Xanthia buys hot dogs that come in packages of six, and she buys hot dog buns that come in packages of eight. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns?

Answer:

  18 °F

Step-by-step explanation:

The means are calculated in the usual way: add up the numbers and divide by the number of them. When there are a bunch of numbers, it is convenient to let a calculator or spreadsheet compute the mean for you.

In the attached, we see the mean for week 1 is 52°, and in week 2, it is 70°. The mean is 70° -52° = 18° greater in week 2.

Answer:

66

Step-by-step explanation:

11,22,33,44, and 55 are 5 consecutive multiples of 11.

11=11(1)

22=11(2)

33=11(3)

44=11(4)

55=11(5)

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

You can see  consecutive multiples of 11 where we don't know the actual multiples will look like:

11n,11(n+1),11(n+2),11(n+3),11(n+4).

Now we are given the sum of the numbers I just mentioned is 220.

This means,

11n+11(n+1)+11(n+2)+11(n+3)+11(n+4)=220

Each term 11n,11(n+1),11(n+2),11(n+3),11(n+4), and 220 all have a common factor of 11 so divide both sides by 11:

1n+1(n+1)+1(n+2)+1(n+3)+1(n+4)=20

1 times anything is still just that anything:

n+n+1+n+2+n+3+n+4=20

Combine the like terms:

n+n+n+n+n+1+2+3+4=20

Simplify the combining:

5n+10=20

Subtract 10 on both sides:

5n     =10

Divide both sides by 5:

 n      =10/5

Simplify right hand side:

 n      =2

So if n=2, then the multiples of 11 in question look like this:

11n=11(2)=22

11(n+1)=11(3)=33

11(n+2)=11(4)=44

11(n+3)=11(5)=55

11(n+4)=11(6)=66

--------------------------Add up to see if sum is actually 220.

Putting into my calculator gives me a result of 220.

We are good.

Now you just have to determine what the greatest of the number 22,33,44,55, and 66 is...

The greatest listed here is 66.

Answer:

900 miles will the plane be from its port when the boat is 54 miles away from its port

Step-by-step explanation:

Given data

boat away = 6 miles

plane away = 100 miles

to find out

How many miles will plane be from port when boat 54 miles away from its port

solution

first we consider x distance while plane travel and boat travel 54 miles

from question given the equation will be

boat away from its port / plane away from its port = boat away from its port/  plane away from its port

54 / x = 6/100

solve this equation

6 × (x) = 54 × 100

x = 54 × 100 / 6

x = 900

so 900 miles will the plane be from its port when the boat is 54 miles away from its port

Answer:

C =56.52 cm

C =56.52 cm

C =56.52 cm

A =254.34 cm^2

Step-by-step explanation:

To find the circumference of a circle, we use

C = 2 * pi *r

where pi is approximated by 3.14 and r is 9

C = 2 * 3.14 *9

C =56.52 cm

To find the area of a circle, we use

A = pi *r^2

where pi is approximated by 3.14 and r is 9

A = 3.14 *9^2

A =254.34 cm^2

Your answer would be 56.52 cm (18π) for the circumference, and 254.34 cm² (81π) for the area.

The formula for the circumference of a circle is 2πr. In this case, we know that the radius is 9, so substituting it in, you get 2 * 9 * π. Solving this, we get 18π as our circumference. Your question states to use 3.14 as π, so the final step is to multiply 18 by 3.14. 18 * 3.14 = 56.52. The unit in this case would just be cm, as we are looking at circumference, or length.

The formula for the area of a circle is πr². Again, we know r = 9, so just substitute it in the equation, to get π9², which can be solved to equal 81π. Then, using 3.14 as π, we get 81 * 3.14 = 254.34 as our final answer. The unit in this case would be cm², as this is concerning area.

Hope this helps!

Answer:

C.

Step-by-step explanation:

The only information you really need in order to determine if this is a right triangle are the slopes of segments AB and BC.  If the slopes of these segments are opposite reciprocals of one another, then the lines are perpendicular, and the angle is a right angle (making the triangle a right triangle!).  Point A has coordinates (-5, 5), B(-3, 2), C(-6, 0).

