What is the logarithmic function modeled by the following table?

x f(x)
8 3
16 4
32 5

Select one:
a. f(x) = logx2
b. f(x) = log2x
c. f(x) = 2 log10x
d. f(x) = x log102

Answer:

y = 3 • ± √2 = ± 4.2426

Step-by-step explanation:

2y2 -  36  = 0  

Step  2  :

Step  3  :

Pulling out like terms :

3.1     Pull out like factors :

  2y2 - 36  =   2 • (y2 - 18)  

Trying to factor as a Difference of Squares :

3.2      Factoring:  y2 - 18  

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

        A2 - AB + BA - B2 =

        A2 - AB + AB - B2 =  

        A2 - B2

Note :  AB = BA is the commutative property of multiplication.  

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 18 is not a square !!  

Ruling : Binomial can not be factored as the difference of two perfect squares.

Equation at the end of step  3  :

 2 • (y2 - 18)  = 0  

Step  4  :

Equations which are never true :

4.1      Solve :    2   =  0

This equation has no solution.

A a non-zero constant never equals zero.

Solving a Single Variable Equation :

4.2      Solve  :    y2-18 = 0  

Add  18  to both sides of the equation :  

                     y2 = 18  

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  

                     y  =  ± √ 18  

Can  √ 18 be simplified ?

Yes!   The prime factorization of  18   is

  2•3•3  

To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 18   =  √ 2•3•3   =

               ±  3 • √ 2  

The equation has two real solutions  

These solutions are  y = 3 • ± √2 = ± 4.2426

Answer:

9 / 20

Step-by-step explanation:

We can form a ratio with the information "for every 25 ml of orange juice you need 30 ml of apple juice and 45 ml of coconut milk"

25 : 30 : 45

Total = 100

Coconut milk = 45 ml

45 / 100 = 9 / 20

Answer:

7/50

Step-by-step explanation:

Experimental and theoretical probability are much different.

With experimental, just read the experiment: the number two was rolled 7 times.

Put that over 50.

Avoid questions with theoretical probability, that's where math comes in.

Final answer:

The experimental probability of rolling a two on a number cube that was rolled 50 times and showed the number two 7 times is 0.14 or 14%.

Explanation:

To calculate the experimental probability of the number cube showing the number two, we divide the number of times two appears by the total number of rolls. In this case, the number cube was rolled 50 times and the number two appeared 7 times. Therefore, the experimental probability, denoted as P(2), is calculated as:

P(2) = Number of times two appears / Total number of rolls

P(2) = 7 / 50

P(2) = 0.14

So, the experimental probability of rolling a two on this number cube, based on the given data, is 0.14 or 14%.

The owner wants people to believe that dance classes are popular so that they sign up for classes. Therefore, option C is the correct answer.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

The most likely reason the owner of the dance studio chain releases the report is to demonstrate the success of their business. By highlighting the fact that participation in dance classes has increased by 5% in each of the past three years, the owner is showing the public that the business is doing well and that they are providing a valuable service. This report may also be used to encourage potential customers to join the studio's classes, which would further increase their profits.

Therefore, option C is the correct answer.

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Answer:

  D.  The graph of g(x) is shifted 2 units up.

Step-by-step explanation:

Adding 2 to the y-coordinate of a point shifts it up by 2 units.

___

The graph of f(x) is all points (x, f(x)). When you add 2 to f(x), you make the graph of g(x) be all points (x, g(x)) = (x, f(x)+2). That is all of the points on the original graph are shifted up by 2 units.

Answer:

D. 272 poles

Explanation:

We are given that:

The top layer has 20 poles, the next down one has 24 poles and the third one has 28 poles

We can note that each layer has 4 poles more that the one above it

Based on this, we can get the number of poles in each layer as follows:

Top layer has 20 poles

Second one has 20 + 4 = 24 poles

Third one has 24 + 4 = 28 poles

Fourth one has 28 + 4 = 32 poles

Fifth one has 32 + 4 = 36 poles

Sixth one has 36 + 4 = 40 poles

Seventh one has 40 + 4 = 44 poles

Eighth one has 44 + 4 = 48 poles

Now, we can get the total number of poles by adding the poles in all layers

This is done as follows:

Total number of poles = 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48

Total number of poles = 272 poles

Hope this helps :)

Answer:

It's choice 2.

