Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = 8x − x^2, y = 12; about x = 2

Answer:

a) If you invest $100 per month starting at age 18, you will have $1,567,088 when you retire. If you wait to start investing at age 35, you will have $278,513 when you retire. Therefore, the difference between the two amounts is:

$1,567,088 - $278,513 = $1,288,575

So, if you invest $100 per month starting at age 18, you will have $1,288,575 more when you retire than if you wait to start investing at age 35.

b) If you wait until age 45 to begin investing for your retirement, you will have $95,317 when you retire. If you had begun investing at age 25, you would have had $788,430 when you retire. Therefore, the difference between the two amounts is:

$788,430 - $95,317 = $693,113

So, if you wait until age 45 to begin investing for your retirement, you will have $693,113 less than if you had begun investing at age 25.

Answer:

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The time is:

The rate is given - 2.3 ft³/min.

Find the volume of cylinder:

Find the required time:

The correct answer is C. The median of Set A is smaller than the median of Set B.

To find the median and mean of a data set, we need to calculate the average value of the data.

For Set A:

The median is the middle value when the data is arranged in numerical order. First, we need to arrange the data set in order:

1, 2, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8

There are 13 values in this data set, so the median is the average of the middle two values:

Median = (3 + 4) / 2 = 3.5

To find the mean, we need to add up all the values and divide by the total number of values:

Mean = (1 + 2 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 6 + 7 + 7 + 8) / 17 = 4

For Set B:

The median is the middle value when the data is arranged in numerical order. First, we need to arrange the data set in order:

4, 6, 50

There are 3 values in this data set, so the median is the middle value:

Median = 6

To find the mean, we need to add up all the values and divide by the total number of values:

Mean = (4 + 6 + 50) / 3 = 20

Therefore, the correct answer is C. The median of Set A is smaller than the median of Set B.

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Answer:

x = [tex]\frac{5}{3}[/tex]

Step-by-step explanation:

4x + 2 = 7X - 3  Subtract 4x from both sides

4x - 4x + 2 = 7x - 4x - 3

2 = 3x - 3  Add 3 to both sides

2 + 3 = 3x -3 + 3

5 =3x  Divide both sides by 3

[tex]\frac{5}{3}[/tex]

Check

4x + 2 = 7x -3  Substitute 5 for x

4([tex]\frac{5}{3}[/tex]) + 2 = 7([tex]\frac{5}{3}[/tex]) -3

[tex]\frac{20}{3}[/tex] + 2 = [tex]\frac{35}{3}[/tex] - 3

[tex]\frac{20}{3}[/tex] + [tex]\frac{6}{3}[/tex] = [tex]\frac{35}{3}[/tex] - [tex]\frac{9}{3}[/tex]

[tex]\frac{26}{3}[/tex] = [tex]\frac{26}{3}[/tex] Checks

Helping in the name of Jesus.

If we write a quadratic in vertex form:

[tex]y=a(x-h)^2+k[/tex]

Then:

[tex]\bold{a}[/tex]   [tex]\longrightarrow[/tex]    is the coefficient of [tex]x^2[/tex]

[tex]\bold{h}[/tex]   [tex]\longrightarrow[/tex]    is the axis of symmetry.

[tex]\bold{k}[/tex]   [tex]\longrightarrow[/tex]    is the max/min value of the function.

Also:

If [tex]a > 0[/tex] then the parabola will be of the form [tex]\cup[/tex] and will have a minimum value.

[tex]a < 0[/tex] then the parabola will be of the form [tex]\cap[/tex] and will have a minimum value.

For the given functions:

[tex]a < 0[/tex]

[tex]f(x)=-(x-1)^2+5[/tex]        this has a maximum value of [tex]\bold{5}[/tex]

[tex]a > 0[/tex]

[tex]f(x)=(x+2)^2-3[/tex]           this has a minimum value of [tex]\bold{-3}[/tex]

The sun is elevated at a about 36.7 degree angle.

We can utilize the tangent function, which connects the opposite and adjacent sides of a right triangle, to get the angle of elevation of the sun:

tan(theta) = adjacent or opposite

The other side in this instance is the basketball backboard's 12.5-foot height. The distance from one side to the other, or 17.33 feet, is the length of the backboard's shadow. We thus have:

theta (theta) = 12.5 / 17.33

We can use the inverse tangent (or arctan) of both sides to find theta:

arctan(12.5 / 17.33) = theta

Calculating the answer, we obtain:

36.7 degrees are theta.

