Mark makes a pattern that starts with 5 and uses the rule "subtract 1, and then multiply by 3. " Which expression can be used to find the third number in Markâs pattern?

A. 5â1â3â1â3

B. 3(5â1)+3(5â1)

C. 3[3(5)â1]

D. 3[3(5â1)â1]


Choose one correct answer

The height of the tree is approximately equal to the tangent of 768 radians multiplied by 50 meters, rounded to the nearest meter.

To find the height of the tree, we can use trigonometry. Let's assume that the height of the tree is represented by the variable "h."

In a right triangle formed by the forester, the base of the tree, and the top of the tree, the tangent of the angle is equal to the opposite side (height of the tree, h) divided by the adjacent side (distance from the base of the tree, 50 m).

Using the tangent function, we can write:

tan(angle) = [tex]\frac{h }{ 50}[/tex]

We can rearrange this equation to solve for h:

h = tan(angle) * 50

Now we can substitute the given angle into the equation and calculate the height:

h = tan(768) * 50

Calculating this in degrees might result in an error because the tangent function expects the angle to be in radians.

Therefore, we need to convert the angle to radians before evaluating the tangent.

There are 180 degrees in pi radians, so we can convert the angle as follows:

angle_radians = [tex]\frac{768 \* \pi }{ 180}[/tex]

Substituting this value into the equation:

h = tan(angle_radians) * 50

Now we can calculate the height using a calculator:

h ≈ [tex]tan(\frac{768 \pi }{ 180}) 50[/tex]

After evaluating this expression, we get the height of the tree. Rounding the answer to the nearest meter will give us the final result.

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The quadratic function in vertex form is y = (8/9)(x - 5)^2 + 7

The vertex form of a quadratic function is given by:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

In this case, we are given that the vertex is (5, 7), so we can write:

y = a(x - 5)^2 + 7

To find the value of a, we can use one of the points on the parabola.

Let's use the point (2, 15):

15 = a(2 - 5)^2 + 7

8 = 9a

a = 8/9

Substituting this value of a into the equation above, we get:

y = (8/9)(x - 5)^2 + 7

Therefore, the quadratic function in vertex form is y = (8/9)(x - 5)^2 + 7

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The statement "the series converges to 31.25" is true about the given series.

Given series is 25 + 5 + 1 + ....

We can clearly see that given series is infinite geometric series.

First term is a=25

common ratio is r = 5/25

= 1/5

We know that the formula of sum of an infinite geometric series is

S = a / (1 - r)

S = 25 / (1 - 1/5)

S = 25/(4/5)

S = (25*5)/4

S = 125/4

S = 31.5

Therefore, the sum of the infinite geometric series is 31.25.

Since, the sum of the series is a finite number, we can say that the series converges.

Therefore, the statement "the series converges to 31.25" is true about the given series.

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The area of the 125 degree sector for a circle with a radius of 12 m is approximately 158 square meters.

To find the area of a 125 degree sector of a circle with a radius of 12 m, we need to use the formula for the area of a sector:

Area of sector = (θ/360) x πr², where θ is the central angle of the sector, r is the radius of the circle, and π is a constant equal to approximately 3.14.

Substituting the given values, we get: Area of sector = (125/360) x π x 12² = (0.3472) x π x 144 = 158.03

Rounding to the nearest whole number, we get the area of the sector as 158 square meters. Therefore, the area of the 125 degree sector for a circle with a radius of 12 m is approximately 158 square meters.

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The mean absolute deviation of the given data set is approximately 2.16.

To find the mean absolute deviation (MAD), we first need to calculate the mean of the data set:

Mean = (10 + 10 + 9 + 8 + 10 + 5 + 6 + 4 + 8 + 4) / 10 = 7.4

Next, we calculate the absolute deviation of each data point from the mean:

|10 - 7.4| = 2.6

|10 - 7.4| = 2.6

|9 - 7.4| = 1.6

|8 - 7.4| = 0.6

|10 - 7.4| = 2.6

|5 - 7.4| = 2.4

|6 - 7.4| = 1.4

|4 - 7.4| = 3.4

|8 - 7.4| = 0.6

|4 - 7.4| = 3.4

Then, we find the average of these absolute deviations:

MAD = (2.6 + 2.6 + 1.6 + 0.6 + 2.6 + 2.4 + 1.4 + 3.4 + 0.6 + 3.4) / 10 ≈ 2.16

Therefore, the mean absolute deviation of the given data set is approximately 2.16.