The slope of segment AB:

[tex]m=\frac{2-5}{-3-(-5)}=-\frac{3}{2}[/tex]

The slope of segment BC:

[tex]m=\frac{0-2}{-6-(-3)}=\frac{2}{3}[/tex]

As you can see, the slopes are opposite reciprocals of one another so angle ABC is a right angle, and triangle ABC is a right triangle.  Choice C is the one you want.

Answer:

Yes, two sides are perpendicular and the side lengths fit the Pythagorean theorm

Step-by-step explanation:

Because it's the answer

Answer:

  A. `x= 0`, `y= (3)/(2)`

Step-by-step explanation:

Dividing the first equation by 4 gives ...

  x = 0

Substituting that into the second equation gives ...

  3·0 +6y - 9

  y = 9/6 = 3/2 . . . . divide by 6, reduce the fraction

The solution of the set of equations is ...

  x = 0, y = 3/2

_____

The question here asks "what is the solution of y ?". That answer is y = 3/2.

Answer:

D

Step-by-step explanation:

A: Difference means minus. A has no minuses anywhere. Not the answer.

B: Could be true if you allow irrational numbers. I'm guessing you are not allowed to give (2x - sqrt(11)y)(2x + sqrt(11)y). So B is not the answer

C: Take out the 7 as a common factor. 7(x^2 - 3y^2) If you allow C, you have to allow B so C is not the answer.

D: answer (5x - y)(5x + y)

Answer:

  B.)  32

Step-by-step explanation:

The diagonals of a parallelogram bisect each other, so ...

  BE = DE

  y+10 = 4y - 8 . . . substitute the given expressions

  18 = 3y . . . . . . . . add 8-y

  6 = y . . . . . . . . . . divide by 3

Then BE = y+10 = 16 and ...

  BD = 2×BE = 2×16

  BD = 32

Answer:

B.) 32

Step-by-step explanation:

Given parallelogram ABCD, diagonals AC and BD intersect at point E, AE = 2x, BE=y+10, CE=x+2 and DE=4y - 8, the length of BD is 32.

BD = 2×BE = 2×16

Answer: She will have $700 left over.

Step-by-step explanation: Since we know that a rectangle has two sides with the measurement, we can add the sides. 37+37+25+25=124. The fencing in 124 feet in total. Multiply the 124 feet by the price per foot. 124 x 75 =9,300. Subtract the price from your total amount of money. 10,000 - 9,300 = $700. She will have $700 left over.

Answer:

there would be $700 left over

Step-by-step explanation:

Answer:

  B.  9, 9x, 18x

Step-by-step explanation:

The value in each box is the product of the row heading and column heading. You can find the missing column heading by dividing the box value (162) by the row heading (18).

Answer:

The answer is B coz its completes the factorization

Answer:

[tex]0.060[/tex]

Step-by-step explanation:

In a two tailed test the probability of occurrence is the total area under the critical range of values on both the sides of the curve (negative side and positive side)

Thus, the probability values for a two tailed test as compared to a one tailed test is given by the under given relation -

[tex]p-value = P(Z< -\frac{\alpha }{2} )+P(Z >\frac{\alpha}{2})[/tex]\

Here [tex]P\frac{\alpha}{2} = 0.030[/tex]

Substituting the given value in above equation, we get -

probability values for a two tailed test

=[tex]0.030 + 0.030\\= 0.060[/tex]

Answer:

S = -9,372 ⇒ 1st answer

Step-by-step explanation:

* Lets revise the geometric series

- There is a constant ratio between each two consecutive numbers

- Ex:

# 5  ,  10  ,  20  ,  40  ,  80  ,  ………………………. (×2)

# 5000  ,  1000  ,  200  ,  40  ,  …………………………(÷5)

* General term (nth term) of a Geometric series:

 U1 = a  ,  U2  = ar  ,  U3  = ar2  ,  U4 = ar3  ,  U5 = ar4

 Un = ar^(n-1), where a is the first term, r is the constant ratio between

 each two consecutive terms

- The sum of first n terms of a geometric series is calculate from

  [tex]S_{n}=\frac{a(1-r^{n})}{1-r}[/tex]