As a result, the sun is elevated at a about 36.7 degree angle.

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The missing value is 1.9096

According to SOH CAH TOA identity:

[tex]\text{sin} \ \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}[/tex]

[tex]\text{sin} \ \theta = \dfrac{\text{O}}{\text{H}}[/tex]

Given the following

Substitute to have:

[tex]0.1736 = \dfrac{\text{O}}{11}[/tex]

[tex]\text{O} = 11 \times 0.1736[/tex]

[tex]\text{O} = 11 \times 1.9096[/tex]

Hence the missing value is 1.9096

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The probability that the first die lands on an even number and the second die is less than 3 is 1/6.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

There are six possible outcomes for each die roll, so the total number of outcomes for rolling two dice is 6 x 6 = 36.

There are three even numbers on a die: 2, 4, and 6. The probability of rolling an even number on the first die is 3/6 or 1/2.

There are two numbers less than 3 on a die: 1 and 2. The probability of rolling a number less than 3 on the second die is 2/6 or 1/3.

To find the probability of both events happening together, we multiply their probabilities:

P(even number on first die and less than 3 on second die) = P(even on first die) x P(less than 3 on second die)

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

= 1/6

Therefore, the probability that the first die lands on an even number and the second die is less than 3 is 1/6.

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Answer:

Part = 24 students

Whole = 60 students

Percent = 40%

Step-by-step explanation:

When dealing with percents, we can use the formula

P%x = y, where x is the whole and y is the part.

Since the 24 students is part of the entire group of 6th graders (whole), our y value is 24.

Now, we can solve for x to find how many students are in the 6th grade.

[tex]0.40x=24\\x=60[/tex]

Option D is correct, y≥3/4x+10  and y≥-1/2x-3  are the system of inequalities of the given graph

Let us find the equations of the lines

y=3/4x+10

and y=-1/2x-3

A relationship between two expressions or values that are not equal to each other is called 'inequality.

As we observe the two lines, the lines are solid

y=-1/2x-3

y=3/4x+10 the shading is outside the origin

y≥3/4x+10

and  y=-1/2x-3 the shading is outside the origin

y≥-1/2x-3

Hence, option D is correct, y≥3/4x+10  and y≥-1/2x-3  are the system of inequalities of the given graph

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The estimated population after one year is 160 and the estimated population after two years is 256.

we can multiply the initial population by 1.60 (100% + 60%) to account for the 60% growth rate:

Population after one year = 100 x 1.60 = 160

Since the sanctuary can sustain up to 800 deer, the actual population after one year would be the minimum of 160 and 800, which is 160.

To estimate the population after two years, we can repeat the process using the population after one year as the new initial population:

Population after two years = 160 x 1.60 = 256

Again, since the sanctuary can sustain up to 800 deer, the actual population after two years would be the minimum of 256 and 800, which is 256.

Therefore, the estimated population after one year is 160 and the estimated population after two years is 256.

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[tex](\stackrel{x_1}{40}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{20}~,~\stackrel{y_2}{-2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-2}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{20}-\underset{x_1}{40}}} \implies \cfrac{ -2 }{ -20 } \implies \cfrac{ 1 }{ 10 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{0}=\stackrel{m}{ \cfrac{ 1 }{ 10 }}(x-\stackrel{x_1}{40})\implies {\Large \begin{array}{llll} y=\cfrac{1}{10}x+4 \end{array}}[/tex]

We can rewrite the expression as ln(x-6) - 11*ln(x)/8, which can then be evaluated by using a calculator.

Logarithmic functions can be used to describe relationships between two variables that involve exponential rates of change.

The logarithmic expression 8ln (x-6) - 11 In x can be written as a single logarithm whose coefficient is 1 by using the properties of logarithms.

First, we rearrange the terms to put the ln terms together: 8ln (x-6) - 11lnx = ln(x-6) - 11lnx/8.

We can then use the property of logarithms that states that

log(ab) = log a + log b.

This allows us to rewrite the expression as ln(x-6/x¹¹/8). Finally, we can simplify the expression by using the property of logarithms that states that log(a/b) = loga - logb. This allows us to rewrite the expression as ln(x-6) - ln(x¹¹/8). The final expression is ln(x-6) - ln[(x¹¹/8)], which is a single logarithm whose coefficient is 1.

If we were to evaluate the expression, we could use the property of logarithms that states that logaᵇ = b*log a.

