Please show detailed algebraic work.

To prove these statements, we will use the formulas for the volume and surface area of a sphere and a cylinder.
The volume of a sphere is given by the formula V = 4/3 πr³, where r is the radius of the sphere.
The volume of a cylinder is given by the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder.

(a) If a sphere is inscribed in a right circular cylinder whose height is equal to the diameter of the sphere, then the height of the cylinder is 2r and the radius of the base is r. Therefore, the volume of the cylinder is
V = πr²(2r) = 2πr³.
Now, we can compare the volume of the cylinder to the volume of the sphere:
V_cylinder/V_sphere = (2πr³)/(4/3 πr³) = (2/4)(3/1) = 3/2.
Therefore, the volume of the cylinder is 3/2 the volume of the sphere.
(b) The surface area of a sphere is given by the formula A = 4πr².
The surface area of a cylinder, including its bases, is given by the formula
A = 2πrh + 2πr², where r is the radius of the base and h is the height of the cylinder.
If a sphere is inscribed in a right circular cylinder whose height is equal to the diameter of the sphere, then the height of the cylinder is 2r and the radius of the base is r. Therefore, the surface area of the cylinder is
A = 2πr(2r) + 2πr² = 4πr² + 2πr² = 6πr².
Now, we can compare the surface area of the cylinder to the surface area of the sphere:
A_cylinder/A_sphere = (6πr²)/(4πr²) = (6/4) = 3/2.
Therefore, the surface area of the cylinder, including its bases, is 3/2 the surface area of the sphere.

To know more about volume and surface area refer here:

https://brainly.com/question/15585435

#SPJ11

In the illustration, there are 2 9-point stars and 8 5-point stars.

Let x be the number of 5-pointed stars and y be the number of 9-pointed stars. Each 5-pointed star has 5 points and each 9-pointed star has 9 points, so the total number of points in the drawing is:

5x + 9y = 66

There are two variables in the equation, so we need another equation to solve for x and y. The problem states that there are a total of 10 stars in the drawing, so:

x + y = 10

We can represent this system of linear equations using a matrix:

[5 9] [x] [66]

[1 1] [y] = [10]

This is a matrix equation of the form Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix. We can solve for x and y by finding the inverse of the coefficient matrix A:

A^-1 = 1/(5-9) [-9 5]

[ 1 -1]

Multiplying both sides by A^-1, we get:

x = A^-1 b

x = 1/(5-9) [-9 5] [66]

[1 -1] [10]

x = [-3 8]

Therefore, there are 8 5-pointed stars and 2 9-pointed stars in the drawing.

To learn more about system of linear equations refer to:

brainly.com/question/19549073

#SPJ4

The only value of x for which f(x) = 0 is x = 36. So, f(x) = -x + 36 does not have a real root, since there is no value of x that makes the function equal to zero.

What is a function ?

A function is a mathematical concept that describes the relationship between a set of inputs (domain) and a set of outputs (range), such that each input has a unique output. In other words, it is a rule that assigns to each input exactly one output.

A real root of a function refers to a value of x for which the value of the function is equal to zero. So, to determine whether f(x) = -x + 36 has a real root, we need to find the value(s) of x for which f(x) = 0.

Setting f(x) = 0, we have:

-x + 36 = 0

Adding x to both sides, we get:

36 = x

Therefore, the only value of x for which f(x) = 0 is x = 36. So, f(x) = -x + 36 does not have a real root, since there is no value of x that makes the function equal to zero.

To learn more about Function from given link.

https://brainly.com/question/12431044

#SPJ1

Answer:

M<1 would be 79 and M<2 would be 101

Step-by-step explanation:

The best histogram to represent the data collected by the surfer is a frequency polygon.

A frequency polygon is a line graph that shows the frequency distribution of a set of data points. It is used to compare the data points and identify any trends or patterns in the data.