Step-by-step explanation:

y=19.485x+86.912

The 19.485 is the slope of the graph of this equation.  This gives the rate of change of the amount of the bill (above $86.912)  for  each added resident (x).

Answer:

[tex]\displaystyle x=\frac{-8\pm\sqrt{(8)^2-4(4)(-221)}}{2(4)}\ \text{; x = 6.5 and x = -8.5}[/tex]

Step-by-step explanation:

Subtract the right side of the given equation to put it into standard form:

  4x² +8x -221 = 0

Then the coefficients used in the quadratic formula are ...

When these are filled into the form ...

[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

the result is as shown above.

Answer:

$9.50 - for adults

$7.50 - children

Step-by-step explanation:

All you have to do is multiply $4.75 by 2 since there are two adults and multiply $2.50 by 3 since there  are three children.

Let [tex]x[/tex] be the amount (mL) of the pure vinegar the chef will use, and [tex]y[/tex] the amount of dressing. She wants to end up with a 310 mL mixture, so

[tex]x+y=310[/tex]

For each mL used of the dressing, 0.07 mL is vinegar, and the chef wants to end up with a 19% vinegar mixture, so

[tex]x+0.07y=0.19(x+y)=58.9[/tex]

Now

[tex]x+y=310\implies y=310-x[/tex]

[tex]\implies x+0.07(310-x)=58.9[/tex]

[tex]\implies0.93x+21.7=58.9[/tex]

[tex]\implies0.93x=37.2[/tex]

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

[tex]\implies y=270[/tex]

Answer:

4 hours

Step-by-step explanation:

0:8

6:12

12:16

18:20

24:24

Answer: 4 hours

Step-by-step explanation:

Answer:

  see below

Step-by-step explanation:

Since there are fewer sequences than functions, we'll identify the matchup according to the sequence.

11, 33, 55, 77, ...

The first term is 11. The terms have a common difference of 33 -11 = 22. That is, each term is 22 more than the previous one. The appropriate recursive function is ...

__

-18, -108, -648, -3888, ...

The first term is -18. The terms obviously do not have a common difference, but their common ratio is -648/-108 = -108/-18 = 6. That is, each term is 6 times the previous one. Then the appropriate recursive function is ...

__

-18, 3, 24, 45, ...

The first term is -18. The terms have a common difference of 3-(-18) = 21. That is, each term is 21 more than the previous one. The appropriate recursive function is ...

Answer:

11, 33, 55, 77, ...=f(n) = f(n - 1) + 22 for n = 2, 3, 4, ...

-18, -108, -648, -3,888, ...=f(n) = 6 · f(n - 1) for n = 2, 3, 4, ...

-18, 3, 24, 45, ...=f(n) = f(n - 1) + 21 for n = 2, 3, 4, ...

Step-by-step explanation:

Answer:

Pemdas

Step-by-step explanation:

Parenthesis

Exponents

Multiplication

Division

Addition

Subtraction

You go from left to right and solve in the order called Pemdas.

To use a system of equations to find the solution algebraically, follow these steps: identify the unknowns and knowns, write down the equations, solve for one variable, substitute the expression into the other equation(s), solve for the remaining variables, and check your answer(s) for reasonableness.