We can rewrite the expression as ln(x-6) - 11*ln(x)/8, which can then be evaluated by using a calculator.

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To represent vector a + b using the head-tail method, we can draw vector a + b with its tail at the origin (0,0) and its head at the point (3,-5).

The head-tail method is a graphical technique used to add or subtract two vectors by placing the tail of one vector at the head of the other vector.

So, to represent vector a + b using the head-tail method, we start by drawing vector a with its tail at the origin (0,0) and its head at the point (4,-3). Then, we draw vector b with its tail at the head of vector a, which is the point (4,-3), and its head at the point (-1,-2). The resulting vector from the tail of a to the head of b is vector a + b.

To find the coordinates that can be used to draw vector a + b, we add the corresponding components of vectors a and b. So, (a + b) = (4 + (-1), -3 + (-2)) = (3,-5). Therefore, we can draw vector a + b with its tail at the origin (0,0) and its head at the point (3,-5).

See attached graph.

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Answer:

C. (310 − 10) ÷ 15 = s

Step-by-step explanation:

The correct answer is C. (310 − 10) ÷ 15 = s. This is because we can use the following steps to find the equation:

First, we need to find how many pieces of candy were given out in total. Since there are 10 pieces left over in the bag, and the bag originally contained 310 pieces, we can subtract 10 from 310 to get 300 pieces given out.

Next, we need to find how many pieces of candy each student received. Since 15 pieces of candy were given to s students, we can divide 300 by 15 to get the number of students. This gives us 20 students.

Finally, we need to write an equation that relates the number of students (s) to the number of pieces of candy given out (300) and the number of pieces of candy per student (15). We can do this by using the inverse operations of subtraction and division. We can multiply both sides by 15 and then add 10 to both sides to get the equation:

(310 − 10) ÷ 15 = s

This equation can be used to find the number of students if we know the number of pieces of candy in the bag, the number of pieces left over, and the number of pieces per student.

But wait, there's more! If you act now, you can also get a bonus question for free! Here it is:

If s students each received 15 pieces of candy, and there are 10 pieces left over in the bag, how many cavities will they have after eating all the candy?

There are a few possible ways to approach this question, but here is one:

- First, we need to find how many pieces of candy each student ate. Since they received 15 pieces each, and there are no leftovers, we can assume they ate all of them. So each student ate 15 pieces.

- Next, we need to find how many cavities each piece of candy causes. This may vary depending on the type and quality of the candy, but let's assume that each piece causes 0.1 cavities on average. This means that for every 10 pieces of candy, one cavity is formed.

- Finally, we need to multiply the number of pieces each student ate by the number of cavities each piece causes, and then add up all the results for all the students. This will give us the total number of cavities for the whole group. We can use this formula:

c = (15 × 0.1) × s

where c is the number of cavities and s is the number of students.

Using this formula, we can plug in the value of s that we found

c = 1.5 × 20

c = 30

✧☆*: .｡. That's all folks, have fun with math! (✧ω✧) .｡.:*☆✧

Answer:

20

Step-by-step explanation:

Avg length of a pencil= 19cm

length of cylinder=380cm

pencils produced=?

num of pencils produced= length of cylinder ÷ avg length of a pencil

= 380÷19

=20

therefore, number of pencils produced are 20

RECHECK

since avg length of pencil × num of pencils produced must be equal to the length of the given cylinder

19×20=380cm

therefore, our answer is correct

mark me as brainiest with 50 points would mean alot

The expression/ equation equivalent to - 3/5 (7 - 3 1/3) is: (- 3/5) (7) + (- 3/5)(- 3 1/3). Option D.

We're given the expression: - 3/5 (7 - 3 1/3)

To find the equivalent expression, first, we should convert the mixed number (3 1/3) to an improper fraction. To do this, multiply the whole number (3) by the denominator (3) and add the numerator (1). This gives us:

3 x 3 + 1 = 9 + 1 = 10

So, 3 1/3 as an improper fraction is 10/3. Now, the expression becomes:

3/5 (7 - 10/3)

Now we can use the distributive property, which states that a(b + c) = ab + ac, to simplify the expression further:

3/5 x 7 + (- 3/5) x (- 10/3)

This expression corresponds to option d:

d. (- 3/5) (7) + (- 3/5)(- 3 1/3) when converted to an improper fraction as (- 3/5)(- 10/3)

The above answer is based on the question below as seen in the picture;

What expression is equivalent to - 3/5 (7 - 3 1/3)?

a) (- 3/5) (-7) + (- 3/5)(-3 1/3)

b. - (- 3/5) (7) - (- 3/5)(3 1/3)

c. - (- 3/5) (7) - (- 3/5)(- 3 1/3)

d. (- 3/5) (7) + (- 3/5)(- 3 1/3)

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We can determine that Gavon sleeps for 3360 minutes this week using the laws of mensuration.