* Lets solve the problem

∵ The series is geometric

∵ a1 = -12

∴ a = -12

∵ a5 = -7500

∵ a5 = ar^4

∴ -7500 = -12(r^4) ⇒ divide both sides by -12

∴ 625 = r^4 take root four to both sides

∴ r = ± 5

∵ r = 5 ⇒ given

∵ [tex]Sn=\frac{a(1-r^{n})}{1-r}[/tex]

∵ n = 5

∴ [tex]S_{5}=\frac{-12[1-(5)^{5}]}{1-5}=\frac{-12[1-3125]}{-4}=3[-3124]=-9372[/tex]

* S = -9,372

Answer:

Option A is the correct choice.

Step-by-step explanation:

We have been given a histogram and we are asked to choose the correct statement about our given histogram.

Upon looking at our given histogram, we can see that our given data set is skewed to right. This means that means that the mean of the given data will be greater than median as our given data set has a long tail towards right or our data set is positively skewed.

Therefore, option A is the correct choice.

Answer:

I'm pretty sure its 1

Step-by-step explanation:

because if y = 1. 1 by the power of 3 is 1, and y by the power 2 is 1. 1- 10 is -9 . -9 times 1 equals -9, and -9 equals -9 therefore it's a true statement

Step-by-step explanation:

y³ (y² − 10) = -9y

Move everything to one side:

y³ (y² − 10) + 9y = 0

Factor out common term:

y (y² (y² − 10) + 9) = 0

Distribute:

y (y⁴ − 10 y² + 9) = 0

Factor:

y (y² − 1) (y² − 9) = 0

Solve:

y = 0, ±1, ±3

Since y > 0, the two possible values for y are 1 and 3.

Answer: A.2x+30=90

wwww

Answer:

  D.  X-7

Step-by-step explanation:

The table tells you that when x=7, g(x) = 0. In order for g(7) to be zero, at least one factor must be zero when x=7. The only factor on the list that is zero when x=7 is (x-7).

To see if (x - a) is a factor of g(x), a polynomial function, you check if g(a) = 0 from the values in the table. Unfortunately, without the table of values, we cannot definitively determine which of the options must be a factor of the function g(x).

In order to determine which of the options given must be a factor of the function g(x), we need to understand a property about polynomial functions and their factors. If (x - a) is a factor of a polynomial, then the function g(a) = 0. This means that (a,0) is a point on the graph of the function.

Unfortunately, without the values of x and g(x) from the table above we cannot definitively conclude which of the options A, B, C, or D must be a factor of the polynomial g(x). However, if for example, g(1) = 0 in the table, then (x - 1) or option A would be a factor of g(x). The same logic applies to the other options.

Remember to always check any similar question using the table of values provided to determine if a given expression is a factor of a function!

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Answer:

Step-by-step explanation:

The vertex order ABC is clockwise in the original figure and also in the first image: A'B'C'. The altitude from AC to B is up in the original and left in the first image, indicating a rotation 90° CCW.

  The first transformation is a rotation 90° CCW.

The vertex order of A''B''C'' is CCW, indicating a reflection. The direction of the altitude from A''C'' to B'' is still to the left, so the reflection must be over a horizontal line. We find the x-axis bisects the segments A'A'', B'B'', and C'C'', confirming that it is the line of reflection.

  The second transformation is reflection across the x-axis.

_____

Algebraically, the transformations are ...

  1st: (x, y) ⇒ (-y, x)

  2nd: (x, y) ⇒ (x, -y)

Both together: (x, y) ⇒ (-y, -x).

Answer:

see explanation

Step-by-step explanation:

To determine the magnitude of the scale factor, calculate the ratio of corresponding sides of image to original, that is

scale factor = [tex]\frac{A'B'}{AB}[/tex] = [tex]\frac{2}{5}[/tex]

ΔA''B''C'' is a reflection of ΔA'B'C' in the y- axis ( corresponding vertices are equidistant from the y- axis )

Step-by-step answer:

This is a problem involving subtraction of fractions.