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The police dog spends 1/6 of its workday at the police station.

To find the fraction of the police dog's workday spent at the police station, we need to add up the fractions of time spent in each location and subtract them from 1, since the dog spends the rest of the day at the police station.

Fraction of time spent in police car =  [tex]1/3[/tex]

Fraction of time spent in public = [tex]1/2[/tex]

To add these fractions, we need to find a common denominator:

[tex]1/3 = 2/6\\1/2 = 3/6[/tex]

So, the fraction of the dog's day spent at the police station is:

[tex]1 - (2/6 + 3/6) = 1 - 5/6[/tex]

                        = [tex]1/6[/tex]

Therefore, the police dog spends 1/6 of its workday at the police station.

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If two fair dice are rolled, the probability that the total showing is either even or less than five is 17/36. The correct answer is B.

There are 36 possible outcomes when rolling two dice, since each die has 6 possible outcomes.

To find the probability that the total showing is either even or less than five, we can first find the probability that the total showing is even, and then add to that the probability that the total showing is less than five, excluding the outcomes where the total showing is even.

To find the probability that the total showing is even, we can consider the following possibilities:

both dice show even numbers (probability 1/4)

both dice show odd numbers (probability 1/4)

So the probability of rolling an even total is 1/4 + 1/4 = 1/2.

To find the probability that the total showing is less than five, excluding the outcomes where the total showing is even, we can consider the following possibilities:

So the probability of rolling a total less than five, excluding the outcomes where the total showing is even, is 1/36 + 1/18 + 1/18 + 1/12 + 1/12 + 1/9 + 1/9 + 1/9 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 11/36.

Therefore, the probability that the total showing is either even or less than five is 1/2 + 11/36 = 17/36.

So the correct answer is (b) 17/36.

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The system of equations can be used to determine the gallons of juice in container A and the gallon of juice in container B is  A + B = 28 and A + 6B = 88

Total amount of juice in both the vats is 28

Let the first container = A

second container = B

A + B = 28

Orange juice in A = 1/8 of A

Orange juice in B = 3/4 of B

Total orange juice in container A and B = 11

(1/8)A  + (3/4)B = 11

A + 6B = 88

Hence the two equations will be A + B = 28 and A + 6B = 88

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The component of the weight parallel to the inclined plane is approximately 34.72 pounds, and the component perpendicular to the inclined plane is approximately 196.96 pounds.

To analyze the situation, we need to break down the weight of the object into its components parallel and perpendicular to the inclined plane.

Given:

Weight of the object = 200 pounds

Angle of the inclined plane with the horizontal = 10 degrees

First, we find the component of the weight parallel to the inclined plane. This component can be determined using trigonometry:

Component parallel to the inclined plane = Weight * sin(angle)

Component parallel to the inclined plane = 200 pounds * sin(10 degrees)

Component parallel to the inclined plane ≈ 200 pounds * 0.1736

Component parallel to the inclined plane ≈ 34.72 pounds

Next, we find the component of the weight perpendicular to the inclined plane:

Component perpendicular to the inclined plane = Weight * cos(angle)

Component perpendicular to the inclined plane = 200 pounds * cos(10 degrees)

Component perpendicular to the inclined plane ≈ 200 pounds * 0.9848

Component perpendicular to the inclined plane ≈ 196.96 pounds

Therefore, the component of the weight parallel to the inclined plane  and the component perpendicular to the inclined plane is approximately 34.72 pounds and 196.96 pounds respectively.

