A Microwaveable Cup-of-soup Package Needs To Be Constructed In The Shape Of Cylinder To Hold 600 Cubic (2024)

Mathematics High School

Answers

Answer 1

To minimize the cost of the package, we need to find the dimensions that minimize the cost function.

The cost function is the sum of the cost of the side and bottom (made of syrofoam) and the cost of the top (made of paper). Let r be the radius and h be the height of the cylinder. Then the cost function is:

C(r, h) = 0.02(2πrh + πr^2) + 0.05(πr^2)

We need to find the values of r and h that minimize this function subject to the constraint that the volume of the cylinder is 600 cubic centimeters. That is:

V = πr^2h = 600

We can solve for h in terms of r from the volume equation:

h = 600/(πr^2)

Substituting this expression for h in the cost function, we get:

C(r) = 0.02(2πr(600/(πr^2)) + πr^2) + 0.05(πr^2)

= 0.04(600/r) + 0.05πr^2

To minimize C(r), we take the derivative with respect to r and set it equal to zero:

dC/dr = -0.04(600/r^2) + 0.1πr = 0

Solving for r, we get:

r = (300/π)^(1/3) ≈ 5.17 cm

Substituting this value of r into the volume equation, we get:

h = 600/(πr^2) ≈ 2.17 cm

Therefore, the dimensions of the cylinder that minimize the production cost are r ≈ 5.17 cm and h ≈ 2.17 cm, and the minimum cost is:

C(r, h) ≈ $1.24

So, the minimum cost of producing a microwaveable cup-of-soup package in the shape of a cylinder with a volume of 600 cubic centimeters is about $1.24, with a radius of about 5.17 cm and a height of about 2.17 cm.

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Related Questions

The solid below is made from cubes.
Find its volume.
1 yd

Answers

The Volume of the solid which is made of small-cubes is 15 yd³.

The "Volume" of a cube is the measure of the amount of space that the cube occupies in three-dimensional space. It is calculated by multiplying the cube's three side lengths.

In the figure, we can see that, the solid consists of 5×3 = 15 cubes,

The side-length of each small cube is = 1 yd,

So, the volume of each small-cube is = (side)×(side)×(side),

⇒ Volume = 1 yd³.

To find the volume of the entire solid, we multiply the volume of single-cube, by the "total-number" of cubes,

So, Volume of Solid is = 15×1 = 15 yd³.

Therefore, the volume of the solid is 15 yd³.

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In one week, a bakery made 1,244 trays of 26 chocolate chip cookies. The bakery also baked 694 trays of 19 sugar cookies. How many cookies did the bakery bake during the week?

Answers

The total number of cookies baked during the week if a bakery baked 1,244 trays of 26 chocolate chip cookies and also 694 trays of 19 sugar cookies are 336530.

To calculate the total number of cookies we have to find the number of chocolate chip and sugar cookies.

The total number of chocolate cookies is given by the product of the number of cookies in a tray and the number of trays.

Trays of chocolate chip cookies = 1244

Number of chocolate chip cookies per tray = 26

Total number of chocolate chip cookies = 1244 * 26

= 32344

Similarly, the total number of sugar cookies is given by the product of the number of cookies in a tray and the number of trays.

Trays of sugar cookies = 694

Number of sugar cookies per tray = 19

Total number of chocolate chip cookies = 694 * 19

= 13186

Total number of cookies = total number of chocolate chip cookies + sugar cookies

= 32344 + 13186

=336530

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determine the vertex and direction of opening of the parabola for the following quadratic equation: y equals 3 x squared minus 18 x minus 10

Answers

The vertex is (3,-37) and the direction of the opening is upwards. To determine the vertex and direction of the opening of the parabola for the quadratic equation y = 3x^2 - 18x - 10, we first need to put it in vertex form.

Completing the square, we have:

[tex]y = 3(x^2 - 6x) - 10[/tex]
y = 3(x^2 - 6x + 9) - 10 - 27

(adding and subtracting 27, which is 3 times 9, inside the parentheses)

[tex]y = 3(x - 3)^2 - 37[/tex]

Now we can see that the vertex is (3,-37), since the equation is in the

form y = a(x - h)^2 + k, where (h,k) is the vertex.

We can also see that the parabola opens upwards, since the coefficient of x^2 is positive.

Therefore, the vertex is (3,-37) and the direction of opening is upwards.