The x-axis of the histogram represents the tide height in feet and the y-axis represents the frequency, or the number of days the tide rose to that level. As can be seen from the data provided, the tide height varied from 5-24 feet over the 15-day period, so this is the range of values that should be used for the x-axis. The frequency of each tide height is then calculated and plotted on the y-axis.

To create the frequency polygon, the first step is to create a frequency table. The next step is to plot the data points on the graph, connecting them with straight lines. The final step is to draw a line to connect the first and last data points.

The resulting frequency polygon is a good representation of the data because it clearly shows the range of tide heights over the 15-day period and the frequency of each height.

A frequency polygon histogram best represents the data collected by the for how far the tide rose, in feet, up the beach over a 15-day period.

Know more about histogram here:

https://brainly.com/question/2962546

#SPJ11

Answer:

Step-by-step explanation:

Answer is D I hope i helped

The equation of the quadratic function is f(x) = (x + 5)² - 2

A quadratic function is a second-degree polynomial function of one variable, which can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and x is the variable.

From the question, we have the table of values which represents a quadratic function f(x).

A quadratic equation is represented as

f(x) = a(x - h)² + k

Where, Vertex = (h, k)

If we plot the graph according to the given table, the vertex will be,

(h, k) = (-5, -2)

Substitute, (h, k) = (-5, -2) in f(x) = a(x - h)² + k

So, we have, f(x) = a(x + 5)² - 2

Also, from the graph, we have the point (0, 23)

This means that

a(0 + 5)² - 2 = 23

25a = 25

a = 1

Substitute a = 1 in f(x) = a(x + 5)² - 2

f(x) = (x + 5)² - 2

Hence, the equation is f(x) = (x + 5)² - 2

To know more about polynomial, visit:

https://brainly.com/question/1496352

#SPJ1

Speed cubing is a competitive activity in which participants attempt to solve Rubik's Cubes and other twisty puzzles as quickly as possible.

In speed cubing competitions, competitors race against the clock to solve the puzzles in the shortest amount of time. Speed cubing became popular in the early 2000s, largely due to the internet and online communities dedicated to the hobby. These online forums provided a platform for cubers to share techniques, compete with one another, and organize local competitions. With the growth of the internet, speed cubing quickly spread around the world and became a global phenomenon. The first world championship for speed cubing was held in 1982, but the competition did not gain widespread attention until the early 2000s. Today, there are numerous international organizations dedicated to organizing speed cubing competitions, including the World Cube Association (WCA), which hosts events for several different types of twisty puzzles, such as the 2x2, 3x3, 4x4, and 5x5 Rubik's cubes, as well as other popular puzzles like the Pyraminx and Megaminx. Speed cubing has become a highly competitive and popular activity,

To learn more about Rubik's cubes click here

brainly.com/question/22645392

#SPJ4

Answer:

A = 45 units²

Step-by-step explanation:

the lined figure is composed of a rectangle and a triangle

area of rectangle WFSC is calculated as

area = length × width

       = FW × WC

      = 2 × 9

     = 18 units²

the area of Δ is calculated as

area = [tex]\frac{1}{2}[/tex] base × perpendicular height

       = [tex]\frac{1}{2}[/tex] × FS × perpendicular length from N to base FS

       = [tex]\frac{1}{2}[/tex] × 9 × 6

      = 4.5 × 6

      = 27 units²

total area (A) is then the sum of the 2 figures , that is

A = 18 + 27 = 45 units²

The probability that the first customer will sit in a seat in the back row is 4/9.

A customer is a person or business that buys goods or services from another business.

There are a total of 45 seats in the theater. Out of those, 20 seats are in the back row.

So the probability that the first customer who enters the theater will sit in a seat in the back row is:

P(back row) = 20/45

Simplifying the fraction gives:

P(back row) = 4/9

Therefore, the probability that the first customer will sit in a seat in the back row is 4/9.

Learn more about fraction here : brainly.com/question/28699958

#SPJ1

By answering the presented question, we may conclude that Kayden surface area used 116.4 square centimeters of wrapping paper as a result.

