Write the explicit formula for each geometric sequence.
Then determine the 8th term of each sequence.
Show work below.
Part A) 4, 16, 64, 256,...

Answer:

x = 25

Step-by-step explanation:

Following the Pythagoras' Theorem,

 a² + b² = c²

24² + 7² = c²

         c² = 625

       ∴ c = 25

In this case, since the length of the hypotenuse is x cm, ∴ c = x.

∴ x = 25

Answer:

25

Step-by-step explanation:

According to Pythagoras theorem,

H²=P²+B²

(H-hypotenuse, P-perpendicular, B-Base)

X²=24²+7²

X²=576+49

X²=625

X²=25²

X=25

The greatest number of blocks that can be packed into a box that is 6 units tall, 7 units wide, and 8 units deep without exceeding the height of the box = 84

Let h represents the height, w represents the width and l represents the length of the box.

The box is 6 units tall, 7 units wide, and 8 units deep.

⇒ h = 6 units, w = 7 units and l = 8  units

The volume of the box would be,

V = l × w × h

V = 6 × 7 × 8

V = 336 cu. units

The four block has  unit-cubes.

The volume of unit each unit cube would be,  1 × 1 × 1 = 1 cubic unit

so, the volume of 4 blocks would be,

V₁ = 4 × 1

V₁ = 4 cu. units

The number of blocks that can be placed in the box would be,

n = V / V₁

n = 336 / 4

n = 84

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The complete question is:

There is an ample supply of identical blocks (as shown). Each block is constructed from four 1 × 1 × 1 unit-cubes glued whole-face to whole-face. What is the greatest number of such blocks that can be packed into a box that is 6 units tall, 7 units wide, and 8 units deep without exceeding the height of the box?

The equations that have x = -2 as their solution are B and C

How to find the solution x  = - 2

A. 4x=4+2x

4x - 2x = 4

2x = 4

x = 4 / 2

x = 2

B 2(x+5)=x+8

2x + 10 = x + 8

2x - x = 8 - 10

x = -2

C.  5+3x=-1

5 + 1 = - 3x

6 = -3x

x = -2

D 3x-5=1

3x = 5 + 1

3x = 6

x = 2

E  19=2(x-6)+3

19 = 2x - 12 + 3

19 = 2x - 9

19 + 9 = 2x

28 = 2x

x = 28/2

x = 14

The equations where x=-2 is a solution are 5+3x=-1 , 2(x+5)=x+8

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Answer:

Stacy needs to sell at least 23 prints in order for her revenue to equal her expenses.

Explanation: To calculate the total expenses, we need to add the fixed cost of equipment ($38) to the variable cost of each print ($2.5). So the total expenses per print is $2.5 + $38 = $40.50. To calculate the number of prints needed to break even, we divide the total expenses by the revenue per print. 40.50 / 12 = 3.375 which is approximately 3.4 and if we round it up to 4 ,then we will get 4*12 = 48 and this is less than the fixed cost of 38. So, we need to increase the number of print to 23 in order to her revenue to equal her expenses.

Step-by-step explanation:

Answer: a. The probability of drawing a single card from a 52 card deck that is a diamond or a 10 is 13/52 or 1/4. There are 13 diamonds in a standard deck of 52 cards and 4 of them are 10s. So, the chance of drawing a diamond 10 is 4/52 and the chance of drawing a diamond is 13/52. The probability of drawing a diamond or a 10 is the sum of the individual probabilities, which is (4/52) + (13/52) = 17/52 or 1/3.

b. The probability of drawing a single card from a 52 card deck that is a spade or a queen is 13/52 or 1/4. There are 13 spades in a standard deck of 52 cards and 4 of them are queens. So, the chance of drawing a spade queen is 4/52 and the chance of drawing a spade is 13/52. The probability of drawing a spade or a queen is the sum of the individual probabilities, which is (4/52) + (13/52) = 17/52 or 1/4.

Step-by-step explanation:

Answer:

To find the coordinates of P, we can use the concept of proportional division of line segments. Given that AP:BP = 2:5, it means that the ratio of the length of AP to BP is 2:5.

