7. Suppose a student stacks 1000 congruent circles one on top of the other, and the teacher
1000 similar circles, with each circle a little smaller than the one below it.
a. What is the name of the solid formed by the stack of congruent circles?
b. Relate the dimensions of one of the congruent circles to the dimensions of this solic
c. What is the name of the solid formed by the stack of similar circles?
d. Relate the height of the stack of similar circles to a dimension of this solid.

The question is about translating the graph of the function y = 3/x. The graph of y = 3/(x - 1) + 3 results from shifting the parent function to the right by 1 unit and upwards by 3 units on the coordinate grid.

The question pertains to the translation of a rational function on the coordinate grid and is related to graph transformations. The function y = 3/x is known as the parent function, and y = 3/(x - 1) + 3 is the transformed function. To illustrate the translation, we can understand the changes in terms. Here x - 1 indicates a horizontal shift to the right by 1 unit, and +3 signifies a vertical shift upwards by 3 units. The new graph represents these translations on the coordinate system, which is a two-dimensional representation with x and y-axes as described in the reference information provided.

Rewrite 1/2 to have a common denominator with 2/6

1/2 x 3 = 3/6

Now you have 1 and 3/6 - 2/6  = 1 and 1/6

Now rewrite 2/3 to have a common denominator with 1/6: 2/3 x 2 = 4/6

Subtract 1/6 from 4/6:

4/6 - 1/6 = 3/6 = 1/2

You will need to add 1/2

To get from the difference between 1 and 1/2 and 2/6 to 1 and 2/3, one must add 1/2 after ensuring both differences have a common denominator.

To find out what should be added to the difference between 1 and 1/2 and 2/6 to get 1 and 2/3, we need to perform a few steps involving fractions. First, let's convert 1 and 1/2 into an improper fraction by multiplying 1 (the whole number) by 2 (the denominator of the fractional part) and adding 1 (the numerator of the fractional part), which gives us 3/2. Now, to find the difference between 3/2 and 2/6, we must have a common denominator, which is 6 in this case. Multiplying the numerator and denominator of 3/2 by 3 gives us 9/6.

Now we subtract 2/6 from 9/6 to get the difference, which is 7/6 or 1 and 1/6. To find what needs to be added to this difference to get 1 and 2/3, we must also express 1 and 2/3 as an improper fraction. Multiplying 1 (the whole number) by 3 (the denominator) and adding 2 (the numerator) gives us 5/3.

To compare 7/6 with 5/3, we need the same denominator, which is 6. So we multiply the numerator and denominator of 5/3 by 2 to get 10/6. Subtracting 7/6 from 10/6, we find the missing number to be 3/6, which simplifies to 1/2.

Answer:

x = 9

Step-by-step explanation:

When 2 chords intersect inside a circle then the product of the measures of the parts of one chord is equal to the product of the measures of the parts of the other chord, that is

2 × x = 3 × 6, that is

2x = 18 ( divide both sides by 2 )

x = 9

Answer:

f(x) = 1/2x + 2

y - 3 = 1/2(x - 2)

y - 1 = 0.5(x + 2)

Step-by-step explanation:

The equations that represent the graphed line are f(x) = 1/2x + 2, f(x) = 1/2x + 1, and y - 1 = 1/2(x + 2).

The equation of a line in slope-intercept form is given by y = mx + b, where m is the slope of the line and b is the y-intercept. In the given options, the equations that represent the graphed line are:

These equations have the same slope and y-intercept as the graphed line.

Answer:

The number of calories in a peanut treat is 30.

The number of calories in a coconut treat is 25.

Step-by-step explanation:

So we have Ben ate 2 peanut treats (let p represent the calories in each peanut treat) and 4 coconut treats (let c represent the calories in each coconut treat).  The sum of those calories is 160.

This is the equation for Ben:

2p+4c=160

I'm going to use the same representation here.  p for the number of calories in each peanut treat and c for the number of calories in each coconut treat.