1. Identify the unknowns and knowns.

2. Write down the equations that represent the given information.

3. Solve one of the equations for one variable in terms of the other.

4. Substitute this expression into the other equation(s), replacing the variable.

5. Solve the resulting equation(s) to find the value(s) of the remaining variable(s).

6. Check your answer(s) to ensure they make sense in the context of the problem.

the answer is 10. just use the distance formula which is the square root of (x2-x1)^2+(y2-y1)^2

Answer: 10 units

Step-by-step explanation:

The distance  from point (a,b) and (c,d) is given by formula below:-

[tex]D=\sqrt{(c-a)^2+(d-b)^2}[/tex]

Therefore, the distance between the points (5, 1) and (-3, -5) will be :

[tex]D=\sqrt{(-3-5)^2+(-5-1)^2}\\\\\Rightarrow\ D=\sqrt{(-8)^2+(-6)^2}\\\\\Rightarrow\ D=\sqrt{64+36}\\\\\Rightarrow\ D\sqrt{100}\\\\\Rightarrow\ D=10\text{ units}[/tex]

Hence, the distance between the points (5, 1) and (-3, -5) = 10 units

Answer:

exponential; y = 89 • 0.62^x

Step-by-step explanation:

Answer:

Option B exponential y = 89 · 0.62x

Step-by-step explanation:

The table shows the estimated number of deer living in a forest over a five year period.

Year            Number of deers

0                     89

1                      55

2                     34

3                     21

4                     13

Now we have to find the model representing this situation. Difference in number of deer, in the forest.

We can see there is a common ratio between each successive term r = [tex]\frac{55}{89}[/tex] = 0.618

r = [tex]\frac{34}{55}[/tex] = 0.618

so it can be represented by an exponential model.

[tex]y=a (r) ^{x}[/tex]

[tex]y=89(62) ^{x}[/tex]

Option B is the answer.

Answer:

Step-by-step explanation:

When there are many instances of the same calculation, it is convenient to let a spreadsheet or graphing calculator do them. The formula can be entered once and used many times. See the attachment for an example.

1. The surface area of a box with dimensions L, W, D can be written as ...

  S = 2(LW +LD +WD) = 2(LW +D(L+W))

Then the surface area of the left (original) box is ...

  S = 2(2·12 + 8(2+12)) = 2(24 +112) = 272 . . . . square inches

The surface area of the right (proposed) box is ...

  S = 2(4·3 +14(4+3)) = 2(12 +98) = 220 . . . . square inches

The volume of the original box, at 2·12·8 = 192 in³ is greater than the volume of the proposed box (3·4·14 = 168 in³), so the customer gets less cereal with the redesigned box. I don't dig it.

__

2. The previous question shows the formula and an example of the calculation. The attachment shows the numbers for this question.

__

3. The attachment shows the numbers for this question.

Answer:

Surface area increases by a factor of 4

Step-by-step explanation:

Given the linear ratio = a : b, then

the area ratio = a² : b²

Here the linear ratio = 1 : 2, hence

area ratio = 1² : 2² = 1 : 4 ← increase by factor of 4

Answer:

5 liters

Step-by-step explanation:

Divide 35 liters evenly between the 7 pitchers and you'll have 5 liters in each pitcher.

To find the amount of lemonade in each pitcher, divide the total amount of lemonade by the number of pitchers. In this case, each pitcher will contain 5 liters of lemonade.

To find the amount of lemonade in each pitcher, we divide the total amount of lemonade by the number of pitchers. In this case, Bob has 35 liters of lemonade and 7 pitchers. So, to find the amount of lemonade in each pitcher, we divide 35 by 7.

35 ÷ 7 = 5 liters of lemonade

Therefore, there will be 5 liters of lemonade in each pitcher.

If Bob has 35 liters of lemonade and he distributes it equally into 7 juice pitchers, we need to perform a division to find out how much lemonade will be in each pitcher. The calculation is straightforward:

Divide the total volume of lemonade by the number of pitchers.

35 liters ÷ 7 pitchers = 5 liters per pitcher.

Therefore, each juice pitcher will contain 5 liters of lemonade.

Answer:

8/5

Step-by-step explanation:

The slope of two parallel lines is always the same, just are in different locations on the x or y axis.

Final answer:

The surveying method that is most likely to produce a representative sample of the yogurt shop's customers is true random sampling.

