The top of a hill rises 64 feet above checkpoint -213. What is the altitude of the hill?

Answer:

a) Common ratio r = [tex]\boxed{2}[/tex]

b) Formula for common ratio:
[tex]\dfrac{a_{n+1}}{a_n} , n = 1, 2, 3, 4....\\[/tex]

Formula for nth term:
[tex]a_n = a_1r^{n-1}[/tex]

(see explanation below for part b)

c) [tex]a_{10} = \boxed{1024}[/tex]

d) Recursive formula: [tex]\boxed{a_{n+1} = 2a_n}[/tex]

Step-by-step explanation:

a) The common ratio r for a geometric sequence is the ratio of any term(except the first) to the previous term. And this is a constant for that sequence

If we have a sequence [tex]a_1, a_2, a_3, a_4, a_5,....\\[/tex]

[tex]r = \dfrac{a_2}{a_1} = \dfrac{a_3}{a_2} = \dfrac{a_4}{a_3} = \dfrac{a_5}{a_4}[/tex]

Here [tex]a_1[/tex] is the first term in the sequence

The sequence we have here is
2, 4, 8, 16

So
[tex]r = \dfrac{4}{2} = \dfrac{8}{4} = \dfrac{16}{8} = 2\\[/tex]
Answer to a)

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b) This part is confusing. Are they asking for a formula for the common ratio or a formula for the nth term? You can write both or check with your teacher

The explicit formula for the common ratio is

[tex]\dfrac{a_{n+1}}{a_n} , n = 1, 2, 3, 4....\\[/tex]

The explicit formula for the nth term, [tex]a_n[/tex] of a geometric sequence is
[tex]a_n = a_1r^{n-1}[/tex]

where [tex]a_1[/tex] is the first term of the sequence and r is the common ratio
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c)
The formula for finding the nth term of a geometric sequence is
[tex]a_n = a_1r^{n-1}\\\\[/tex]

Therefore

[tex]a_{10} = a_1 r^{(10-1)} = a_1 \times r^9\\\\[/tex]

[tex]a_1 = 2, r = 2\\\\[/tex]

So
[tex]a_{10} = 2 \cdot 2^9 \\\\= 2 \cdot 512\\\\= 1024\\[/tex]ANSWER c)

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d)

A recursive formula is just an expression that denotes the relationship between one term and the next

We know that
[tex]r= \dfrac{a_{n+1}}{a_n} \\\\[/tex]

Multiplying both sides by [tex]a_n[/tex],

[tex]ra_n = a_{n+1}[/tex]

Or

[tex]a_{n+1} = ra_n\\\\[/tex]

For the specific case of r = 2 we get the recursive formula as

[tex]a_{n+1} = 2a_n[/tex]

ANSWER d)

The given system of equations has infinitely many solutions. Option A is correct.

Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.

Here,
Given equation,
- 6x - 2y = 10 - - - -1

y = - 3x - 5  - - - -2

From equation 1
-6x - 2y = 10
Reform the equation,
-2y = 6x + 10

Divide the above equation by -2,
y = -3x - 5

Since the above equation is equivalent to equation 2, therefore the system of equations has infinitely many solutions.

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The plumber bought 5 pieces of copper wire and 5 pieces of plastic wire.

What is an equation?

Two expressions are combined in an equation using an equal symbol ("="). The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals sign.

Length of each piece of copper wire = 6 m

Length of each piece of plastic wire = 1 m

Total number of pipes bought = 10

Total length of pipes = 35 m

Let the number of copper pipes be x.

Let the number of plastic pipes be y.

The equation for number of pipes will be -

x + y = 10...(1)

The equation for length of pipes will be -

6x + y = 35...(2)

Subtracting equation (1) from (2) -

5x = 25

x = 5

Substituting the value of x in equation (1) -

5 + y = 10

y = 10 - 5

y = 5

Therefore, the number of copper and plastic wires is 5 each.

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We can simplify the expression to get:

2³ + 3*(2) + 14/2 = 21

The correct option is E.

Here we want to simplify the expression below:

2³ + 3*(2) + 14/2

The additions are the last thing we need to do, we can replace:

2³ = 8

3*(2) = 6

14/2 = 7

Replacing that in the expression we will get:

2³ + 3*(2) + 14/2 = 8 + 6 + 7

8 + 6 + 7 = 21

So the correct option is E.