To solve the problem, we find out the increase of the fraction of container and equate it to the amount of water added.  Then we find the amount of water contained in the whole container (fraction = 1)

7 gallons = 4/5 - 3/10 = 8/10 - 3/10 =5/10 = 1/2 container

therefore, multiply by two on both sides,

14 gallons = 1 container

So container can hold 14 gallons.

The volume of a 3D object is the amount of space it contains. A fish tank, for example, is three feet long, one foot wide, and two feet tall. To get the volume, multiply the length by the breadth by the height, which is 3x1x2, or six. As a result, the fish tank has a volume of 6 cubic feet.

Volume of cuboidal container- LBH

Given after poring 7 gallons tank is 4/5 full

Hence 7 gallon=4/5x

x=7*5/4=35/4gallon

Hence container can hold 35/4 gallon i.e=8.75gallon

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Answer:

11 years old

Step-by-step explanation:

Let Jimmy's age be represented as x. His big brother's age is x+5 and his sister's age is 2x. Adding these gives us 4x+5=29. Solving for x gives us 6. His brother's age is 6+5=11.

11 years old is Jimmy's brother.

Let Jimmy's age be represented as x. His big brother's age is x+5 and his sister's age is 2x. Adding these gives us 4x+5=29. Solving for x gives us 6. His brother's age is 6+5=11.

J=Jimmy's age; S=sister's age=2J; B=brother's age=J+5

.

J+S+B=29

J+(2J)+(J+5)=29

4J+5=20

4J=24

J=6

Jimmy is 6 years old.

B=J+5=6+5=11

ANSWER: Jimmy's brother is 11 years old.

.

CHECK:

S=2J=2(6)=12

Jimmy's sister is 12 years old.

.

J+S+B=29

6+12+11=29

29=29.

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Answer:

Y

Step-by-step explanation:

The answer is no, the result of the joint test for the null hypothesis that both [tex]\( \beta_1 = 0 \) and \( \beta_2 = 0 \)[/tex] is not necessarily implied by the results of two separate tests for each coefficient.

To understand why, let's consider the two scenarios:

1. Separate Tests: When we conduct two separate tests for [tex]\( \beta_1 = 0 \) and \( \beta_2 = 0 \)[/tex], we are looking at the significance of each predictor independently. We might find that neither [tex]\( \beta_1 \) nor \( \beta_2 \)[/tex] is significantly different from zero on its own. However, this does not account for the potential multicollinearity between [tex]\( X_1 \) and \( X_2 \)[/tex]. Multicollinearity can result in high variance of the coefficient estimates, leading to insignificant t-tests even if the predictors have a joint effect on the response variable.

2. Joint Test (F-test): The joint test, typically conducted using an F-test, assesses whether both [tex]\( \beta_1 \) and \( \beta_2 \)[/tex] are simultaneously equal to zero. This test takes into account the correlation between [tex]\( X_1 \) and \( X_2 \)[/tex] and evaluates the combined effect of both variables on the response variable. It is possible that while neither variable alone is significant, together they might have a significant effect.

The F-test for the joint hypothesis is based on the reduction in the sum of squared residuals when including [tex]\( X_1 \) and \( X_2 \)[/tex] in the model compared to a model with only the intercept (reduced model). The test statistic is calculated as:

[tex]\[ F = \frac{(\text{SSR}_{\text{reduced}} - \text{SSR}_{\text{full}}) / k}{\text{SSR}_{\text{full}} / (n - p - 1)} \][/tex]

where:

- [tex]\( \text{SSR}_{\text{reduced}} \)[/tex] is the sum of squared residuals from the reduced model.

- [tex]\( \text{SSR}_{\text{full}} \)[/tex] is the sum of squared residuals from the full model.

- [tex]\( k \)[/tex]is the number of restrictions (in this case, 2, since we are testing two coefficients).

- [tex]\( n \)[/tex] is the number of observations.

-  [tex]\( p \)[/tex] is the number of predictors in the full model (not including the intercept).

The degrees of freedom for the numerator are k and for the denominator are [tex]\( n - p - 1 \)[/tex].