Explanation:

The surveying method that is most likely to produce a representative sample of the yogurt shop's customers is the true random sampling method. This method involves randomly selecting participants from the entire population of yogurt shop customers, ensuring that each customer has an equal chance of being selected. This helps to minimize bias and ensure that the sample is representative of the entire customer population.

For example, the manager can generate a list of all the customers who have made purchases at the yogurt shop over a specific period of time, and then use a random number generator or a random selection method to choose a certain number of customers from the list. This will ensure that the selected participants represent the diversity of the yogurt shop's customer base.

The most appropriate surveying method for the yogurt shop manager to obtain a representative sample of customers is systematic sampling.

To obtain a representative sample of the yogurt shop's customers, the most appropriate surveying method would be a systematic sampling approach. This involves selecting every nth customer who visits the shop during different times of the day and different days of the week.

By using a systematic sampling method, the manager can ensure that the sample includes customers from various demographic groups, such as different age ranges, genders, and visiting patterns. This approach reduces the potential for bias that may arise from convenience sampling methods, where only customers who are readily available or willing to participate are surveyed.

Additionally, the systematic sampling method allows the manager to capture the preferences of both regular and occasional customers, as well as those who visit during peak and off-peak hours. This comprehensive representation of the customer base increases the likelihood that the survey results will accurately reflect the preferences of the yogurt shop's overall customer population.

Systematic sampling is more time-consuming and resource-intensive than convenience sampling, but it is a more reliable method for obtaining a representative sample of the yogurt shop's customers, which is crucial for making informed decisions about introducing new flavors that will appeal to a wide range of customers.

Answer:

The jar of jelly that costs $2.32 for 16 ounces is the better buy, because the unit rate is less

Step-by-step explanation:

step 1

Find the units rate

One jar of jelly costs $2.32 for 16 ounces

so

The unit rate is equal to [tex]\frac{2.32}{16}= 0.145\frac{\$}{ounce}[/tex]

Another jar costs $2.03 for 13 ounces

so

The unit rate is equal to [tex]\frac{2.03}{13}= 0.156\frac{\$}{ounce}[/tex]

step 2

Compare the unit rates

[tex]0.145\frac{\$}{ounce} < 0.156\frac{\$}{ounce}[/tex]

therefore

The jar of jelly that costs $2.32 for 16 ounces is the better buy, because the unit rate is less

The jelly that costs $2.32 for 16 ounces is the better buy. The unit rate for this jar of jelly is $0.145 or approximately $0.15 per ounce. The unit rate for the second jar of jelly is $0.156 or approximately $0.16 per ounce

Answer: Third option.

Step-by-step explanation:

You need to find the surface area with the formula for calculate the surface area of a rectangular prism:

[tex]SA=2(wl + lh + hw)[/tex]

Where w is the width, l is the length, and h is the height.

You can observe in the figure that:

[tex]l=5cm\\w=4cm\\h=2cm[/tex]

Then, substituting the values into the formula  [tex]SA=2(wl + lh + hw)[/tex], you get that  minimum amount of cardboard used for the wrapper is:

[tex]SA=2[(4cm)(5cm)+(5cm)(2cm)+(2cm)(4cm)]\\SA=76cm^2[/tex]

Answer:

third option

Step-by-step explanation:

Answer:

Sometimes you to plot some points to get a good approximation of the location of extreme values.

Approximating some points, especially influential points or outliers, helps to get a better grasp of the location of the extreme values. This is crucial in fields like calculus, statistics and graphing, and it assists in identifying patterns and trends. Additionally, substitution of 'x' values in an equation can help estimate 'y' values.

Sometimes in mathematics, particularly in fields such as calculus, statistics, and graph theory, you need to approximate some points for a good understanding of the location of extreme values. This practice is especially useful when identifying influential points or outliers within a data set, as these points significantly alter the slope or fitness of a regression line. When you find such points, you can exclude them from your calculations to get a more accurate overview of the general pattern or trend.

In the case of graphing, selection of an appropriate scale for both axes is also crucial. The scale should reflect all your data while also making it easy to identify any trends within it. Too large a scale can make data changes hard to see, while a too fine scale necessitates more space for the graph and can crowd information.