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You have 7 different video games. How many different ways can you arrange the games side by side on a shult? You can arrange the 7 different video games in different ways

Answers

There are 5,040 different ways to arrange the 7 video games side by side on a shult.

To calculate the number of different ways you can arrange the 7 different video games on a shult, you can use the formula for permutations, which is n!/(n-r)!. In this case, n is the total number of games (7) and r is the number of games being arranged at once (also 7, since you are arranging all of them).

So the calculation would be:

7! / (7-7)! = 7! / 0! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040

Therefore, there are 5,040 different ways you can arrange the 7 different video games side by side on a shult.

To arrange the 7 different video games side by side on a shult, you can use the concept of permutations. In this case, there are 7 video games, and you want to arrange all of them. So, the number of ways to arrange the video games can be calculated using the formula:

Number of arrangements = 7! (7 factorial)

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

Therefore, there are 5,040 different ways to arrange the 7 video games side by side on a shult.

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What is area and perimeter?

Answers

Answer:

Perimeter is the distance around the outside of a shape. Area measures the space inside a shape.

Step-by-step explanation:

Your welcome ;)

Select True or False for each statement.

Answers

1. (1/64)^–⅚ = 32 True

2. √12 - 2/5 × √75 = 4√3 = False

3. √9a + √98b -a+√2b = 2a+10√b = 0 =True

4. 1/(√5-√6)² = 11-2√30/241 = False

How did we arrive at these assertions?

1. (1/64)^–⅚ = 32

write in exponential form

2⁵ = 2⁵, hence, true

2. √12 - 2/5 × √75 = 4√3

Simplify the expressions

2√3-2√3 = 4√3

Eliminate the opposites

0 = 4√3

The statement is false

3. √9a + √98b -a+√2b = 2a+10√b = False

Evaluate

3a + √98b-a + √2b = 2a+10√b

Collect like terms

2a + √98b+ √2b = 2a+10√b

Cancel equal terms

98b+ √2b = 10√b Simplify the equation

Answers

The expected winning and the value of that prize, which gives an expected winning is c. $3.13 per ticket.

The expected winnings per ticket in the raffle can be calculated by adding the products of the probability of the expected value of winning each prize. The predicted wins per ticket may be computed as the total of each prize's value multiplied by its probability of winning, i.e.

Expected winnings per ticket

= (1/800) × $400 + (4/800) × $200 + (12/800) × $125 + (30/800) × $60 + (753/800) × $0

If we condense the equation, we obtain:

Calculating the total wins anticipated per ticket

= $0.50 + $1.00 + $1.88 + $0.75 + $0

Expected winnings per ticket = $3.13

Therefore, if you buy a ticket, the expected winnings per ticket are $3.13.

Answers

a) Only the first statement is true. As the sample size N becomes large, the sample mean of IID random samples from a population becomes more precise and approaches the true population mean.

However, there is no direct relationship between the sample size and the distribution of the sample mean.
The second statement is only true if the population is normally distributed. If the population is not normal, the distribution of the sample mean may not be normal, and the central limit theorem may not apply. Therefore, option c) is not the correct answer. Option d) is also not correct as the first statement is true.

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2) The same liquor store owner wants to do a similar comparison but for high end wines to see if there is a difference. The owner samples 16 white wines finding an average of $45.13 (s=5.10) and samples 16 red wines and finds an average of $48.69 (s=5.23). Use alpha=0.05.
2a) What is the standard error?
2b) What is the test statistic?
2c) What is the p-value?
2d) What can you conclude about cost of high end wines?

Answers

Answer:

2a) The standard error is given by:

SE = sqrt[(s1^2/n1) + (s2^2/n2)]

= sqrt[(5.10^2/16) + (5.23^2/16)]

= 2.32

2b) The test statistic is given by:

t = (x1 - x2) / SE

= (45.13 - 48.69) / 2.32

= -1.53

2c) The p-value for a two-tailed test with alpha = 0.05 and degrees of freedom = 30 (n1 + n2 - 2) is 0.1384.

2d) Since the p-value (0.1384) is greater than the level of significance (0.05), we fail to reject the null hypothesis that there is no difference in the cost of high end red and white wines. Therefore, we cannot conclude that there is a significant difference in the cost of high end wines.

Write an equation to match the graph.

Answers

The equation which matches the absolute value function graph given as required is; y = -1 |x + 1| + 1.