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The function for the population of Madison t years from now is:[tex]p(t) = 25000 (1.12)^{t}[/tex]

To write a function that gives the population P(t) in Madison t years from now, considering the town has an initial population of 25,000 and an annual increase factor of 1.12, you can use the formula:

[tex]p(t) = P_{0}  (1 + r)^{t}[/tex]

Where:
- P(t) is the population at time t
- P_0 is the initial population (25,000)
- r is the annual increase factor (1.12 - 1 = 0.12)
- t is the number of years

So, the function for the population of Madison t years from now is:
[tex]p(t) = 25000 (1.12)^{t}[/tex]

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The volume of a rectangular prism is calculated by multiplying the length, width, and height. In this case, we are given the length and width, but not the height. So, we cannot calculate the exact volume without knowing the height.

To find the volume of a rectangular prism, we need to multiply its length, width, and height.

Given:

Length = 2 units

Width = 7 1/3 units

To calculate the volume, we first need to convert the mixed fraction to an improper fraction.

7 1/3 = (7 * 3 + 1) / 3 = 22/3 units.

Now, we can calculate the volume:

Volume = Length * Width * Height

= 2 units * (22/3 units) * Height.

Since the height is not provided, we cannot calculate the exact volume without that information. However, if you provide the height of the rectangular prism, I can help you find the volume by substituting the value into the formula.

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To evaluate the integral of (3x+6)/(2x²-3)dx, we can use partial fraction decomposition:

(3x+6)/(2x²-3) = A/(x-√(3)/2) + B/(x+√(3)/2)

Multiplying both sides by the denominator and simplifying, we get:

3x+6 = A(x+√(3)/2) + B(x-√(3)/2)

Setting x = √(3)/2, we get:

3√(3)/2 + 6 = B(√(3)/2-√(3)/2) = 0

So B = -2√(3). Setting x = -√(3)/2, we get:

-3√(3)/2 + 6 = A(-√(3)/2+√(3)/2) = 0

So A = 2√(3). Therefore, we have:

(3x+6)/(2x^2-3) = 2√(3)/(x-√(3)/2) - 2√(3)/(x+√(3)/2)

Integrating each term, we get:

∫(3x+6)/(2x²-3)dx = 2√(3)ln|x-√(3)/2| - 2√(3)ln|x+√(3)/2| + C

where C is the arbitrary constant.
To evaluate the integral of the function 3x + 6 with respect to x, we will use the integral symbol and find the antiderivative:

∫(3x + 6) dx

To find the antiderivative, we will apply the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the constant rule, which states that the integral of a constant is the constant times the variable:

(3 * (x^(1+1))/(1+1)) + (6 * x) + C

Simplifying the expression:

(3x²)/2 + 6x + C

Here, C is the arbitrary constant. So, the evaluated integral of 3x + 6 is:

(3x²)/2 + 6x + C

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The work done to lift a 90 kg weight by 75 cm is approximately 661.5 joules.

Para calcular el trabajo realizado al elevar una pesa de 90 kg a una altura de 75 cm sobre el piso, necesitamos conocer la fuerza necesaria para levantar la pesa y la distancia que recorre la pesa para llegar a su altura máxima.

El trabajo se define como la energía necesaria para mover un objeto a través de una distancia determinada, y se calcula como el producto de la fuerza aplicada y la distancia recorrida.

En este caso, la fuerza necesaria para levantar la pesa es igual a su peso, que se calcula como su masa multiplicada por la aceleración debido a la gravedad (9.8 m/s²):

peso = masa x gravedad

    = 90 kg x 9.8 m/s²

    = 882 N

La distancia que recorre la pesa para elevarse 75 cm (o 0.75 m) es:

distancia = 0.75 m

Por lo tanto, el trabajo realizado al elevar la pesa es:

trabajo = fuerza x distancia

       = 882 N x 0.75 m

       = 661.5 J

Por lo tanto, se requiere un trabajo de 661.5 J para elevar una pesa de 90 kg a una altura de 75 cm sobre el piso.

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Based on the information provided, the probability that Ben wins the board game is 30%.