In summary, the results from separate t-tests for [tex]\( \beta_1 \) and \( \beta_2 \)[/tex] do not necessarily inform us about the joint significance of these coefficients. It is entirely possible for the separate tests to show non-significance while the joint F-test shows significance, indicating that the predictors have a joint effect on the dependent variable even if their individual effects are not significant. Conversely, it is also possible for the separate tests to show significance for one or both coefficients, while the joint test does not show significance, suggesting that the combined effect of the predictors is not significant.

Answer:

Step-by-step explanation:

speed x time = distance

s (1.5) = 105

1.5s=105

s = 70mph

d(t) = 70t

Answer:

The equation that relates y and x is [tex]y=x+612 mi[/tex], and the equation that relates x and d is [tex]4d=x[/tex].

Step-by-step explanation:

Step 1: First we know that the total distance is equal to y. The distance traveled in 4 hours equals x, and the distance from point x to y equals 612 miles. Adding x to the remaining 612 miles gives the total distance y.

[tex]y=x+612 mi[/tex]

Step 2: To know the relationship between x and d, we must first raise the speed during the journey to x.

[tex]v=\frac{x}{4h}[/tex]

Then, we set the speed d e equal v:

[tex]v=\frac{d}{h}[/tex]

[tex]\frac{x}{4h} = \frac{d}{h}[/tex]

Clearing x we get:

[tex]x=4h * \frac{d}{h}[/tex]

[tex]x=4d[/tex]

Have a nice day!

Answer:

So the approximate relative minimum is (0.4,-58.5).

Step-by-step explanation:

Ok this is a calculus approach.  You have to let me know if you want this done another way.

Here are some rules I'm going to use:

[tex](f+g)'=f'+g'[/tex]       (Sum rule)

[tex](cf)'=c(f)'[/tex]          (Constant multiple rule)

[tex](x^n)'=nx^{n-1}[/tex] (Power rule)

[tex](c)'=0[/tex]               (Constant rule)

[tex](x)'=1[/tex]                (Slope of y=x is 1)

[tex]y=4x^3+13x^2-12x-56[/tex]

[tex]y'=(4x^3+13x^2-12x-56)'[/tex]

[tex]y'=(4x^3)'+(13x^2)'-(12x)'-(56)'[/tex]

[tex]y'=4(x^3)'+13(x^2)'-12(x)'-0[/tex]

[tex]y'=4(3x^2)+13(2x^1)-12(1)[/tex]

[tex]y'=12x^2+26x-12[/tex]

Now we set y' equal to 0 and solve for the critical numbers.

[tex]12x^2+26x-12=0[/tex]

Divide both sides by 2:

[tex]6x^2+13x-6=0[/tex]

Compaer [tex]6x^2+13x-6=0[/tex] to [tex]ax^2+bx+c=0[/tex] to determine the values for [tex]a=6,b=13,c=-6[/tex].

[tex]a=6[/tex]

[tex]b=13[/tex]

[tex]c=-6[/tex]

We are going to use the quadratic formula to solve for our critical numbers, x.

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

[tex]x=\frac{-13 \pm \sqrt{13^2-4(6)(-6)}}{2(6)}[/tex]

[tex]x=\frac{-13 \pm \sqrt{169+144}}{12}[/tex]

[tex]x=\frac{-13 \pm \sqrt{313}}{12}[/tex]

Let's separate the choices:

[tex]x=\frac{-13+\sqrt{313}}{12} \text{ or } \frac{-13-\sqrt{313}}{12}[/tex]

Let's approximate both of these:

[tex]x=0.3909838 \text{ or } -2.5576505[/tex].

This is a cubic function with leading coefficient 4 and 4 is positive so we know the left and right behavior of the function. The left hand side goes to negative infinity while the right hand side goes to positive infinity. So the maximum is going to occur at the earlier x while the minimum will occur at the later x.

The relative maximum is at approximately -2.5576505.

So the relative minimum is at approximate 0.3909838.

We could also verify this with more calculus of course.

Let's find the second derivative.

[tex]f(x)=4x^3+13x^2-12x-56[/tex]

[tex]f'(x)=12x^2+26x-12[/tex]

[tex]f''(x)=24x+26[/tex]

So if f''(a) is positive then we have a minimum at x=a.