What is the equation which matches the graph?

As evident from the task content, the equation which matches the given graph is to be determined.

On this note, recall that the absolute value function takes the form;

f (x) = a |x - h| + k where the vertex is; (h, k).

Therefore, since the vertex is; (-1, 1); we have;

f (x) = a |x + 1| + 1

To find a; use point (0, 0)

0 = a |0 + 1| + 1

a = -1.

Ultimately, the equation which matches the graph is; y = -1 |x + 1| + 1.

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a bag of marbles contains 6 red and 2 white marbles. if two marbles are selected, what is the probability that one is red and the other is white

Answers

The probability that one marble is red and the other is white when two marbles are selected from the bag is 3/7

To find the probability that one marble is red and the other is white when two marbles are selected from a bag containing 6 red and 2 white marbles, you can use the following formula:

P(Red and White) = P(Red first) * P(White second) + P(White first) * P(Red second)

In this case:

P(Red first) = 6/8 (since there are 6 red marbles and a total of 8 marbles)
P(White second) = 2/7 (after removing one red marble, there are 2 white marbles and a total of 7 marbles left)

P(White first) = 2/8 (since there are 2 white marbles and a total of 8 marbles)
P(Red second) = 6/7 (after removing one white marble, there are 6 red marbles and a total of 7 marbles left)

Now, substitute these values into the formula:

P(Red and White) = (6/8) * (2/7) + (2/8) * (6/7) = (12/56) + (12/56) = 24/56

Simplify the fraction:

P(Red and White) = 3/7

So, the probability that one marble is red and the other is white when two marbles are selected from the bag is 3/7.

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mr. ng can complete a job in x hours while mr. luciano can complete the same job in y hours. how much of the job can they complete if they work together for k hours?

Answers

They can complete k(1/x + 1/y) portion of the job if they work together for k hours.

We have,

If Mr. Ng can complete a job in x hours and Mr. Luciano can complete the same job in y hours, their individual rates of work are given by 1/x and 1/y, respectively.

Working together, their combined rate of work is the sum of their individual rates, which is 1/x + 1/y.

If they work together for k hours, the amount of the job they can complete is given by the product of their combined rate and the time they work, which is k(1/x + 1/y).

Thus,

They can complete k(1/x + 1/y) portion of the job if they work together for k hours.

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Eli cut out a 12-inch square. Then he cut off 2-inch squares from each corner of his large square. Which expression can be used to find the remaining area of the larger square?

Answers

The expression which can be used to find the remaining area of the larger square is calculated to be A = (12 in)² - 4(2 in)²

If Eli cuts off a 2-inch square from each corner of his 12-inch square, the new dimensions of the square will be 12 - 2 - 2 = 8 inches. Therefore, the remaining area of the larger square is:

(8 in) x (8 in) = 64 square inches

We can also express this mathematically as:

(12 in)A = (12 in)² - 4(2 in)² - 4(2 in)^2 = 144 sq in - 16 sq in = 128 sq in

So the expression that can be used to find the remaining area of the larger square is:

A = (12 in)² - 4(2 in)²

where A is the remaining area of the larger square.

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H. :P - 0.65 and H.: p > 0.65 where p = the proportion of students who were quarantined at some point during the Fall Semester of 2020. Identify the correct explanation for a Type II error. Conclude the percent was higher than 65%, but it was not higher. Conclude the percent was higher than 65% and it was higher. Did not conclude the percent was higher than 65%, but it was higher. Did not conclude the percent was higher than 65% and it was not higher.

Answers

A Type II error occurs when we fail to reject a null hypothesis that is actually false. In this case, the null hypothesis is that the proportion of students who were quarantined at some point during the Fall Semester of 2020 is equal to or less than 0.65.

The alternative hypothesis is that the proportion is greater than 0.65. If we make a Type II error, we fail to reject the null hypothesis when it is actually false, meaning we do not conclude that the proportion is higher than 0.65 even though it actually is higher.

Therefore, the correct explanation for a Type II error, in this case, we would be: "Did not conclude the percent was higher than 65%, but it was higher."

Answers

The volume of one ball is approximately 4,058.67 cubic inches.

We have,

The volume of a sphere is given by the formula:

V = (4/3)πr³

where r is the radius of the sphere.

Since the diameter of the ball is given as 18 inches, the radius is half of the diameter, which is 9 inches.