To calculate the probability of Ben winning the board game, let's start by checking the information provided:

Probability for Justin to win: 20% or 0.2

Probability for Cam to win: 50% or 0.5

Now, the total probability is always equivalent to 100% or 0.1. Based on this, let's calculate now the probability that Ben wins the game.

1 - (0.2 + 0.5)  

1 - 0.7 = 0.3

Note: Here is the complete question:

Justin, Cam, and Ben are playing a board game where exactly one player will win. Ben estimates that Justin has a %20 percent chance of winning each game and that Cam has a %50 percent chance of winning each game. What is the probability that Ben will win the board game?

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The average walking rate for Path A is 2.5 mph, path B's is 2 mph and path C's is 3 mph.

Average walking rate = distance/time

for path A = 0.5/12 = 0.0416 mpm

To convert miles per meter into miles per hour it we will multiply the equation by 60 because 1 hour has 60 min

so, for path A = 0.0416 × 60 = 2.5 mph

Similarly , for path B = (1.5/45) × 60 = 2 mph

For path C = (0.75/15)×60 = 3 mph

Table to show the average walking rate

Path     Distance(mi)     Time(min)      Walking rate (mph)

  A              0.5                   12                    2.5

  B              1.5                    45                    2

  C              0.75                 15                     3

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The artist used a total of $0.33 worth of paint for the project.

The artist purchased 2 liters of paint, which is equivalent to 2,000 milliliters of paint. This amount of paint was used to complete a project, and after the project was finished, there were 350 milliliters of paint left over.

To determine how much paint was used for the project, we subtract the amount of leftover paint from the total amount of paint purchased, which gives us 2,000 - 350 = 1,650 milliliters of paint used for the project.

The cost of the paint is 2 cents per milliliter, which is equivalent to $0.02/100 milliliters or $0.0002 per milliliter. To determine the cost of the paint used for the project, we multiply the amount of paint used by the cost per milliliter.

Therefore, the cost of 1,650 milliliters of paint used for the project can be calculated by multiplying 1,650 milliliters by $0.0002 per milliliter, which gives us $0.33.

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The ratios that have a unit rate of 3 include the following:

A. 15/2 cups: 2 1/2 cups

C. 2 cups: 2/3 cups

F. 2 1/2 cups: 5/6 cups

In Mathematics, the unit rate is sometimes referred to as unit ratio and it can be defined as the quantity of material that is equivalent to a single unit of product or quantity.

15/2 cups : 5/2 cups

15/2 ÷ 5/2 : 5/2 ÷ 5/2

15/2 × 2/5  : 1

3 : 1  (True)

1 cup: 1/4 cups

1 × 4 : 1/4 × 1

4 : 1   (False)

2/3 cups: 1 cup

2/3 × 3/2  : 1 × 3/2

1 : 3/2   (False)

3 3/4 cups: 2 cups

(4 × 3 + 3)/4 : 2

15/4 : 2

15/8  : 2/2

15/8 : 1    (False).

2 cups: 2/3 cups

2 × 3/2 : 2/3 × 3/2

3  : 1   (True).

2 1/2 cups: 5/6 cups

(2 × 2 + 1)/2 : 5/6

5/2  : 5/6

5/2 × 6/5 : 5/6 × 6/5

3 : 1  (True).

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Complete Question:

Which ratios have a unit rate of 3? Choose all that apply.

15/2 cups: 2 1/2 cups

1 cup: 1/4 cups

2/3 cups: 1 cup

3 3/4 cups: 2 cups

2 cups: 2/3 cups

2 1/2 cups: 5/6 cups

The values of U and d for an arithmetic sequence with U20 = 100 and U25 = 115 is U =  43 and d = 3.

The formula for the nth term of an arithmetic sequence: Un = U1 + (n-1)d

We know that U20 = 100 and U25 = 115, so we can set up two equations using the formula above:

U20 = U1 + 19d = 100    

U25 = U1 + 24d = 115

We now have two equations with two variables (U1 and d) that we can solve for.