Let's first find the vector that represents the direction of AB, which is the vector pointing from A to B. We can find this by subtracting the coordinates of A from the coordinates of B:

(10,13) - (-4,-8) = (14, 21)

This is the vector <14,21>

Now, we can find the point P, which is located two fifths of the way from A to B, by multiplying the vector <14,21> by 2/7 and adding the result to the coordinates of A:

P = A + (2/7) * <14,21>

= (-4,-8) + (2/7) * <14,21>

= (-4,-8) + <8,6>

= (4,-2)

So, the coordinates of P are (4,-2)

We can observe that point P is two fifth of the way from A to B.

Jackie made 12 bracelets each day and the equation used to represent this is 8x = 96. The solution has been obtained using the concept of simplification.

What is simplification?

In mathematics, the operation and interpretation of a function to make it simple or easier to grasp is known as simplifying, and the process is known as simplification.

We are given that Jackie made bracelets for 8 days and when he was done he had 96 bracelets with him.

Let 'x' be the number of bracelets made by Jackie each day

So, we get

⇒8x = 96

⇒x = 96/8

⇒x = 12

Hence, Jackie made 12 bracelets each day and the equation used to represent this is 8x = 96.

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Question: Jackie made bracelets for 8 days. When he was done he had 96 bracelets Write equation to express how many bracelets Jackie made each day.

The fraction 19/8 on the graph is located between the integers 2 and 3, exactly at the decimal 2.375.

A fraction is converted to decimal applying a proportion, which is the division of the numerator of the fraction by the denominator of the fraction.

The fraction in this problem is given as follows:

19/8.

For which:

Hence the equivalent decimal is given as follows:

19/8 = 16/8 + 3/8 = 2 + 0.375 = 2.375.

Which is between 2 and 3 on the number line.

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x = 20.4 it take to fill the entire tank.

Calculate a tank's total capacity and the volume that is filled in gallons and litres for tanks like water and oil storage. assumes the tank's interior measurements.

You can enter metric dimensions in metres (m) or centimetres, or U.S. dimensions in feet (ft) or inches (in) (cm). Results are given in fluid US gallons, fluid Imperial (UK) gallons, fluid cubic feet (ft3), fluid metric litres (fl), and fluid cubic metres (m3).

x = total volume of the entire tank.

We can suggest the following equation:

6 minutes

6 minutes to fill 3.4

x=6*3.4=20.4

x=20.4

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The value of x for the expression is 73.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

log (4x + 8) - log 3 = 2

[ log (a/b) = log a - log b ]

So,

log (4x + 8) - log 3 = 2 can be written as,

log (4x + 8)/3 = 2

Taking exponents on both sides.

(4x + 8)/3 = [tex]10^2[/tex]

4x + 8 = 100 x 3

4x + 8 = 300

4x = 292

x = 73

Thus,

The value of x is 73.

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Answer: To prove that triangle ABE is congruent to triangle CBD, we will use the SAS (Side-Angle-Side) Congruence Theorem.

This theorem states that if two triangles have two sides and the included angle congruent, then the triangles are congruent.

Given:

BE ≅ BD (congruent sides)

AD ≅ CE (congruent sides)

We have congruent sides AB and BC, and congruent angles A and C (included angles)

By SAS congruence theorem, we can prove that:

Triangle ABE ≅ triangle CBD

Therefore, triangle ABE is congruent to triangle CBD

Alternatively, if we use the ASA (Angle-Side-Angle) congruence theorem, where we have congruent angles and congruent sides between those angles, we can also prove that triangle ABE ≅ triangle CBD

Step-by-step explanation:

The function f(x) = 1/4x³ is constant when the intercept is 0

Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer.

the graph of a polynomial function is a smooth continuous curve.

The y-intercept is the point where the graph crosses the y-axis and can be found by substituting x = 0. The x-intercepts are points where the graph crosses the x-axis and can be found by making the function equal to zero and solving for x.

therefore f(0) = 1/4 0³ = 0

therefore the y- intercept of the function is 0

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Answer:

In the product 12 to the power of 3 * 11 to the power of 3, the exponents (3 and 3) indicate that 12 is multiplied by itself 3 times, and 11 is multiplied by itself 3 times.