Arnold ate 3 p's and 2 c's for a sum of 140 calories:

Alrnoldo's equation is:

3p+2c=140

So we have the system:

2p+4c=160

3p+2c=140

I'm going to divide the first equation by 2 because each term is divisible by 2:

p+2c=80

3p+2c=140

This system is setup for elimination because the equations are in the same form and the second column have the same variable expression, 2c.  So we are going to subtract to eliminate the variable c, that is 2c-2c=0.

p+2c=80

3p+2c=140

---------------------Subtract!

-2p+0=-60

-2p    =-60

Divide both sides by -2:

  p      =30

So if p=30 and p+2c=80 , then 30+2c=80.

So let's solve:

30+2c=80 for c

Subtract 30 on both sides:

     2c=50

Divide both sides by 2:

     c=25

The number of calories in a peanut treat is 30.

The number of calories in a coconut treat is 25.

Answer:

2 cm base, 3 cm height

Step-by-step explanation:

Let

x -----> the height of one of the identical triangles pieces of the plane

y ----> the base of one of the identical triangles pieces of the plane

we know that

x=3 cm

2y+2=6

solve for y

2y=6-2

2y=4

y=2 cm

therefore

The dimensions of one of the identical triangles pieces of the plane are 2 cm base and 3 cm height

Answer:

The correct option is 1. The dimensions of one of the identical triangular pieces of the plane are 2 cm base and 3 cm height.

Step-by-step explanation:

From the given figure it is noticed that the length of parallel sides of the trapezoid are 6 cm and 2 cm.

The height of triangular pieces is same as the height of trapezoid. So the height of triangular pieces is 3 cm.

Let the measure of base of the triangular pieces be x.

[tex]x+2+x=6[/tex]

Combine like terms.

[tex]2x+2=6[/tex]

Subtract 2 form both the sides.

[tex]2x+2-2=6-2[/tex]

[tex]2x=4[/tex]

Divide both sides by 2.

[tex]x=\frac{4}{2}[/tex]

[tex]x=2[/tex]

The base of the triangles is 2 cm.

The dimensions of one of the identical triangular pieces of the plane are 2 cm base and 3 cm height. Therefore the correct option is 1.

Step-by-step explanation:

[tex]\sqrt[n]{a^m}=a^\frac{m}{n}\\\\\large\huge\boxed{a^\frac{4}{9}=\sqrt[9]{a^4}}[/tex]

The radical expression which represents a to the four ninths power is:

         Ninth root of a to the fourth power.

We are asked to find the radical expression for the word phrase:

             a to the four ninths power.

i.e. mathematically it could be written as:

[tex]a^{\dfrac{4}{9}}[/tex]

Now, we know that:

[tex]a^{\dfrac{m}{n}}=(a^m)^{\dfrac{1}{n}}=\sqrt[n]{a^m}[/tex]

Here we have:

[tex]m=4\ \text{and}\ n=9[/tex]

Hence, the expression cold be written as:

[tex]a^{\dfrac{4}{9}}=\sqrt[9]{a^4}[/tex]

Answer:

-4

Step-by-step explanation:

plug 2 into the equation.

f(2)=4(2)-12

    = 8-12

    = -4

I hope this helped!

Answer:

f(2) =-4

Step-by-step explanation:

f(x)=4x-12

Let x=2

f(2) = 4(2) -12

      =8-12

       =-4

Answer:

[tex]x=7 \frac{3}{7}[/tex] gallons

Step-by-step explanation:

So we are given:

70 miles   ->  1 gallon is used.

520 miles ->  x gallons is used.

Set up a proportional to solve.

I lined up everything already above:

[tex]\frac{70}{520}=\frac{1}{x}[/tex]

Cross multiply:

[tex]70(x)=520(1)[/tex]

[tex]70x=520[/tex]

Divide both sides by 70:

[tex]x=\frac{520}{70}[/tex]

Reduce the fraction (divide top and bottom by 10):

[tex]x=\frac{52}{7}[/tex]

How many 7's are in 52?  7 because 7(7)=49

How much is left over after seven 7's go into 52? 52-49=3

So the answer as a mixed fraction is:

[tex]x=7 \frac{3}{7}[/tex]

To find out how much gas was in the motorcycle's tank, divide the total distance driven by the fuel efficiency. In this case, 520 miles / 70 miles per gallon equals 7 2/5 gallons of gas.