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The smiley face is solved to get a value of 72

The solution to the value of the smiley face is solved by forming some equations

Using the third row that have only squares

8 square - 6 square + 3 square = 60

2 square + 3 square = 60

5 square = 60

dividing through by 5

square = 60 / 5 = 12

hence a square is 12

Using the second row

12 square + 5 star = 234

substituting a square = 12

12 * 12 + 5 star = 234

144 + 5 star = 234

5 star = 234 - 144

5 star = 90

dividing through by 5

star = 90 / 5 = 18

hence a star is 18

using the first row

square + 3 triangle + star = 57

substituting square = 12 and star = 18

12 + 3 triangle + 18 = 57

3 triangle + 30 = 57

3 triangle = 57 - 30

3 triangle = 27

dividing through by 3

triangle = 27 / 3 = 9

hence triangle = 9

solving for the smiley face

substituting the values of the figures

18 (9 + 18 - 12) + 12 (18 + 9 + 9) = smiley face

solving the parenthesis

smiley face = 18 (15) + 12 (36)

smiley face = 18 * 15 + 12 * 36

smiley face = 702

hence the smiley face is 702

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The equation of the line passing through the point (0, 2) and (2, 12) is y = 5x + 2

An equation is an expression composed of variables and numbers linked together by mathematical operations.

The slope intercept form of a linear equation is:

y = mx + b

Where m is the slope (rate of change) and b is the y intercept.

Let us assume the line passes through the point (0, 2) and (2, 12), hence:

y - 2 = [(12 - 2)/(2 - 0)](x - 0)

y = 5x + 2

The equation of the line is y = 5x + 2

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The value of {d} for F = 4 is equivalent to {d} = 7.9.

Given is that {F} is inversely proportional to {d} to the power of 2.

We can write the inverse relation as -

F [tex]$\alpha[/tex] 1/d²

Rewriting the relation replacing constant, we get -

F = K/d²

For {d} = 6, {F} = 7. We can write -

F = K/d²

7 = K/36

K = 36 x 7

K = 252

Now, we get the relation as -

F = 252/d²

For {F = 4}, we can write -

4 = 252/d²

d² = 252/4

d² = 63

d = 7.9

Therefore, the value of {d} for F = 4 is equivalent to {d} = 7.9.

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Answer: Which is impossible, indicating that the system is inconsistent. Therefore, based on the analysis performed with the elimination method, the system has no solution.

Step-by-step explanation:

Multiplying the first equation by 2 we get:

Now, once we have amplified the original equations, the sum of the first equation and the second equation leads to:

Which is impossible, indicating that the system is inconsistent. Therefore, based on the analysis performed with the elimination method, the system has no solution.

Answer:

8232 South African Rand

Step-by-step explanation:

The question cuts off at "if the current exchange" so the exchange rate is not specified. At the time of writing this answer, the echnage rate from one sterling pound to South African Rand is 1=20.58. So, we just need to multiply 20.58*400 to get 8232.

Answer:

3/2

Step-by-step explanation:

8 ratio 12 would be 2/3. but to get the right ratio sequence, we switch up the fraction to give the right answers. so 3/2 as in 12 ratio 8.

The sum of numbers 4 and - 9 is, - 5.

And, A weather station is a collection of instruments that measure atmospheric conditions to help study the weather and climate of a specific location.

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

Given that;

The numbers are,

⇒ 4 and - 9

Now, Sum of the numbers 4 and - 9 as;

⇒ 4 + (-9)

⇒ 4 - 9

⇒ - 5

Thus, The sum of numbers 4 and - 9 is, - 5.

And,  A weather station is a collection of instruments that measure atmospheric conditions to help study the weather and climate of a specific location.

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Answer:

( - inf, 5) (9, inf )

Step-by-step explanation:

The arrow(s) at the end of the line indicate that the highlighted green line does not stop at a specific number, hence the use of infinity signs (inf). If the arrow is going to the left on a line, then the value would be negative infinity. If it is going to the right, then it would be positive.

As for the space in between the green arrows pointing separate ways, there are two points, one which is positioned at the number 5 and the other at the number 9. Because there are two separate points which are both solid green, the use of parentheses is needed. The final answer is, as mentioned, ( - inf, 5) (9, inf ).

10 Km²  is the area of the triangular section that the forest managers plant with trees.

A sector is the area between any two circle radii and any arcs that connect them. Segment: A line segment in geometry is a section of a line that has two clearly defined end points and includes every point on the line in between them.