Substituting this value into the formula for the volume of a sphere, we get:

V = (4/3)π(9³) = 4,058.67 cubic inches (rounded to two decimal places)

Therefore,

The volume of one ball is approximately 4,058.67 cubic inches.

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a publisher reports that 43% of their readers own a particular make of car. a marketing executive wants to test the claim that the percentage is actually different from the reported percentage. a random sample of 200 found that 38% of the readers owned a particular make of car. is there sufficient evidence at the 0.10 level to support the executive's claim?

Answers

Therefore, we do not have sufficient evidence to support the executive's claim that the true proportion of readers who own the particular make of car is different from the reported proportion of 43% at the 0.10 level of significance.

To determine whether there is sufficient evidence to support the executive's claim that the percentage of readers who own a particular make of car is different from the reported percentage of 43%, we need to conduct a hypothesis test.

The null hypothesis H0 is that there is no difference between the true proportion of readers who own the particular make of car and the reported proportion of 43%:

H0: p = p0

The alternative hypothesis Ha is that the true proportion of readers who own the particular make of car is different from the reported proportion:

Ha: p ≠ p0

We can use a z-test for proportions to test this hypothesis, since the sample size is sufficiently large and the sampling distribution of the sample proportion can be approximated by a normal distribution.

test statistic is calculated as:

z = (x/n - p0) / √(p0*(1-p0)/n)

Substituting the values from the problem statement, we get:

z = (0.38 - 0.43) / √(0.43*(1-0.43)/200)

= -1.51

At the 0.10 level of significance, the critical values for a two-tailed test are ±1.64. Since our calculated z-value of -1.51 is not in the rejection region, we fail to reject the null hypothesis.

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A girl read a book that had 1,159 pages. She read 72 pages on the first day. She then read 40 pages per day until she finished the book. Which equation can be used to find how many more days, d, it took her to read the book after the first day?

Answers

Answer:

The girl read 1,159 - 72 = 1,087 pages after the first day.

Let d be the number of days after the first day that she took to finish the book.

The total number of pages she read on these d days is 40d.

The equation that can be used to find how many more days it took her to read the book after the first day is:

40d = 1,087

Solving for d:

d = 1,087/40

d ≈ 27.175

Therefore, it took her approximately 27 more days to finish reading the book after the first day.

a math teacher gives two different tests to measure students' aptitude for math. scores on the first test are normally distributed with a mean of 23 and a standard deviation of 5.1. scores on the second test are normally distributed with a mean of 68 and a standard deviation of 10.4. assume that the two tests use different scales to measure the same aptitude. if a student scores 30 on the first test, what would be his equivalent score on the second test? (that is, find the score that would put him in the same percentile.)

Answers

The student's equivalent score on the second test would be approximately 82.2, placing them in the same percentile as their score on the first test.

To find the equivalent score on the second test, we need to first find the student's percentile rank on the first test.

Using a z-table, we can find that a score of 30 on the first test has a z-score of (30-23)/5.1 = 1.37. This means that the student scored higher than approximately 91% of the other students who took the first test.

Next, we need to find the score on the second test that corresponds to the same percentile rank. To do this, we can use the formula:

z = (x - μ) / σ

where z is the z-score corresponding to the desired percentile rank, x is the corresponding score on the second test, μ is the mean of the second test (68), and σ is the standard deviation of the second test (10.4).

Substituting the z-score we found earlier (1.37) and solving for x, we get:

1.37 = (x - 68) / 10.4

Multiplying both sides by 10.4 and adding 68, we get:

x = 83.29

Therefore, if a student scores 30 on the first test, their equivalent score on the second test would be approximately 83.29. This means that if they scored 83.29 or higher on the second test, they would be in the same percentile as they were on the first test (in this case, approximately the 91st percentile).
To find the equivalent score on the second test, follow these steps:

1. Determine the student's z-score on the first test:
z-score = (student's score - mean) / standard deviation
z-score = (30 - 23) / 5.1 ≈ 1.37

2. The z-score of 1.37 represents the student's percentile ranking on the first test. Now we need to find the equivalent score on the second test that corresponds to the same percentile.

3. Convert the z-score back to a raw score using the second test's mean and standard deviation:
Equivalent score = (z-score * standard deviation) + mean
Equivalent score = (1.37 * 10.4) + 68 ≈ 14.2 + 68 = 82.2

So, the student's equivalent score on the second test would be approximately 82.2, placing them in the same percentile as their score on the first test.