To determine how much gas started in the tank of the motorcycle that gets 70 miles per gallon and drove 520 miles before running out of gas, we use the formula:

Answer:

H-L Theorem

Step-by-step explanation:

They tell you that they are right triangles, and you have your congruency marks. If they never had right angle indicators, this theorem would never work.

I hope this helps you out, and as always, I am joyous to assist anyone at any time.

Answer:

1, 3, 5

Step-by-step explanation:

The sum of an arithmetic series is:

S_n = n (a_1 + a_n) / 2

100 = n (1 + 19) / 2

n = 10

The nth term of an arithmetic sequence is:

a_n = a_1 + d (n − 1)

19 = 1 + d (10 − 1)

d = 2

The common difference is 2.  So the first three terms are 1, 3, and 5.

Answer:

I don't remember well, and I'm not sure if this is correct, but, I think the answer is  This is beacuse ∠FCD are facing with ∠CDG meaning it would be a same side interior angle.

Answer:

Step-by-step explanation:

If you extend BF until it meets AG and consider the the triangle CD and the meeting point, then angle FCD is an interior angle.

If on the other hand it might just be the supplement of DCB which would make it neither.  I can see why you want  us to take a shot at it. That is a seat of my pants answer. If someone has a better reason, take that one.

The volume of the prism is [tex]\(V = 648 \, \text{cubic units}\).[/tex]

The formula for the volume (V) of a rectangular prism is given by [tex]\(V = lwh\),[/tex] where (l) is the length, (w) is the width, and (h) is the height. In this case, the provided dimensions are length (l = 18), width (w = 9), and height (h = 4). Substituting these values into the formula: [tex]\[ V = 18 \times 9 \times 4 = 648 \, \text{cubic units}. \][/tex] Therefore, the volume of the prism is [tex]\(648 \, \text{cubic units}\).[/tex]

Understanding the concept of volume in three-dimensional geometry is vital. The formula [tex]\(V = lwh\)[/tex] reflects the relationship between the length, width, and height of a rectangular prism. Multiplying these dimensions provides the total space enclosed by the prism in cubic units. In this instance, with a length of 18 units, a width of 9 units, and a height of 4 units, the volume calculation yields [tex]\(648 \, \text{cubic units}\),[/tex] representing the spatial capacity of the given prism.

Calculating the volume of prisms is a fundamental skill in geometry and has practical applications in various fields, including architecture and engineering. This calculation allows us to quantify the amount of space a three-dimensional object occupies, providing essential information for design and analysis. In this case, the resulting volume, [tex]\(648 \, \text{cubic units}\),[/tex] represents the capacity of the rectangular prism specified by the given dimensions.

Step-by-step explanation:

Write a proportion:

3 buckets / 1 gallon = 1 bucket / x gallons

Cross multiply and solve:

3x = 1

x = 1/3

You need 1/3 of a gallon of water.

Answer:

Step-by-step explanation:

Givens

I foot of sill = 1.75

35 feet of sill = x

Paid with 100 dollars.

Solution

First part

Solve the proportion

1/35 = 1.75/ x                 Cross multiply

1*x = 35 * 1.75

x = 61.25

=========

Second part

Get the change.

Change = 100 - 61.25

Change  = 38.75

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

The change =

3 tens                30.00

1 five                   5.00

3 ones                3.00

3 quarters           .75

Total                 38.75

Hey there!

The area of a trapezoid is h(a+b/2), where h is the height and a and b are the two bases. We already have our area, so let's solve for our height in the equation.

40= x(16/2)

40=8x

x=5

Therefore, the height is B) five units.

I hope this helps!

Answer:

answer is 5

Step-by-step explanation:

For this case we have a fucnion of the form [tex]y = f (x)[/tex]

Where:

[tex]f (x) = 4x-1[/tex]

By definition, the rank of a function is given by:

The set of the real values that the variable y or f (x) takes.