When two or more numbers with the same base are multiplied together, the exponents can be added, because it's the same as multiplying the base numbers together. For example, (12^3) * (12^2) = 12^(3+2) = 12^5

However, in this case, the bases of 12 and 11 are different, and therefore, the exponents cannot be added. The product of 12 to the power of 3 * 11 to the power of 3 is different from (1211) to the power of 3, the result is (1211)^3= 1332^3

Answer:

$2 for each candy bar

Step-by-step explanation:

16 - 2 = 14

14 / 7 = 2

Answer:

$2

Step-by-step explanation:

16-2=14

14/7=2

he spent $14 total on candy bars, divide that by 7, and each candy bar costs $2

A longitudinal study is a type of research method used to observe changes in a given behavior or condition of a group of individuals over a period of time.

It is especially useful in resolving the limitation of a traditional correlational study in that it can provide evidence for the direction of a possible causal relationship between the variables. This is because the study is conducted over a period of time, which allows for the observation of changes in the variables over time, and the ability to determine which variable is causing the changes in the other. This can be done through the use of statistical tests, such as regression analysis, which can measure the strength of the correlation and the direction of the relationship. Additionally, longitudinal studies can also help to determine whether the sample size is sufficient, as well as whether the relationship between the variables is statistically significant.

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According to the diagram,

The two triangles are right-angle triangles.

ΔFUN and ΔCAR

where angle CAR and angle FUN are 90° each.

According to RHS congruency

NU = CA (each 3 cm)

FUN = CAR (each 90°)

FN = CR (each 5cm)

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The current computer's speed to price ratio is 400% of the older computer's speed to price ratio.

The speed to price ratios, along with the percentage, is obtained applying the proportions in the context of this problem.

Five years ago, the ratio was given as follows:

0.5x/2y = 0.25x/y.

The current ratio is given as follows:

x/y.

The current ratio is 4 times greater than the ratio of five years ago, hence it's a percentage of 400% the old ratio.

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Answer:

Step-by-step explanation:

According to the normal distribution and the central limit theorem the ballot measure is greater than 0.50.

The term called normal distribution is defined as a symmetrical plot of data around its mean value, where the width of the curve is defined by the standard deviation.

Here we know that t a random sample of 500 voters from one region, region a, of the city and the ballot measure in region a is 0.47.

Here we have to find the mean and the standard deviation are calculated as,

=> mean = 0.47

=> Standard deviation = √0.47(1-0.47)/500

When we simplify this then we get,

=> Standard deviation = 0.0223

By using the Central Limit Theorem the probability is calculated as,

=> Z = (0.5 - 0.47) / 0.0223

=> Z = 1.34

When we use the z table, then we get the p - value as 0.9099

Then the probability is calculated as,

=> P = 1 - 0.9099

=> P = 0.0901

Hence 9.01% probability that the proportion of voters in the sample of Region A who support the ballot measure is greater than 0.50.

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The mean period of the oscillation is 1.257 s and the standard deviation is 0.123 s.

To find the mean and standard deviation of the data, we can use the following formulas:

The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest

Standard deviation = [tex]\frac{\sqrt{((sum of (data point - mean)^{2}) } \\}{number of data points - 1}[/tex]

First, let's find the mean:

Mean = (1.88 + 0.97 + 1.43 + 1.26 + 1.61 + 1.19 + 1.17 + 1.28 + 1.25 + 1.48) / 10

Mean = [tex]\frac{12.57}{10}[/tex]

Mean = 1.257

The mean is 1.257.

Next, let's find the standard deviation:

Standard deviation = ([tex]\sqrt{(((1.88 - 1.257)^{2}+(0.97 - 1.257)^{2} (1.43 - 1.257)^{2}+(1.26 - 1.257)^{2}+(1.61 - 1.257)^{2}}[/tex][tex]+\sqrt{+(1.17 - 1.257)^{2}+(1.28 - 1.257)^{2}+(1.48 - 1.257)^{2}}[/tex])/9

Standard deviation =[tex]\sqrt{\frac{0.134}{9} }[/tex]

Standard deviation = [tex]\sqrt{0.015}[/tex]

Standard deviation = 0.123

Therefore, the mean period of the oscillation is 1.257 s and the standard deviation is 0.123 s (sample standard deviation).

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The measure of angle MAC is 90°.

What does the isosceles triangle mean?

When two of a triangle's sides are congruent, or the same length, the triangle is said to be isosceles. This means that if you draw a line through the middle of an isosceles triangle, it will split in half, with the two parts having equal-length sides.