The surface area of an object indicates the overall space occupied by its surface. The surface area of a three-dimensional form is the entire amount of space that surrounds it. The surface area of a three-dimensional form refers to its full surface area. By summing the areas of each face, the surface area of a cuboid with six rectangular faces may be computed. As an alternative, you may use the following formula to name the box's dimensions: 2lh + 2lw + 2hw = surface (SA). Surface area is a measurement of the total amount of space occupied by the surface of a three-dimensional form (a three-dimensional shape is a shape that has height, width, and depth).

Then, we must establish the size of the pyramid's square base. The net shows that the base has a side length of 6 cm.

sqrt((5.7 cm)2 + (3 cm)2) = 6.7 cm slant height (rounded to one decimal place)

Each triangle face has the following area:

(1/2) x width x height

(half) × 6 cm x 6.7 cm

= 20.1 cm2 4 times 20.1 cm2 = 80.4 cm2

The area of the square face is:

6 cm × 6 cm = 36 cm2 side x side

As a result, the total surface area of the net (together with the amount of wrapping paper used by Kayden) is:

80.4 cm2 plus 36 cm2 equals 116.4 cm2.

Kayden used 116.4 square centimeters of wrapping paper as a result.

To know more about surface area visit:

https://brainly.com/question/2835293

#SPJ1

Answer:

The volume of the cake is:

12 inches x 9 inches x 2 inches = 216 cubic inches

Dividing the volume of the cake by the number of pieces, we get:

216 cubic inches / 18 = 12 cubic inches per piece

Therefore, the volume of each piece is 12 cubic inches.

Answer:

12 inches cubed

Step-by-step explanation:

First we have to find the total volume so we can split it up. We can do this by using the formula LWD = V, where l = length, w = width, and d = depth.

Here, the length is 12, the height is 2, and the depth is 9. We apply the formula:

(12)(2)(9) = 216

Now, we have to split it into 18 separate pieces. We can do this by dividing the total volume (216 inches cubed) by 18.

216/18 = 12

Therefore, each equal piece will be 12 inches cubed

The probability of getting exactly one tail would be = 1/2

Probability can be defined as the possibility of an event occuring or an outcome of an event.

The number of time the fair coin was tossed = 2 times

From the sample space, the total number of samples = 4.

The number of single heads = 2

The probability of getting a single head (H) = 2/4 = 1/2

Learn more about probability here:

https://brainly.com/question/25289437

#SPJ1

The statement "Two chords in the same circle are congruent if and only if the associated central angles are supplementary" is false because  two chords in the same circle are congruent if and only if they are equidistant from the center of the circle

Two chords in the same circle are congruent if and only if they are equidistant from the center of the circle. However, two chords having associated central angles that are supplementary will always be equal in length only if the chords are diameters of the circle.

For any other pair of chords with supplementary central angles, their lengths will depend on their distance from the center of the circle. Therefore, the statement "Two chords in the same circle are congruent if and only if the associated central angles are supplementary" is not true in general.

Learn more about chord here

brainly.com/question/17080243

#SPJ4

The angle between the beam and the road is equal to 5.71°.

A headlamp is a lamp that is mounted on the front of a car to light the way. Although headlights and headlamps are frequently used interchangeably, in the most formal sense, headlight refers to the device's beam of light and headlamp to the device itself.

We given that :- h= 2inch and r= 20.0feets.

where, h is beam drops and r is the headlights distance.

To find the angle between the beam and the road we have to use the equation:-

tan theta = [tex]\frac{h}{r} = \frac{2 inch}{20.0 ft} = \frac{1}{10} = 0.1[/tex]

Calculating tan theta to find theta which is an angle between the beam and the road.

Theta= [tex]tan^{-1} (0.1)= 5.71[/tex]

hence, the angle between the beam and the road is 5.71 degree.