Additionally, in calculus and algebra, to estimate the 'y' values for various 'x-values', one can substitute the 'x' values into the equation. This helps to offer a clearer insight into the correlation of variables within a function.

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I'll match them for you, but to find the inverse of an equation, all you must do is

[tex]f(x)^{-1}  = 5x[/tex]  →  [tex]f(x) = \frac{x}{5}[/tex]

[tex]f(x)^{-1} = \frac{x^{3}}{2}[/tex]  →  [tex]f(x) = \sqrt[3]{2x}[/tex]

[tex]f(x)^{-1} = x + 10[/tex]  →  [tex]f(x) = x - 10[/tex]

[tex]f(x)^{-1} = \frac{3(x+17)}{2}[/tex]  →  [tex]f(x) = \frac{2x}{3} -17[/tex]

Hope I help ! :)

ANSWER

[tex] \boxed {f(x)= \frac{2x}{3} - 17\to \: f ^{ - 1} (x)=\frac{3x + 51}{2}}[/tex]

[tex] \boxed {f(x) = x - 10 \to {f}^{ - 1} (x) = x + 10 }[/tex]

[tex] \boxed {f(x) = \sqrt[3]{2x} \to {f}^{ - 1} (x) = \frac{ {x}^{3} }{2} }[/tex]

[tex] \boxed {f(x) = \frac{x}{5} \to{f}^{ - 1} (x) = 5x}[/tex]

EXPLANATION

1.

Given :

[tex]f(x) = \frac{2x}{3} - 17[/tex]

Let

[tex]y =\frac{2x}{3} - 17[/tex]

Interchange x and y.

[tex]x=\frac{2y}{3} - 17[/tex]

Solve for y.

[tex]x + 17=\frac{2y}{3} [/tex]

[tex]3x + 51=2y[/tex]

[tex]y=\frac{3x + 51}{2} [/tex]

[tex]f ^{ - 1} (x)=\frac{3x + 51}{2} [/tex]

2.

Given: f(x)=x-10

Let y=x-10

Interchange x and y.

x=y-10

Solve for y.

y=x+10

This implies that,

[tex] {f}^{ - 1} (x) = x + 10[/tex]

3.

Given:

[tex]f(x) = \sqrt[3]{2x} [/tex]

Let

[tex]y=\sqrt[3]{2x} [/tex]

Interchange x and y.

[tex]x=\sqrt[3]{2y} [/tex]

solve for y.

[tex] {x}^{3} = 2y[/tex]

[tex]y = \frac{ {x}^{3} }{2} [/tex]

[tex] {f}^{ - 1} (x) = \frac{ {x}^{3} }{2} [/tex]

4.

Given:

[tex]f(x) = \frac{x}{5} [/tex]

Let

[tex]y = \frac{x}{5} [/tex]

Interchange x and y.

[tex]x = \frac{y}{5} [/tex]

Solve for y.

[tex]y = 5x[/tex]

[tex] {f}^{ - 1} (x) = 5x[/tex]

Answer:

60

Step-by-step explanation:

Angles AKB and EKD are vertical angles and congruent.

The measure of angle AKB is the sum of the measures of arcs AB and ED divided by 2.

m<AKB = (110 + 130)/2 = 120

Angles AKB and AKE are supplementary angles, so their measures add to 180 deg.

m<AKE + m<AKB = 180

m<AKE + 120 = 180

m<AKE = 60

Answer:

The average rate of change is : [tex]6.3[/tex]

Step-by-step explanation:

The number of car owners is modeled by the function;

[tex]f(t)=1.1t^2-2.5t+1.5[/tex], where t is the different number of years.

