality. }} \end{gathered}$}[/tex]

[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{Subtract \ 3y \ from \ both \ sides. }} \end{gathered}$}[/tex]

[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{ Subtract \ 14 \ from \ both \ sides. }} \end{gathered}$}[/tex]

[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{Divide \ both \ sides \ by \ 7. }} \end{gathered}$}[/tex]

The total mass of Adrian and Chang at birth is 7.41 kilograms

Total mass of the babies = Adrian weight + Chang weight

= 2.87 kilograms + 4.54 kilograms

= 7.41 kilograms

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Answer:

Step-by-step explanation:

Givens

line: 5y + 2x =  10

point (-2 , 5)

Discussion and Solution

You have to rearrange the given line to get the slope. It isn't nice, but it's not impossible.

5y + 2x = 10                            Subtract 2x from both sides

5y + 2x - 2x = -2x + 10           Simplify

5y = - 2x + 10                          Divide by 5

5y/5 = -2x/5 + 10/5                 Simplify

y = -0.4x + 2            

So the new equation is going to have a slope of - 0.4

So far what you have is

y =  - 0.4x + b

Now use the point to find the y intercept. What you should be thinking about the point is that when x = -2 then y = 5. Substitute that into the new equation.

5 = -0.4*-2 + b

5 = 0.8 + b                                Subtract 0.8 from both sides

5 - 0.8 = 0.8 - 0.8 + b               Simplify

4.2 = b

Answer

y = -0.4x + 4.2

Considering the given proportions, the correct statements are:

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

The table gives these following proportions:

Hence the correct statements are:

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money paid by employer to Alan is $ 155.09

A word problem is a type of mathematics exercise where the majority of the issue's context is supplied in spoken language as opposed to mathematical notation.

money paid for dinner by employer=$20.20

money paid for lunch by employer =$10.70

money paid for breakfast by employer =$6.93

according to question,

in this problem we will simply multiply quantity of dinners with amount

to get our answer.

so as given in problem  Alan scheduled 4 dinners , 5 lunches and 3 breakfast

(4×$20.20)+(5×$10.70)+(3×$6.93)

80.8+53.5+20.79

=$ 155.09

hence money paid by employer to Alan is $ 155.09

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based on the number of marbles in the bag, the number of marble groups possible is 7,140 groups.

the probability of selecting 2 red and 1 yellow is 1/68.

first, find the number of marbles:

= 7 + 8 + 9 + 7 + 5

= 36

the number of marble groups are:

= 36 ! / (3 ! x 33 !)

= 7,140 groups

the probability of picking 2 red and 1 yellow is:

= (( (7! / (2! x 5!)) x ( ( 5! / (1! x 4!))) / 7,140

= 1 / 68

= 1.47%

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Answer:

w= 9

Step-by-step explanation:

[tex] \sqrt{ - 4w + 61} = w - 4[/tex]

Square both sides:

-4w +61= (w -4)²

[tex]\boxed{(a - b)^{2} = a^2 -2ab + b^2 }[/tex]

Expand:

-4w +61= w² -2(w)(4) +4²

-4w +61= w² -8w +16

Simplify:

w² -8w +16 +4w -61= 0

w² -4w -45= 0

Factorize:

(w -9)(w +5)= 0

w -9= 0 or w +5= 0

w= 9 or w= -5 (reject)

Note:

-5 is rejected since we are only taking the positive value of the square root here. If the negative square root is taken into consideration, then w= -5 would give us -9 on both sides of the equation.

Why do we use negative square root?

When solving an equation such as x²= 4,

we have to consider than squaring any number removes the negative sign i.e., the result of a squared number is always positive.

In the case of x²= 4, x can be 2 or -2. Thus, whenever we introduce a square root, a '±' must be used.

However, back to our question, we did not introduce the square root so we should not consider the negative square root value.

Answer: w=9.

Step-by-step explanation:

[tex]\sqrt{-4w+61}=w-4[/tex]

Tolerance range:

[tex]\left \{ {{-4w+61\geq 0} \atop {w-4\geq 0}} \right. \ \ \ \ \left \{ {{4w\leq 61\ |:4} \atop {w\geq 4}} \right. \ \ \ \ \left \{ {{w\leq 15,25} \atop {w\geq 4}} \right. \ \ \ \ \Rightarrow\ \ \ \ \\w\in[4;15,25].[/tex]

Solution:

[tex](\sqrt{-4w+61})^2=(w-4)^2\\-4w+61=w^2-2*w*4+4^2\\-4w+61=w^2-8w+16\\w^2-4w-45=0\\D=(-4)^2-4*1*(-45)\\D=16+4*45\\D=16+180\\D=196.\\\sqrt{D}=\sqrt{196}\\\sqrt{D}=14 \\w_{1,2}=\frac{-(-4)б14}{2} \\w_1=\frac{4-14}{2} \\w_1=\frac{-10}{2} \\w_1=-5\ \notin tolerance\ range.\\w_2=\frac{4+14}{2}\\ w_2=\frac{18}{2} \\w_2=9\ \in tolerance \ range.[/tex]

Answer:

     2√2

Step-by-step explanation:

We can find the relationship of interest by solving the given equation for A, the mean distance.

  [tex]T^3=A^2\\\\A=\sqrt{T^3}=T\sqrt{T}\qquad\text{take the square root}[/tex]

The mean distance of planet X is found in terms of its period to be ...

  [tex]D_x=T_x\sqrt{T_x}[/tex]

The mean distance of planet Y can be found using the given relation ...

  [tex]T_y=2T_x\\\\D_y=T_y\sqrt{T_y}=2T_x\sqrt{2T_x}=(2\sqrt{2})T_x\sqrt{T_x}\\\\D_y=2\sqrt{2}\cdot D_x[/tex]

The mean distance of planet Y is increased from that of planet X by the factor ...

  2√2

Answer:

3, 7, 14

Step-by-step explanation:

small brain