Firstly count how many cells are there

First line = 6

Second line = 1

Last line = 3

So the area of the section that the forest managers plant with tree is

6 + 1 + 3 = 10 Km²

( count more than half a grid or not count )

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The solution of the equation (x - 5)² + 3(x - 5) + 9 = 0 is x = -7 ± √35/2. Option B

From the information given, we have the equation as;

(x - 5)² + 3(x - 5) + 9 = 0

To solve the equation, take the following steps;

(x -5)(x-5) + 3x - 15 + 9 = 0

x² - 5x - 5x + 25 + 3x - 15 + 9 = 0

Now, collect like terms

x² - 5x - 5x + 3x + 25 - 15 + 9 = 0

add or subtract the like terms

x² - 7x + 21 = 0

The quadratic formula is is given as;

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

substitute the values

x = -(7) ± [tex]\frac{(-7)^2 - 4(1)(21)}{2(1)}[/tex]

expand the bracket

x = -7 ± [tex]\sqrt{\frac{49 - 84}2} }[/tex]

x = -7 ± √35/2

Hence, the value is x = -7 ± √35/2

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Hope it helps.

If you have any query, feel free to ask.

Answer:

Kathy is 48 years old and Samantha is 12.

Step-by-step explanation:

This is a challenging question that requires some interpretation.  Let K and S stand for the current ages of Kathy and Samatha.

We are told that:  K+S = 60

And, in 8 years:  K+8 and S+8

"Kathy will be 5 years older than 3 times as old as Samantha was 3 years ago"  ["3 years ago:"  from now or in 8 years.  I'll interpret it as from now, or (S-3)]

(K+8) = 3(S+8-3)+5   ["5 years older than 3 times as old as Samantha was 3 years ago"]

(K+8) = 3(S+5)+5

K+8 = 3S+20

 K = 3S+12

---

Rearrange the firts equation to isolate K

K = 60-S

Now use this definition of K in the previous equation:

K = 3S+12

60-S = 3S+12

4S = 48

S = 12

K+S = 60

K+12 = 60

K = 48

=====

CHECK

Does K + S = 60?   48+12 = 60 YES

In 8 years, will Kathy will be 5 years older than 3 times as old as Samantha was 3 years ago?

   K = 48+8 = 56

   S = 12+8 = 20

Three years ago S was 9.

(K+8) = 3(S+8-3)+5

(K+8) = 3(12+8-3)+5

K + 8 = 3(17)+5

K = 48

===

Kathy is 48 years old and Samantha is 12.

Answer:

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

The distance is the product of the speed and time.

Given:

Find the distance:

Answer:

131.25 feet.

Step-by-step explanation:

We can use the concept of similar triangles to find the length of the monument's shadow.

Since the pole and the monument are casting the shadows, we can assume that the ratio of the height of the pole to the length of its shadow is equal to the ratio of the height of the monument to the length of its shadow.

12/5 = 315/x

Where x is the length of the monument's shadow.

We can cross-multiply and solve for x:

12x = 5315

x = 5*315/12

x = 131.25

So, the length of the monument's shadow is approximately 131.25 feet.

Answer:

162 miles.

Step-by-step explanation:

To find the total distance Nicole cycled, we can multiply the number of laps by the length of each lap.

The total distance Nicole cycled is:

9 laps * 9540 ft/lap = 858,600 ft

To convert this distance to miles, we can use the conversion factor that there are 5280 feet in a mile.

858,600 ft / 5280 ft/mile = 162 miles

So the total distance Nicole cycled is 162 miles.

A function to calculate the average attendance for all of the locations for the year 2018 would be: =average(F5:F11).

To enter a function to calculate the average attendance for the year 2022 we have to follow the following steps.

Note: This question is incomplete because the information is incorrect, and some information is missing. Here is the complete information and the correct information:

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the plot is attached below and the slope is 4.

What is the slope of line?

Given two line-side coordinates, use the slope formula to determine the slope of the line. The slope is defined as the ratio of the change in the y values to the change in the x values using the formula m=(y2-y1)/(x2-x1). x1 and y1 are represented by the first point's coordinates. The second points are located at x2, y2, and their locations. Whichever point you choose to classify as the first and which as the second doesn't matter.

We know that formula two find the slope of the line joining two points which is

The equation y = 4x + 3 models the data below.

x 1   2  3  4   5    6   7   8

y 8 13 16 21 26 29 33 37

The plot will be using this data.

Hence the plot is attached below and the slope is 4.