The angles on either side of the congruent side of an isosceles triangle are also congruent. As a result, in the ABC triangle, mBAM = mCAB = mNAC.

Angle MAC is congruent to itself since angles BAM and NAC are complementary to it and are also congruent with each other.

Therefore, m∠MAC = m∠MAC = 180 - m∠BAM = 180 - m∠NAC.

Since m∠BAM = m∠NAC, we can conclude that

m∠MAC = 180 - m∠NAC = 90°

Hence, the measure of angle MAC is 90°.

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The total payment will be $90 + $15x and the job would have to be 1/15 hour long for the payment to cancel out Wanda's expenditures.

What is Fraction?

Fraction is a mathematical expression that represents a part of a whole. It is written as a ratio of two numbers, where the first number is the numerator and the second number is the denominator. For example, 1/2 is a fraction that represents one-half of a whole.

The total cost of equipment and employee wages for a business is the sum of the cost of the equipment and the cost of the employee's wages, which is calculated by multiplying the employee's hourly wage by the number of hours worked.

To determine how much the job would pay, we need to calculate the total cost of the equipment and the server's wages.

The cost of the equipment is $89 and the server's wage is $16 per hour.

To calculate the total cost of the server's wages, we need to know how long the job is. Let x be the number of hours the server works.

Therefore, the total cost of the server's wages is $16x

The job pays a base of $90 in addition to $15 per hour that the server works. So the total payment is

90 + 15x

To cancel out Wanda's expenditures, The total payment should be equal to the total cost of the equipment and the server's wages. So, we can set up the equation:

$90 + $15x = $89 + $16x

To solve for x, we can subtract $89 from both sides:

$15x = $1

Finally, we can divide both sides by 15

x= 1/15

So, the job would have to be 1/15 hours long for the payment to cancel out Wanda's expenditures.

It should be noted that this answer is not realistic, since time cannot be expressed in fractions of hours. Also, the job payment would be enough to cover the costs but not to generate any profit for Wanda.

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Answer:

y = 1/2x + 7/6

Step-by-step explanation:

The average rate of change from 2014 to 2017 is 554.

It is the average amount by which the function changed per unit throughout that time period. It is calculated using the slope of the line linking the interval's ends on the graph of the function.

Given:

f(x) = 12,000[tex](1.036)^x[/tex]

Year              2008  2011      2014    2017    2020

Population  12,000 13,343 14,836 16,498 18,344

Now, the average rate of change from 2014 to 2017

= f(2017)- f(2014)/(2017- 2014)

=  12,000[tex](1.036)^{2017[/tex] -  12,000[tex](1.036)^{2014[/tex]/3

= 1662/ 3

= 554.

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Answer:

Step-by-step explanation:

a. To find out how much floor space each box will cover, we need to multiply the volume of the box by the height of the stack of items. The volume of the box is x3 + 10x2 + 29x + 20 cubic units and the height of the stack of items is x + 5 units. So the floor space covered by each box is (x3 + 10x2 + 29x + 20) * (x + 5) square units.b. To find the maximum number of boxes Darren can place on the floor of the storage unit, we need to divide the total floor area of the storage unit by the floor space covered by each box. The total floor area of the storage unit is x* + 5x° + x7 _ 20x - 14 square units and the floor space covered by each box is (x3 + 10x2 + 29x + 20) * (x + 5) square units, then we can divide the total floor area by the floor space covered by each boxx* + 5x° + x7 _ 20x - 14 / (x3 + 10x2 + 29x + 20) * (x + 5) = maximum number of boxesc. To find out how much of the floor space is not covered by a box, we can subtract the floor space covered by the maximum number of boxes from the total floor area of the storage unit.Total floor area - floor space covered by the maximum number of boxes = remaining floor space.Please note that this is a polynomial equation and the solution will be different depending on the value of x.

a) Floor space of each box = ([tex]x^3 + 10x^2 - 29x + 20[/tex]) / (x + 5)

b) Maximum number of boxes = ([tex]x^4 + 5x^3 + x^2 - 20x - 14[/tex]) / (([tex]x^3 + 10x^2 - 29x + 20[/tex]) / (x + 5))

c) Uncovered floor space = [tex]x^4 + 5x^3 + x^2 - 20x - 14 - ((x^3 + 10x^2 - 29x + 20) / (x + 5) * Maximum number of boxes)[/tex]

We have a storage unit with a given floor area and boxes with specified volumes and heights.