To know more about angle go through:-

https://brainly.com/question/25716982

#SPJ1

The product of two binomials is the result of multiplying each term from the first binomial by each term from the second binomial. The product of the two binomials, (-5x-1) and (2x+8), is[tex]0x^2+(-10x-2x)+(-8x-8)[/tex] which simplifies to -18x-10.

To calculate the product of the two binomials, we multiply the coefficients of each term together and add the exponents of the same base together. The coefficients of the first binomial, -5x and -1, are multiplied together to give a coefficient of -5. The exponents of x, 1 and 0, are added together, giving an exponent of 1. Thus, the first term is[tex]-5x^1[/tex]. The coefficients of the second binomial, 2x and 8, are multiplied together to give a coefficient of 16. The exponents of x, 1 and 0, are added together, giving an exponent of 1. Thus, the second term is[tex]16x^1[/tex].The last terms, -1 and 8, are multiplied together to give a coefficient of -8. The exponents of x, 0 and 0, are added together, giving an exponent of 0. Thus, the third term is[tex]-8x^0[/tex], which simplifies to -8.When all the terms are combined, the product of the two binomials, (-5x-1) and (2x+8), is -18x-10.

Learn more about binomials here:

https://brainly.com/question/28983916

#SPJ1

Answer:

d

Step-by-step explanation:

Answer:

Step-by-step explanation:

This is a geometric means problem. 6 is the altitude, which belongs to both of those smaller triangles; thus, it is the geometric means. We set up the proportion like this:

[tex]\frac{8}{6} =\frac{6}{b+3}[/tex] and we cross multiply to solve.

8(b + 3) = 36 and

8b + 24 = 36 and

8b = 12 so

b = 3/2 or 1.5

Answer:

We can use the principle of conservation of momentum to solve this problem. According to this principle, the total momentum of a system of objects remains constant if there are no external forces acting on the system.

Initially, only the cue ball is moving, and the 8 ball is at rest. Therefore, the initial momentum of the system is:

p_initial = m1 * v1 + m2 * v2

= 0.2 kg * 10 m/s + 0.15 kg * 0 m/s

= 2 kg m/s

After the collision, the cue ball is rolling forward at 1 m/s, and the 8 ball is moving in some direction with some velocity v_final. Therefore, the final momentum of the system is:

p_final = m1 * v1 + m2 * v_final

According to the conservation of momentum principle, p_initial = p_final. Therefore,

2 kg m/s = 0.2 kg * 1 m/s + 0.15 kg * v_final

Solving for v_final, we get:

v_final = (2 kg m/s - 0.02 kg m/s) / 0.15 kg

= 13.33 m/s

Therefore, the velocity of the 8 ball after the collision is 13.33 m/s.

+----------------+

|  0  |  1  | -1  |

+-----+-----+-----+

|  2  | -2  |  3  |

+-----+-----+-----+

| -3  |  4  | -4  |

+-----+-----+-----+

 o            o

The zero pairs are (-2, 2) and (-4, 4), and I've circled them in the model above.

To draw a model using integer chips and circle the zero pairs, start by laying out the positive and negative integer chips side by side in a line. The negative numbers should be on the left and the positive numbers should be on the right. Next, take a second line of chips, also with negative numbers on the left and positive numbers on the right, and arrange it so that the chips on the first line line up with the corresponding chips on the second line. For example, the negative 3 chip on the first line should be paired with the positive 3 chip on the second line. The same should be done for all other chips. Finally, circle any pairs of chips where the sum of the two numbers is equal to 0. This will indicate that they are zero pairs.

For such more questions on integer chips:

brainly.com/question/15327444

#SPJ11

Where the above H0 : p1 = 0.65, p2 = 0.70, p3 = 0.75, note that there is not sufficient evidence to conclude that the proportion of adherence differs between the age groups at the 0.05 level of significance.

Level of significance is the probability of rejecting the null hypothesis when it is actually true. It is typically set at 0.05 or 0.01, indicating a 5% or 1% chance of a Type I error, respectively.