So:

[tex]f (-1) = 4 (-1) -1 = -4-1 = -5\\f (0) = 4 (0) -1 = 0-1 = -1\\f (1) = 4 (1) -1 = 4-1 = 3\\f (2) = 4 (2) -1 = 8-1 = 7\\f (3) = 4 (3) -1 = 12-1 = 11[/tex]

ANswer:

The range is: -5, -1,3,7,11

Answer:

{-5,-1,3,7,11}

Step-by-step explanation:

The range is the output of the function

We have the inputs

f(x) = 4x - 1

f(-1) = 4(-1)-1 = -4-1 = -5

f(0) = 4(0)-1 = 0-1 = -1

f(1) = 4(1)-1 = 4-1 = 3

f(2) = 4(2)-1 = 8-1 = 7

f(3) = 4(3)-1 = 12-1 = 11

The outputs for the given inputs are

{-5,-1,3,7,11}

Answer:

A and X

Step-by-step explanation:

None

Answer:

Therefore, the answer is 3

Step-by-step explanation:

The given expression is:

[tex]\frac{3k}{k-2} + \frac{6}{2-k} \\\\Taking\ LCM\\= \frac{3k(2-k)+6(k-2)}{(k-2)(2-k)} \\=\frac{6k-3k^2+6k-12}{(k-2)(2-k)}\\= \frac{-3k^2+12k-12}{(k-2)(2-k)}\\\\Applying\ formula\\=\frac{-3(k^2-4k+4)}{(k-2)(2-k)}\\=\frac{-3((k)^2-2*2*k+(2)^2)}{(k-2)(2-k)}\\=\frac{-3(k-2)^2}{(k-2)(2-k)}\\=\frac{-3(k-2)}{(2-k)}\\=\frac{3(2-k)}{2-k}\\=3[/tex]

Answer: 2k

Step-by-step explanation:

-k + 3k = (-1 + 3)k = 2k

Answer:

A

Step-by-step explanation:

One and three hundred and four thousand

1+                             3/100+          4/1000

Answer:

≈ 545.42 cm²

Step-by-step explanation:

The area (A) of sector EOG is calculated as

A = area of circle × fraction of circle

   = πr² × [tex]\frac{100}{360}[/tex]

   = π × 25² × [tex]\frac{10}{36}[/tex]

   = 625π × [tex]\frac{10}{36}[/tex]

   = [tex]\frac{625(10)\pi }{36}[/tex] ≈ 545.42 ( to 2 dec. places )

Answer: [tex]\bold{focus: \bigg(2, -7 \dfrac{1}{8}\bigg), \quad directrix: y = -6 \dfrac{7}{8}}[/tex]

Step-by-step explanation:

First, rearrange the equation into vertex form: y = a(x - h)² + k   where

NOTE: p is the distance from the vertex to the focus

      y = -2x² + 8x - 15

y + 15 = -2x² + 8x            → added 15 to both sides

y + 15 = -2(x² - 4x)           → factored out -2 from the right side

y + 15 + (-2)(4) = -2(x² - 4x + 4)    → completed the square

y + 7 = -2(x - 2)²                → simplified

     y = -2(x - 2)² - 7           → subtracted 7 from both sides

Now it is in vertex form where:

Focus = (2, -7 + p)  →  Focus = (2, -7 + (-1/8))   →  [tex]Focus = \bigg(2, -7 \dfrac{1}{8}\bigg)[/tex]

Directrix: y = -7 - p   →  Directrix: y = -7 - (-1/8)   →  [tex]Directrix: y = -6 \dfrac{7}{8}[/tex]

Answer:

The equation is (x+9)^2 + (y-3)^2 = 5

Step-by-step explanation:

The standard form for the equation of a circle is:

(x−h)^2+(y−k)^2=r2

The center of the circle is the midpoint of the diameter.

So the midpoint of the diameter at  (-10, 1) and (-8, 5) can be determined as:

(-10 +(-8))/2 , (1+5)/2

= -10-8/2, 1+5/2

= -18/2 , 6/2

= -9 , 3

Thus(-9,3) is the center of the circle.

Now we will use the distance formula to find the radius of the circle.

r^2=(-10-(-9))^2 + (1-3)^2

r^2=(-10+9)^2 +(-2)^2

r^2=(-1)^2 + (-2)^2

r^2=1 + 4

r^2= 5

Take square root at both sides.

√r^2= √5

r=√5

Now put the values in the 1st equation.