A measurement action is what the word "mensuration" alludes to. Our universe has three dimensions. Both elementary school and high school mathematics depend heavily on the idea of measurement. Additionally, measurement directly affects how we live. When measuring objects, 2D and 3D measurements are practised. Both traditional and non-conventional measurement techniques can be used to determine the size or quantity of something.

According to the query,

Each night, Gavon slept for eight hours.

Gavon slept for eight hours every night for seven days in a row.

Gavon now sleeps for eight hours per night, or eight times sixty.

= 480 minutes.

Gavon now sleeps seven nights a week, thus

7 × 480

= 3360 minutes.

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The complete question is,

Gavon slept 8 hours all 7 nights this week. How many minutes did Gavon sleep this week? Note: 1 hour = 60 minutes​

Answer:

the mass of a dime is 2.268 grams.

Step-by-step explanation:

Given that the mass of a quarter is 5.67 grams, the mass of a dime can be found by multiplying 0.4 to the mass of a quarter.

So,

Mass of a dime = 0.4 × mass of a quarter

= 0.4 × 5.67 grams

= 2.268 grams

Therefore, the mass of a dime is 2.268 grams.

Answer:

Step-by-step explanation:

You are helping your little brother pick up his toy blocks.  The blocks are cubes and each side measures 5 in.  The toy chest that you have to put them in measures 12 1/3  in by 10 and 3/4  in by 12in.  Your brother has 10 blocks to put away, will they all fit in the toy chest?

Since you have 10 blocks to put away, what is the volume of all 10 blocks

Answer:

85

Step-by-step explanation:

check the lower answer

The only solution for log₂(x-1)+log₂(x+5) is x = 3 using logarithms.

Logarithmic properties are rules that allow us to simplify and manipulate logarithmic expressions. They are based on the properties of exponents and are essential in solving logarithmic equations and evaluating logarithmic expressions.

Using the logarithmic property that log a + log b = log ab, we can simplify the left-hand side of the equation to:

log₂((x-1)(x+5)) = 4

2⁴ = (x-1)(x+5)

Simplifying:

16 = x² + 4x - 5

Rearranging:

x² + 4x - 21 = 0

Factorizing:

(x + 7)(x - 3) = 0

Therefore, either x + 7 = 0 or x - 3 = 0. Solving for x in each case:

x + 7 = 0 → x = -7

x - 3 = 0 → x = 3

However, we need to check whether these solutions are valid in the original equation. We cannot take the logarithm of a negative number, so x = -7 is not a valid solution.

Therefore, the only solution is x = 3.

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The total distance to Olivia's house is 50 miles and x = 5

Given data ,

Let the total distance to Olivia's house be = 50 miles

Now , the distance from Anna and bakery = ( 7x + 8 ) miles

The remaining distance = ( 2x - 3 ) miles

So , on simplifying the equation , we get

( 7x + 8 ) + ( 2x - 3 ) = 50

9x + 5 = 50

Subtracting 5 on both sides , we get

9x = 45

Divide by 9 on both sides , we get

x = 5

Hence , the equation is solved and x = 5

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The value of the expression z² + 6(x - z)² is 252.

Given information:

z² + 6(x - z)².

And x = 12 and z = 6.

To find the value of the expression, substitute x = 12 and z = 6 into the expression:

z² + 6(x - z)² = 6² + 6(12 - 6)²

= 36 + 6(6)²

= 36 + 216

= 252

Therefore, when x = 12 and z = 6, the value of the expression z² + 6(x - z)² is 252.

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The complement of angle x degrees can be written as the expression:

(90 - x)°.

The complement of an angle is defined as the angle that, when added to the original angle, will give us a sum in a total of 90 degrees.

This means that, if an angle measure is represented by the variable, x degrees, then its complement is (90 - x) degrees.

By implication, if the measure of an angle is 30 degrees, the complement of 30 degrees would be:

90 - 30 = 60°.