To test whether there is sufficient evidence to reject the null hypothesis, we can use the χ2 goodness of fit test. The null hypothesis is that the proportion of adherence is the same in each age group, with p1 = 0.65, p2 = 0.70, and p3 = 0.75.

The alternative hypothesis is that the proportion of adherence is different in at least one of the age groups.

To apply the χ2 goodness of fit test, we need to calculate the expected frequencies for each age group under the null hypothesis.

The expected frequency for each group is the total sample size (100) times the hypothesized proportion of adherence for that group. Therefore, the expected frequencies for each age group are:

Young: 100 * 0.65 = 65

Medium: 100 * 0.70 = 70

Old: 100 * 0.75 = 75


We can now calculate the test statistic χ2 by summing over the three age groups the squared difference between the observed and expected frequencies divided by the expected frequency:

χ² = ((0.64-65)²/65) + ((0.68-70)²/70) + ((0.70-75)²/75)

= 1.81

The degrees of freedom for the χ2 goodness of fit test are the number of categories minus one, which in this case is 3 - 1 = 2. Using a chi-square distribution table with 2 degrees of freedom and a significance level of 0.05, we find the critical value to be 5.99.

Since the calculated χ2 value of 1.81 is less than the critical value of 5.99, we fail to reject the null hypothesis.

Thus, there is not sufficient evidence to conclude that the proportion of adherence differs between the age groups at the 0.05 level of significance.

Learn more about level of significance:
https://brainly.com/question/4599596
#SPJ1

Answer:

After one year, the amount in the bank account will be:

$200 + ($200 x 0.05) = $200 + $10 = $210.

Therefore, the total in the bank account after 1 year is $210.

Answer:

To find the absolute change in population from 2007 to 2009, we can subtract the initial population in 2007 from the final population in 2009:

Absolute change = Final population - Initial population

= 6,200 - 3,900

= 2,300 people

Therefore, the absolute change in population from 2007 to 2009 is 2,300 people.

To find the percent increase (relative change) in population, we can use the formula:

Percent increase = (Absolute change / Initial population) x 100%

Substituting the values given in the problem, we get:

Percent increase = (2,300 / 3,900) x 100%

= 58.97%

Therefore, the population increased by approximately 58.97% from 2007 to 2009. Rounded to the nearest tenth of a percent, the percent increase is 59.0%.

(a)   1

(b)  0.3

(c)  0.15
(d)  0.27

(a) The cumulative probability distribution of the random variable X for the year 2020 is:
X = 0, P(X<=0) = 0.2
X = 1, P(X<=1) = 0.6
X = 2, P(X<=2) = 0.9
X = 3, P(X<=3) = 1
Graph:


(b) P(1 ≤ X < 3) = P(X<=2) - P(X<=1) = 0.9 - 0.6 = 0.3

(c) The joint probability distribution of the variables X and Y for the year 2020 is:
X = 0, Y = 0, P(X=0, Y=0) = 0.15
X = 0, Y = 1, P(X=0, Y=1) = 0.25
X = 0, Y = 2, P(X=0, Y=2) = 0.05
X = 1, Y = 0, P(X=1, Y=0) = 0.2
X = 1, Y = 1, P(X=1, Y=1) = 0.4
X = 1, Y = 2, P(X=1, Y=2) = 0.2
X = 2, Y = 0, P(X=2, Y=0) = 0.3
X = 2, Y = 1, P(X=2, Y=1) = 0.3
X = 2, Y = 2, P(X=2, Y=2) = 0.2
X = 3, Y = 0, P(X=3, Y=0) = 0.15
X = 3, Y = 1, P(X=3, Y=1) = 0.15
X = 3, Y = 2, P(X=3, Y=2) = 0.15

(d) Treating the answer from question 3(c) as the joint probability distribution in the population, the correlation between X and Y is 0.27.