If f''(a) is negative then we have a maximum at x=a.

Rounding to nearest tenths here:  x=-2.6 and x=.4

Let's see what f'' gives us at both of these x's.

[tex]24(-2.6)+25[/tex]

[tex]-37.5[/tex]  

So we have a maximum at x=-2.6.

[tex]24(.4)+25[/tex]

[tex]9.6+25[/tex]

[tex]34.6[/tex]

So we have a minimum at x=.4.

Now let's find the corresponding y-value for our relative minimum point since that would complete your question.

We are going to use the equation that relates x and y.

I'm going to use 0.3909838 instead of .4 just so we can be closer to the correct y value.

[tex]y=4(0.3909838)^3+13(0.3909838)^2-12(0.3909838)-56[/tex]

I'm shoving this into a calculator:

[tex]y=-58.4654411[/tex]

So the approximate relative minimum is (0.4,-58.5).

If you graph [tex]y=4x^3+13x^2-12x-56[/tex] you should see the graph taking a dip at this point.

Answer:

  3/2

Step-by-step explanation:

Every coordinate of E'F'G'H' is 3/2 times that of EFGH, so the image is 3/2 times the size of the original.

___

For example, E'x/Ex = -3/-2 = 3/2; E'y/Ey = (-3/2)/-1 = 3/2.

The scale factor between quadrilateral EFGH and its image E'F'G'H' can be found by calculating the ratio of the lengths of corresponding sides. In this case, it is approximately 0.707.

In this task, we are asked to find the scale factor between a quadrilateral and its image. The scale factor can be found by dividing the lengths of corresponding sides in the image by the respective side length in the original figure.

First, let's calculate the distance between the vertices E and F in the original figure using the Euclidean distance formula: sqrt[(x2-x1)^2 + (y2-y1)^2] = sqrt[(1+2)^2 + (2+1)^2] = sqrt[9+9] = sqrt[18] = approximately 4.2426.

Then, let's do the same for vertices E' and F' in the image: sqrt[(3/2+3)^2 + (3+3/2)^2] = sqrt[(9/4 + 9/4)+(9/4 + 9/4)] = sqrt[(9/2)+(9/2)] = sqrt[9] = 3.

Your scale factor, then, is the length of EF' divided by the length of EF: 3 / 4.2426, which roughly equals 0.707.

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Answer:

= 1/2[sin2Acos7A + cos2Asin7A + sin2Acos7A - cos2Asin7A]

we cut out

cos2Asin7A - cos2Asin7A

then we have

=1/2[sin2Acos7A + sin2Acos7A ]

= 1/2*2[ sin2Acos7A ]

cut out 2 we get

#sin2Acos7A

Each side of the square would be approximately 9.3 units long.

To predict the side length of the square when the area is 86.6 units, we need to use the model that Sam created. Sam likely developed a mathematical relationship between the area (x) and the length of one side of the square (f(x)). This relationship is typically expressed as a function, such as [tex]\( f(x) = \sqrt{x} \),\\[/tex] where [tex]\( x \)[/tex] represents the area and [tex]\( f(x) \)[/tex]represents the length of one side of the square.

In this model, the side length of the square is equal to the square root of the area. Therefore, to predict the side length when the area is 86.6 units, we substitute this value into the function:

[tex]\[ f(86.6) = \sqrt{86.6} \][/tex]

Now, we can calculate this:

[tex]\[ f(86.6) \approx \sqrt{86.6} \approx 9.3 \][/tex]

So, according to Sam's model, when the area is 86.6 units, the length of one side of the square is approximately 9.3 units. This means that if you were to draw a square with an area of 86.6 units, each side of the square would be approximately 9.3 units long.

With the function [tex](f(x) = \sqrt{x - 5} + 3\),[/tex] the predicted side length for an area of 86 is calculated by evaluating f(86).

This results in a side length of 12 units.

Therefore option b. 12 units is correct.

To find the predicted side length of a square when given the area, let's use the function provided by Sam:

[tex]\[f(x) = \sqrt{x - 5} + 3.\][/tex]

Here, x represents the area, and f(x) represents the predicted side length of the square.