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Step-by-step explanation:

Possible values of the correlation coefficient range from -1 to +1, with -1 indicating a perfectly linear negative, i.e., inverse, correlation (sloping downward) and +1 indicating a perfectly linear positive correlation (sloping upward). A correlation coefficient close to 0 suggests little, if any, correlation

Answer:255

Step-by-step explanation:

Answer:

Step-by-step explanation:

You want the missing length EG and the corresponding value of x in isosceles triangle DEF with midline GH of length 6 parallel to base DF of length 12.

The length of segment GH is half the length of segment DF (6/12 = 1/2), so we know that corresponding sides EG and ED have the same ratio:

  EG/ED = 1/2

  2EG = ED . . . . . . . . . . . . multiply by 2·ED

  2(x -5) = (x-5)+9 . . . . . . substitute given values

  2x -10 = x +4 . . . . . . . . simplify

  x = 14 . . . . . . . . . . . . . add 10-x

Then the length of EG is ...

  EG = 14 -5 = 9

Answer:

Newton's 3rd law of motion

Step-by-step explanation:

Newton's 3rd law of motion states that for every force in nature there is an equal and opposite reaction.

So, assuming the people are both pushing with the same force, their forces are basically canceling each other out, and a net force of 0 is acting uponm the box, which explains why it is not moving.

The domain of the function 5h + 2d = 15 is [0, 3].

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

The function 5h + 2d = 15

Where h is the number of hours and d is the distance remaining after h hours.

Now,

5h + 2d = 15

2d = 15 - 5h

d = (15 - 5h)/2

For h = 0, d = 15/2 = 7.5

For h = 1, d = (15 - 5)/2 = 10/2 = 5 miles.

For h = 2, d = (15 - 10)/2 = 5/2 = 2.5 miles.

For h = 3, d = (15 - 15)/2 = 0 miles

The domain of the function is [0, 3].

Thus,

The domain is [0, 3].

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Answer:

A) 4 batches

B) 1 1/2 batches

C) 10 batches

D) 1 2/3 batches

Step-by-step explanation:

A) A 1/4 goes into 1, 4 times so it would make 4 batches.

B) 1/4 = 2/8 so 3/8 = 1.5/4 and that would make 1.5 batches.

C) since 1 cup of sugar can make 4 batches, 2 cups can make 8 batches and 1/2 cup makes 2, so 8+2= 10 batches

D) 1/4=3/12 that means 5/12 = 1/4+1/6 and 1/6 divided by 1/4 = 2/3 so that means that it can make 1 2/3 batches.

The boat and the whale are about 218.5 miles apart.

Using the Pythagorean theorem to calculate the distance between the boat and the whale.

Let's call the distance between the boat and the whale "d" and the time they moved apart "t".

1st movement:

d = 45t

2nd movement:

d = (45t + 25(t+35))²

To calculate the final distance, we need to square d from the second movement and add the result from the first movement.

d² = (45t)² + (25(t+35))²

We know that the boat and the whale were moving apart for 35 minutes and 25 minutes, so we can substitute t = 35 and t = 25 in the equation.

d² = (4535)² + (2560)²

Taking the square root of the result and get the final distance between the boat and the whale.

d =  ([tex]\sqrt{4535 ^{2} +2560^{2} }[/tex])

d ≈ 218.5 miles

Hence, the boat and the whale are about 218.5 miles apart.

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The equation that models the scenario and number of years for the population to reach 656,000 will be option A.

i.e. y=100(3)^t and 8 years.

What are equations?

Mathematically, an equation can be defined as a statement that supports the equality of two expressions, which are connected by the equals sign “=”. For example, 2x – 5 = 13.

Here,

2x – 5 and 13 are expressions

The sign that connects these two expressions is “=”.

Algebra considers two prominent families of equations, namely polynomial equations and linear equations. Polynomial equations with one variable can be written in P(x) = 0, where P is a polynomial, and ax + b = 0 is the general form of linear equations. Here, a and b are parameters. We can practice geometric or algorithmic methods from linear algebra or mathematical analysis to solve these equations. Also, there are different types of equations, such as:

Linear equations

Quadratic equations

Cubic equations

Quartic equations

Differential equations

Parametric equations

Now,

The equation as the population is compounding annually for the scenario

will be y=P*(1+R)^t

Here p=Initial population

R=rate of increment/ decrement of population

t=time

Hence,

y=100*(1+2)^t                    (R=2 )

y=100*3^t

Time for which the population will be 656,000 will be

656000=100*3^t

6560=3^t

therefore,

t=8 years.

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Circle the equation that models the scenario and the number of years it takes for the population to reach 656,000 is y = 100(3)^x, 8 years.

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