  Learn more about probability

  brainly.com/question/30034780

  #SPJ4

On answering the provided question, we have got that As a result, the scale factor of the dilatation between triangle  ABC and A′′B′′C′′ is 1/2.

An additional four or so parts make a triangle a polygon. Its shape is a simple rectangle. An edged rectangle with the letters ABC on it is known as a triangle. Euclidean geometry yields a single plane and cube when the sides are actually not collinear. Having three parts and three angles makes a triangle a polygon. The corners of a triangle are where the three edges meet. The sum of the angles of a triangle's sides is 180.

By multiplying the individual scale factors of the two dilations, we can determine the scale factor of the dilation between ABC and A′′B′′C.

Each side of ABC is scaled down by a factor of 1/3 in the first dilation with a scale factor of 3. Thus, the corresponding sides of A, B, and C are a third the length of corresponding s of ABC.

In the second dilation, each side of "A" through "C" is scaled up by a factor of 3/2. As a result, the corresponding sides of A′′B′′C′′ are 3/2 times longer than those of A′B′C′.

1/3 × 3/2 = 1/2

As a result, the scale factor of the dilatation between ABC and A′′B′′C′′ is 1/2.

To know more about triangle visit:

https://brainly.com/question/2773823

#SPJ1

The correct option is A. m∠X = 45° as the triangles ABC and XYZ are congruent.

Congruent triangles are two triangles that have the same shape and size. In other words, if two triangles are congruent, then all their corresponding sides and angles are equal. Some methods that allow us to compare the lengths of corresponding sides and the measures of corresponding angles in the two triangles and determine whether they are equal includes: SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), or HL (hypotenuse-leg).

The angle m∠A corresponds to the angle m∠X given that the triangles ABC and XYZ are congruent.

In conclusion, the correct option for the congruent triangles is A. m∠X = 45°.

Know more about congruent triangles here:https://brainly.com/question/14285697

#SPJ1

Answer:

As we do not have a visual representation of the spinner, we cannot give an accurate answer to this question. However, we can provide a general approach to solving such problems.

To find the probability of an event, we divide the number of favorable outcomes by the total number of possible outcomes.

In this case, we need to find the number of favorable outcomes for the spinner landing on a multiple of 3 or in the dark green region. We then divide this by the total number of possible outcomes.

Assuming the spinner is divided into equal regions, we can determine the number of favorable outcomes by counting the number of regions that satisfy the given conditions. Let's say there are m regions that are multiples of 3, and n regions that are in the dark green region. We also assume that these regions do not overlap.

The total number of possible outcomes is simply the total number of regions on the spinner, which we can denote as t. Therefore, the probability of the spinner landing on a multiple of 3 or in the dark green region is given by:

(m + n) / t

To simplify this fraction, we can try to reduce the numerator and denominator to their lowest terms by dividing both by their greatest common factor.

Without knowing the number of regions, however, we cannot give a specific answer to this problem.

Over the course of the 6-year loan, Jack will have paid a total of $32,741.19.Jack took out a 6-year loan for $25,000 to purchase a vehicle.

The loan is compounded monthly, meaning the interest accrued on the loan is added to the total loan balance each month. At the end of the 6-year loan, Jack will have paid a total of $32,741.19.
To calculate the total amount of the loan, we can use the formula:
Total Amount Paid = Principal + Interest
The principal of the loan is $25,000 and the interest rate is 6.25%, compounded monthly. The interest rate can be broken down into a monthly rate of 0.5208% (6.25/12).
Total Amount Paid = $25,000 + (0.5208% x 25000 x 6) = $32,741.19

For more such interest related questions

https://brainly.com/question/25793394

#SPJ11

The third term of the arithmetic sequence presented is given as follows:

14.

An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a constant value, called the common difference (d), to the previous term.

From the sequence in this problem, the terms are given as follows:

More can be learned about arithmetic sequences at https://brainly.com/question/6561461

#SPJ1