(x−h)^2+(y−k)^2=r2

where h = -9, k =3 and r = √5

(x-(-9))^2 + (y-3)^2= (√5)^2

(x+9)^2 + (y-3)^2 = 5

Thus the equation is (x+9)^2 + (y-3)^2 = 5 ....

Answer:

1/3

Step-by-step explanation:

The formula for computing the sum of an infinite geometric series is

[tex]S=\frac{a_1}{1-r}[/tex] where r is between -1 and 1 and [tex]r[/tex] is the common ratio, and [tex]a_1[/tex] is the first term of the series.

So let's plug in:

[tex]S=\frac{0.3}{1-0.1}[/tex]

[tex]S=\frac{0.3}{0.9}[/tex]

[tex]S=\frac{3}{9}[/tex] I multiplied bottom and top by 10.

[tex]S=\frac{1}{3}[/tex] I divided top and bottom by 3.

The sum is 1/3.

The sum of the infinite geometric series is 1 / 3.

When there is a constant between the two successive numbers in the series then it is called a geometric series. In other words, every next term is multiplied with that constant term to form a geometric progression.

Given that the first term a₁ = 0.3 and common ratio r = 0.1. The sum of the geometric series is calculated by using the formula below:-

S = a₁ / ( 1 - r )

S = 0.3 / ( 1 - 0.1 )

S = 0.3 / 0.9

S = 3 / 9

S = 1 / 3

Therefore, the sum of the infinite geometric series is 1 / 3.

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To answer your question the list price for Emily's vehicle would have a $23,655.91 list price

Hope this helped

Aaron

Answer:  The list price of the car was $ 23655.91.

Step-by-step explanation:  Given that Emily bought a new car for $22000 and she paid 93% of the list price.

We are to find the list price of the car.

Let $ x denote the list price of the car.

Then, according to the given information, we have

[tex]93\%\times x=22000\\\\\\\Rightarrow \dfrac{93}{100}\times x=22000\\\\\Rightarrow 93x=22000\times100\\\\\Rightarrow 93x=2200000\\\\\Rightarrow x=\dfrac{2200000}{93}\\\\\Rightarrow x=23655.91.[/tex]

Thus, the list price of the car was $ 23655.91.

Answer:

r is the common ratio (in geometric sequence)

d is the common difference (in arithmetic sequence)

n is the term number

a1 is the value of the first term

an is the value of the nth term

Step-by-step explanation:

For the sequence formula, each term represent as follows,

[tex]r[/tex]: common ratio

[tex]d[/tex]: common difference

[tex]n[/tex]: term number

[tex]a_{n}[/tex]: the value of the [tex]nth[/tex] term

[tex]a_{1}[/tex]: the value of the first term

"Sequence is defined representation of a number in a particular order using some formula."

According to the question,

For the sequence formula,

Each variable is matched with the following term,

[tex]r[/tex]: common ratio

[tex]d[/tex]: common difference

[tex]n[/tex]: term number

[tex]a_{n}[/tex]: the value of the [tex]nth[/tex] term

[tex]a_{1}[/tex]: the value of the first term

Hence, for the sequence formula, each term represent as follows,

[tex]r[/tex]: common ratio

[tex]d[/tex]: common difference

[tex]n[/tex]: term number

[tex]a_{n}[/tex]: the value of the [tex]nth[/tex] term

[tex]a_{1}[/tex]: the value of the first term

Learn more about sequence here

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Answer:

[tex]\large\boxed{\dfrac{1}{4}a^{-3}b^3=\dfrac{b^3}{4a^3}}[/tex]

Step-by-step explanation:

[tex]\dfrac{4a^2b^5}{16a^5b^2}\qquad\text{use}\ \dfrac{x^n}{x^m}=x^{n-m}\\\\=\dfrac{4}{16}a^{2-5}b^{5-2}=\dfrac{1}{4}a^{-3}b^3\qquad\text{use}\ x^{-n}=\dfrac{1}{x^n}\\\\=\dfrac{b^3}{4a^3}[/tex]

Answer:

-4/3

Step-by-step explanation:

because the line are perpendicular, so the slope of re line is -1/(3/4)=-4/3