First, we'll isolate U1 in the first equation:

U1 = 100 - 19d

Then we'll substitute this expression for U1 into the second equation and solve for d:

100 - 19d + 24d = 115

5d = 15

d = 3

Substitute d = 3 in the equation, U1 = 100 - 19d

So, U1 = 100 - 19(3) = 43.

Therefore, the values of U and d for the arithmetic sequence are U= 43 and d = 3.

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p(x) = x + 5, and k = 2. The expression x^2 + 7x + 12 / (x + 2) can be written in the form p(x) + k / (x + 2) as:

x + 5 + 2 / (x + 2)

To express the given expression x^2 + 7x + 12 / (x + 2) in the form p(x) + k / (x + 2), we will perform polynomial division.

1. Divide the numerator (x^2 + 7x + 12) by the denominator (x + 2):

(x^2 + 7x + 12) ÷ (x + 2)

2. Perform long division:

       x + 5
       ________________
x + 2 | x^2 + 7x + 12
       - (x^2 + 2x)
       ________________
             5x + 12
             - (5x + 10)
       ________________
                 2

3. Write the result:

p(x) + k / (x + 2) = x + 5 + 2 / (x + 2)

So, p(x) = x + 5, and k = 2. The expression x^2 + 7x + 12 / (x + 2) can be written in the form p(x) + k / (x + 2) as:

x + 5 + 2 / (x + 2)

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Answer:

(-1/5)x + 8 = -x + 4

(4/5)x + 8 = 4

(4/5)x = -4

x = -5, so y = 9

C. There is one unique solution (-5, 9).

The height of the replica souvenir is approximately 7.55 inches.

To find the height of the replica souvenir of the Statue of Liberty, we'll use the given scale factor of 1 inch = 20 feet. The original height of the statue is 151 feet. Divide the original height by the scale factor:

151 ft / 20 ft/in = 7.55 inches

The height of the replica souvenir is approximately 7.55 inches.

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The total number of cupcakes charity can make in 8 hours is 384

The total number of cupcakes she can make in 45 minutes is 36

Cupcakes she can make in 1 minute = 36/45

Cupcakes she can make in 1 minute = 0.8

Cupcakes she can make in 8 hours

We will convert hours into minutes

1 hour = 60 min

8 hour = 8 × 60 min

8 hour = 480 min

Cupcakes she can make in 8 hours that is 480 min = 480 × 0.8

Cupcakes she can make in 8 hours = 384

Total number of cupcakes she can make is 384

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None of the given options have a focus at (-6, 12) and directrix of x = -12,so none of the option is correct.

To find the equation with a focus at (-6, 12) and directrix of x = -12, we can use the general equation for a parabola with a vertical axis of symmetry:

(y - k)^2 = 4p(x - h)

where (h, k) is the focus and x = h - p is the directrix.

Given the focus (-6, 12) and directrix x = -12, we can determine the value of p:

p = h - (-12) = -6 - (-12) = 6

Now, we can plug in the values of h, k, and p into the equation:

(y - 12)^2 = 4(6)(x + 6)

Simplify the equation:

(y - 12)^2 = 24(x + 6)

Now, let's compare this equation to the given options:

1. (y - 12)^2 = 1/12 (x + 9)
2. (y - 12)^2 = -1/12 (x + 9)
3. (y - 12)^2 = 12 (x + 9)
4. (y - 12)^2 = -12 (x + 9)

None of the given options match the equation we found. Therefore, none of the given options have a focus at (-6, 12) and directrix of x = -12

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The mean is 35 minutes and the median is 34 minutes.

let's first calculate the mean, median, and mode for Sarah's exercise times: 34, 30, 45, 30, 40, 38, and 28 minutes.