Thus, an angle and its complement will always give a total sum of 90 degrees.

Therefore, the expression we can write for the complement of angle x is:

90 - x

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The quadratic function in standard form that models the graph is f(x) = 4x² + x + 1.

A quadratic function is a function that can be represented by a second-degree polynomial equation of the form f(x) = ax² + bx + c, where a, b, and c are constants and x is the variable. The graph of a quadratic function is a parabola, which can open upward or downward depending on the sign of the coefficient a.

The coefficient a determines the shape and direction of the parabola. If a is positive, the parabola opens upward, and if a is negative, it opens downward. The vertex of the parabola, which is the highest or lowest point depending on the direction, is located at the point (-b/2a, f(-b/2a)).

To write a quadratic function in standard form, we need to use the general equation of a quadratic function:

f(x) = ax² + bx + c

We know that the parabola passes through the points (-1, 4) and (0, 1), so we can use these two points to form a system of equations and solve for a, b, and c.

Substituting the x and y values of the first point (-1, 4), we get:

4 = a(-1)² + b(-1) + c

4 = a - b + c (equation 1)

Substituting the x and y values of the second point (0, 1), we get:

1 = a(0)² + b(0) + c

1 = c (equation 2)

We can use equation 2 to solve for c, which is already known to be 1:

c = 1

Substituting this value of c into equation 1, we get:

4 = a - b + 1

a - b = 3 (equation 3

To solve for a and b, we can use any method we like, such as substitution or elimination. Let's use substitution. Solving equation 3 for b, we get:

b = a - 3

Substituting this expression for b into equation 1, we get:

4 = a - (a - 3) + 1

4 = 4

a = 4

Now that we know a = 4, we can substitute this value into the expression we found for b:

b = a - 3

b = 4 - 3

b = 1

So the quadratic function in standard form that models the graph is:

f(x) = 4x² + x + 1.

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Step-by-step explanation:

Let x be the number of ounces of 20% saline solution needed, and y be the number of ounces of 70% saline solution needed to obtain 25 ounces of a 50% saline solution.

To solve the problem, we can use the fact that the amount of salt in the final mixture must equal the sum of the amounts of salt in the two original solutions. That is:

0.2x + 0.7y = 0.5(25)

Simplifying this equation gives:

0.2x + 0.7y = 12.5

We also know that the total amount of solution is 25 ounces, so:

x + y = 25

We now have two equations with two variables, which we can solve using algebra. Rearranging the second equation to solve for x in terms of y, we get:

x = 25 - y

Substituting this expression for x into the first equation, we get:

0.2(25 - y) + 0.7y = 12.5

Multiplying out the terms and simplifying gives:

5 - 0.2y + 0.7y = 12.5

0.5y = 7.5

y = 15

Substituting this value for y into the equation x = 25 - y, we get:

x = 25 - 15 = 10

Therefore, we need 10 ounces of the 20% saline solution and 15 ounces of the 70% saline solution to obtain 25 ounces of a 50% saline solution.

The height of the smaller pyramid is 0.01875 miles.

The height of the smaller pyramid is obtained applying the proportions in the context of the problem.

The ratio between the volumes is given as follows:

20000/10240 = 1.953125.

The heights are measured in units, while the volume is measured in cubic units, hence the proportion to obtain the value of x is given as follows:

h/0.015 = cubic root of 1.953125

h/0.015 = 1.25

h = 1.25 x 0.015

h = 0.01875 miles.

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The simplified form of the given expression is -50(12x⁴-5³+12x²+2x+1).

The given expression is 50-50x[3-3x{4-4x²}-5+5x²].

Here, 50-50x[3-12x+12x³-5+5x²]

= 50-50x[-2-12x+12x³+5x²]

= 50+100x+600x²-600x⁴-250³

= -600x⁴-250³+600x²+100x+50

= -50(12x⁴-5³+12x²+2x+1)

Therefore, the simplified form of the given expression is -50(12x⁴-5³+12x²+2x+1).

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The algebraic expression to express "Multiply y by 6, then divide the product by 7" is 6y/7.

We have,

The algebraic expression that represents "multiply y by 6, then divide the product by 7" is:

(6y) / 7

In this expression,

The product of y and 6 is divided by 7.

Thus,

The algebraic expression to express "Multiply y by 6, then divide the product by 7" is 6y/7.

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Answer:

$71.58

Step-by-step explanation:

$859 / 12 = $71.58