We'll determine the floor space each box covers, the maximum number of boxes Darren can place on the floor, and how much floor space is left uncovered when he places the maximum number of boxes without overlap.

a) Floor space covered by each box:

To find the floor space covered by each box, we divide the volume of the box by its height. This gives us the area each box covers on the floor.

The floor space of each box can be calculated by dividing its volume by its height.

The volume of each box is given by the expression [tex]x^3 + 10x^2 - 29x + 20[/tex], and its height is x + 5. Therefore, the floor space covered by each box can be represented as:

Floor space of each box = Volume of the box / Height of the box

Floor space of each box = [tex](x^3 + 10x^2 - 29x + 20) / (x + 5)[/tex]

b) Maximum number of boxes Darren can place:

To find the maximum number of boxes Darren can place on the floor, we need to divide the total floor area by the floor space covered by each box.

The maximum number of boxes Darren can place is determined by dividing the total floor area, given by [tex]x^4 + 5x^3 + x^2 - 20x - 14[/tex], by the floor space covered by each box. We already calculated the floor space of each box as [tex](x^3 + 10x^2 - 29x + 20) / (x + 5)[/tex]. Now, we'll perform the division:

Maximum number of boxes = Total floor area / Floor space of each box

Maximum number of boxes = [tex](x^4 + 5x^3 + x^2 - 20x - 14) / ((x^3 + 10x^2 - 29x + 20) / (x + 5))[/tex]

c) Uncovered floor space with maximum number of boxes:

To find the uncovered floor space when Darren places the maximum number of boxes without overlap, we subtract the total floor area covered by the boxes from the total floor area of the storage unit.

The floor space not covered by boxes can be determined by subtracting the total floor area covered by the boxes from the total floor area of the storage unit.

We already know the total floor area of the storage unit is [tex]x^4 + 5x^3 + x^2 - 20x - 14[/tex], and we calculated the floor space covered by each box in part (a). The maximum number of boxes Darren can place was determined in part (b). Now, we'll perform the subtraction:

Uncovered floor space = Total floor area of storage unit - (Floor space of each box × Maximum number of boxes)

Uncovered floor space =[tex]x^4 + 5x^3 + x^2 - 20x - 14 - ((x^3 + 10x^2 - 29x + 20) / (x + 5) * Maximum number of boxes)[/tex]

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Complete Question

Darren is placing shipping boxes in a storage unit with a floor area of

[tex]x^4+5x^3+x^2-20x-14[/tex] square units. Each box has a volume of  [tex]x^3+10x^2-29x+20[/tex].

cubic units and can hold a stack of items with a height of x + 5 units.

a) How much floor space will each box cover?

b) What is the maximum number of boxes Darren can place on the floor of the storage unit?

c) Assume Darren places the maximum number of boxes on the floor of the storage unit, with no overlap. How much of the floor space is not covered by a box?

The point of intersection of the diagonals of the rhombus with vertices is (3, -5)

Function is a type of relation, or rule, that maps one input to specific single output.

When a relation is given in the form of ordered pairs, the first element of each ordered pair is the member of domain set and second element is the member of range set.

Given the rhombus with vertices:

(-1,2), (3,4), (5,8), and (1,6)

The domain is:

{1,3,5,1}

Rhombus is a parallelogram in which diagonals bisect each other, we can see there is repetition in domain i.e. 1 is twice in the set, the given relation is not a function.

Hence, the point of intersection is (3, -5)

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Answer:

512-358= how manyy miles left

Step-by-step explanation:

The expression that shows how many miles Graham has left to hike would be 512 - 358 = 154 miles.

Answer:

B

because if yiy multiple all those you get somewhere around 390 and then you round the number it is 390

The quantity that is measured in square units is area ( option C )

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. Take a pencil and draw a square on a piece of paper. It is a 2-D figure. The space the shape takes up on the paper is called its Area.

The area of a rectangle is given as Length × width . The area of a triangle is 1/2 bh.

Height has only one dimension. it is measured in meters. Perimeter has only one dimension, it is also measure in meters. Volume has three dimensions i.e l × b× h = m³

Therefore Out of the options only area has two dimension i.e l× w= m× m = m²

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