Step 1: Calculate the mean
Add up all the values and divide by the total number of values:
(34 + 30 + 45 + 30 + 40 + 38 + 28) / 7 = 245 / 7 = 35 minutes (rounded)

Step 2: Calculate the median
Arrange the values in ascending order: 28, 30, 30, 34, 38, 40, 45
There are 7 values, so the median is the middle value: 34 minutes

Step 3: Calculate the mode
Determine the value(s) that occur most often: 30 minutes (occurs twice)

Now, should Sarah use the mean or the median to show she exercises for large amounts of time each day? The mean is 35 minutes and the median is 34 minutes. Both values are close and represent the central tendency of the data. However, since the mean is slightly higher than the median, Sarah should use the mean (35 minutes) to show she exercises for a larger amount of time each day.

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The area of the polygon composed of rectangles and triangle is 52 in²

Area is the amount of space occupied by a two dimensional shape or object.

For the first right triangle:

base = 2 in, height = 6 in

Area of first right triangle = 1/2 * base * height = 0.5 * 2 in * 6 in = 6 in²

For the second right triangle:

base = 2 in, height = 6 in

Area of second right triangle = 1/2 * base * height = 0.5 * 2 in * 6 in = 6 in²

For the triangle:

base = (2 + 4 + 2) = 8 in, height = 4 in

Area of triangle = 1/2 * base * height = 0.5 * 8 in * 4 in = 16 in²

For the rectangle:

length = 4 in, width = 6 in

Area of rectangle = length * width = 4 in * 6 in = 24 in²

Area of polygon = 6 + 6 + 16 + 24 = 52 in²

The area of the polygon is 52 in²

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Answer:

(3x+4) ÷ (x+6)

Step-by-step explanation:

3x²-14x-24 = (3x+4) (x-6)

x²-36 = (x+6) (x-6)

= (3x+4) (x-6) ÷ (x+6) (x-6)

Eliminate the (x-6)

= (3x+4) ÷ (x+6)

Therefore, the linearization predicts that f(2.9, 4.1) is approximately 4.142.

To find the linearization of the function f(x, y) = √x^2 + y^2 at the point (3, 4), we need to find the partial derivatives of f with respect to x and y, evaluate them at (3, 4), and use them to write the equation of the tangent plane to the surface at that point.

First, we have:

∂f/∂x = x/√(x^2 + y^2)

∂f/∂y = y/√(x^2 + y^2)

Evaluating these at (3, 4), we get:

∂f/∂x(3, 4) = 3/5

∂f/∂y(3, 4) = 4/5

So the equation of the tangent plane to the surface at (3, 4) is:

z - f(3, 4) = (∂f/∂x(3, 4))(x - 3) + (∂f/∂y(3, 4))(y - 4)

Plugging in f(3, 4) = 5 and the partial derivatives, we get:

z - 5 = (3/5)(x - 3) + (4/5)(y - 4)

Simplifying, we get:

z = (3/5)x + (4/5)y - 1

This is the linearization of f(x, y) = √x^2 + y^2 at the point (3, 4).

To approximate f(2.9, 4.1), we plug in x = 2.9 and y = 4.1 into the linearization:

z = (3/5)(2.9) + (4/5)(4.1) - 1

z ≈ 4.142

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To find the volume and surface area of the composite figure, we need to first find the individual volumes and surface and the right cone, and then add them together.

The volume of a hemisphere with radius r is (2/3)πr^3, and the surface area is 2πr^2.

The volume of a right cone with radius r, height h, and slant s is (1/3)πr^2h, and the surface area is πr^2 + πrs.

In this case, the radius r and slant s are both 6 cm, and the height h of the cone can be found using the Pythagorean theorem: h = √(s^2 - r^2) = √(8^2 - 6^2) = √28 ≈ 5.29 cm.

So, the volume of the hemisphere is (2/3)π(6 cm)^3 = 72π/3 = 24π cubic cm, and the surface area is 2π(6 cm)^2 = 72π square cm.

The volume of the right cone is (1/3)π(6 cm)^2(5.29 cm) = 62.83π/3 ≈ 20.94π cubic cm, and the surface area is π(6 cm)^2 + π(6 cm)(8 cm) = 36π + 48π = 84π square cm.

Therefore, the total volume of the composite figure is 24π + 20.94π = 44.94π cubic cm, and the total surface area is 72π + 84π = 156π square cm.

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