The average rate of change  of f(t)  from t=3 to t=5 is simply the slope of the secant line connecting:

(3, f(3)) and (5,f(5))

Which is given by:

[tex]\frac{f(5)-f(3)}{5-3}[/tex]

Now, we substitute t=3 into the function to get;

[tex]f(3)=1.1(3)^2-2.5(3)+1.5[/tex]

[tex]f(3)=3.9[/tex]

We substitute t=5 into the function to get;

[tex]f(5)=1.1(5)^2-2.5(5)+1.5[/tex]

[tex]f(5)=16.5[/tex]

Therefore the average rate of change is : [tex]\frac{14.5-3.9}{2}=6.3[/tex]

Answer:

15 seconds

Step-by-step explanation:

If you make a table of values for the dog and the squirrel using d = rt, then the rates are easy:  the dog's rate is 150 and the squirrel's is 100.  The t is what we are looking for, so that's our unknown, and the distance is a bit tricky, but let's look at what we know:  the dog is 200 feet behind the squirrel, so when the dog catches up to the squirrel, he has run some distance d plus the 200 feet to catch up.  Since we don't know what d is, we will just call it d!  Now it seems as though we have 2 unknowns which is a problem.  However, if we solve both equations (the one for the dog and the one for the squirrel) for t, we can set them equal to each other.  Here's the dog's equation:

d = rt

d+200 = 150t

And the squirrel's:

d = 100t

If we solve both for t and set them equal to each other we have:

[tex]\frac{150}{d+200} =\frac{100}{d}[/tex]

Now we can cross multiply to solve for d:

150d = 100d + 20,000 and

50d = 20,000

d = 400

But we're not looking for the distance the squirrel traveled before the dog caught it, we are looking for how long it took.  So sub that d value back into one of the equations we have solved for t and do the math:

[tex]t=\frac{100}{d}=\frac{100}{400} =\frac{1}{4}[/tex]

That's 1/4 of a minute which is 15 seconds.

Answer:

4 mins

Step-by-step explanation:

Let x = the distance the squirrel runs before it's caught,

then the dog runs 200 + x.

distance/rate = time

x/100 = (200+x)/150 =>x/2 = (200+x)/3 => 400 +2x = 3x => x = 400

The squirrel runs 400' in 4 minutes.

Answer:

  A)  (20, 0)

Step-by-step explanation:

Double the second equation and add that to the first:

  (0.2x +0.5y) +2(-0.1x +0.3y) = (4) +2(-2)

  1.1y = 0

   y = 0

Substitute this value into either equation to find x.

  0.2x +0.5·0 = 4

  x = 4/0.2 = 20

The solution is (x, y) = (20, 0).

The answer is:

The correct option is:

A. $0.49

From the statement, we know that the iHome is used on average three hours a day, and we are asked to find the cost for a week, so first, we need to calculate the total hours that the iHome is used for, and then, calculate the kilowatt-hour consumption rate.

[tex]TotalTime_{week}=3\frac{hours}{day} *7days=21hours[/tex]

[tex]TotalEnergyConsumption_{week}=180watt*21hours=3780watt.hour[/tex]

Now, we must remember that:

[tex]1Kilowatt=1000watts[/tex]

So,

[tex]3780watts=3780watts.hour*\frac{1KiloWatt}{1000watts}=3.78KiloWatt.hour[/tex]

Then, calculating the cost, we have:

[tex]TotalCost_{week}=0.13\frac{dollar}{killowat.hour}*3.78killowat.hour=0.49(dollar)[/tex]

Hence, we have that the correct option is:

A. $0.49

Have a nice day!

Answer:

1.)[tex]-12+4x=-12+4x[/tex]

2.) [tex]2-2x \neq -8-2x[/tex]

Step-by-step explanation:

The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding all the products.

[tex]a(b+c)=ab+ac[/tex]

For the first equation

[tex]-2(6-2x)=4(-3+x)\\(-2)(6)+(-2)(-2x)=(4)(-3)+(4)(x)\\-12+4x=-12+4x[/tex]

For the second equation

[tex]5-1(2x+3)=-2(4+x)\\5+((-1)(2x)+(-1)(3))= (-2)(4)+(-2)(x)\\5+(-2x-3) = -8+(-2x)\\5-2x-3=-8-2x\\2-2x \neq -8-2x[/tex]

.