Find f(x) for (x = 86):

1. Plug in[tex]\(x = 86\):[/tex]

[tex]\[f(86) = \sqrt{86 - 5} + 3 = \sqrt{81} + 3 = 9 + 3 = 12.\][/tex]

The predicted side length when the area is 86 is 12.

Given the function[tex]\(f(x) = \sqrt{x - 5} + 3\),[/tex]  the predicted side length when [tex]\(x = 86\)[/tex] is option b. 12 units.

Question : After plotting the data where x=area, and f(x)=the length of one side of the square, Sam determined the model to approximate the side of a square was f(x)= *square root sign* x-5+3 Use the model Sam created to predict the side length of the square when the area is 86.

a. 6

b. 12

c. 81

d. 144

Answer:

a

Step-by-step explanation:

Given

x² + 20x + 100 = 36 ( subtract 36 from both sides )

x² + 20x + 64 = 0 ← in standard form

Consider the factors of the constant term ( + 64) which sum to give the coefficient of the x- term ( + 20)

The factors are + 16 and + 4, since

16 × 4 = + 64 and 16 + 4 = + 20, hence

(x + 16)(x + 4) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 16 = 0 ⇒ x = - 16

x + 4 = 0 ⇒ x = - 4

Answer:

f(x)=3-4 In (x-2)=graph 3

f(x)=3-In x=graph 1

f(x)=In (x+1)=graph 4

f(x)= 2In (x+3)= graph 2

Step-by-step explanation:

Use a graph tool to visualize the functions.Attached are the graphed functions respectively.

Answer:

f(x) = 3 - 4㏑(x - 2) ⇒ graph 3

f(x) = 3 - ㏑(x) ⇒ graph 1

f(x) = ㏑(x + 1) ⇒ graph 4

f(x) = 2㏑(x + 3) ⇒ graph 2

Step-by-step explanation:

* Lets look to the graphs and solve the problem

- We will use some points on each graph and substitute in the function

  to find the graph of each function

- Remember: ㏑(1) = 0 and ㏑(0) is undefined

- Lets solve the problem

# f(x) = 3 - 4㏑(x - 2)

- Let x - 2 = 1 because ㏑(1) = 0, then f(x) will equal 3

∵ x - 2 = 1 ⇒ add 2 for both sides

∴ x = 3

- Substitute the value of x in f(x)

∴ f(x) = 3 - 4㏑(3 - 2)

∴ f(x) = 3 - 4㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 3

∴ Point (3 , 3) lies on the graph

- Look to the graphs and find which one has point (3 , 3)

∵ Graph 3 has the point (3 , 3)

∴ f(x) = 3 - 4㏑(x - 2) ⇒ graph 3

# f(x) = 3 - ㏑(x)

- Let x = 1 because ㏑(1) = 0, then f(x) will equal 3

- Substitute the value of x in f(x)

∴ f(x) = 3 - ㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 3

∴ Point (1 , 3) lies on the graph

- Look to the graphs and find which one has point (1 , 3)

∵ Graph 1 has the point (1 , 3)

∴ f(x) = 3 - ㏑(x) ⇒ graph 1

# f(x) = ㏑(x + 1)

- Let x = 0 because ㏑(1) = 0, then f(x) will equal 0

- Substitute the value of x in f(x)

∴ f(x) = ㏑(0 + 1) = ㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 0

∴ Point (0 , 0) lies on the graph

- Look to the graphs and find which one has point (0 , 0)

∵ Graph 4 has the point (0 , 0)

∴ f(x) = ㏑(x + 1) ⇒ graph 4

# f(x) = 2㏑(x + 3)

- Let x + 3 = 1 because ㏑(1) = 0, then f(x) will equal 0

∵ x + 3 = 1 ⇒ subtract 3 from both sides

∴ x = -2

- Substitute the value of x in f(x)

∴ f(x) = 2㏑(-2 + 3) = 2㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 0

∴ Point (-2 , 0) lies on the graph

- Look to the graphs and find which one has point (-2 , 0)

∵ Graph 2 has the point (-2 , 0)

∴ f(x) = 2㏑(x + 3) ⇒ graph 2