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A card is drawn from a standard deck of playing cards and is not replaced. Then a second card is drawn. Find the probability the first
card is a jack of spades and the second card is black. Express your answer as a fraction in simplest form.

Answers

The probability of drawing a jack of spades on the first draw and a black card on the second draw is 13/1326.

The probability of drawing a jack of spades from a standard deck of playing cards is 1/52 since there is only one jack of spades in the deck of 52 cards.

Once the jack of spades has been drawn, there are 51 cards left in the deck, of which 26 are black (13 clubs and 13 spades) and 25 are red (13 diamonds and 12 hearts).

The probability of drawing a black card as the second card, given that the first card was the jack of spades, is 26/51.

Now the probability of both events occurring, we multiply the probabilities of the individual events:

P(jack of spades and second card is black) = (1/52) x (26/51)

= 13/1326

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What is the mapping formula expressed by the vector that translates JKLM to J’K’L’M’.

(x,y) → (x - 2, y - 3)
(x,y) → (x +1, y - 4)
(x,y) → (x +1, y + 4)
(x,y) → (x - 2, y +3)

Answers

Answer:

The answer to your problem is, D. (x,y) → (x - 2, y +3)

Step-by-step explanation:

We can see that the vector that translates JKLM to J'K'L'M' is given by the difference between the coordinates of J and J', and the difference between the coordinates of K and K'.

The differences can be called, "a" and "b", respectively. Then we can write:

a = J' - J = (-1) - 0 = -1

b = K' - K = 3 - (-1) = 4

Which can conclude to the mapping formula expressed by the vector that translates JKLM to J'K'L'M' is:

(X,Y) → (X - 1, Y + 4)

Thus the answer to your problem is, D. (x,y) → (x - 2, y +3)

Answer these questions?

Answers

The dimensions of this rectangle are 6 units by 7 units.

The coordinates of the new point is (3, -3).

The length of the line segment with end points A and B is 7 units.

The length of the line segment with end points C and D is 11 units.

How to determine the dimensions of this rectangle?

In order to determine the dimensions of this rectangle, we would use the distance formula. In Mathematics, the distance between two (2) points that are on a coordinate plane can be calculated by using this formula:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance AB = √[(5 + 1)² + (4 - 4)²]

Distance AB = √[36 + 0]

Distance AB = 6 units

For the width, we have:

Distance AC = √[(-1 + 1)² + (4 + 3)²]

Distance AC = √[0 + 49]

Distance AC = 7 units.

In Mathematics and Geometry, a reflection over the x-axis is modeled by this transformation rule;

(x, y) → (-x, y)

New point = (-3, -3) → (3, -3).

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Complete Question:

The coordinates of the vertices of a rectangle are (-1, 4), (5, 4), (5, -3), and (-1, -3). What are the dimensions of this rectangle?

(Chapter 13) If |r(t)| = 1 for all t, then r'(t) is orthogonal to r(t) for all t.

Answers

The statement is true. This means that r'(t) is orthogonal (perpendicular) to r(t) for all t.

If |r(t)| = 1 for all t, then r(t) is a unit vector for all t. Differentiating both sides of this equation with respect to t, we get:

|r(t)|' = 0

Using the chain rule and the fact that the magnitude of a vector is the square root of the dot product of the vector with itself, we have:

|r(t)|' = (r(t) · √r(t))

= (2r(t) · r'(t)) / (2|r(t)|)

= r(t) · r'(t) / |r(t)|

= r(t) · r'(t)

Since |r(t)|' = 0, we have:

r(t) · r'(t) = 0

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pls hlp thx your the best

Answers

The total cost for 13 rides is 46 $.

Amount to be paid as entry fee = 20 $

Amount to be paid in each ride = 2 $

Therefore, total cost for 13 rides, P = 20 + (2 x 13)

P = 46 $

Total cost for r rides, P' = (20 + 2r) $

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Factor out the greatest common factor from the following polynomial.
10b³ +5b² +4
I
OA. 10b³ +5b²+4=
3
(Factor completely.)
OB. The polynomial has no common factor other than 1.

Answers

The polynomial 10b³ + 5b² + 4 has no common factor other than 1.

Factoring out the greatest common factor from the polynomial.

From the question, we have the following parameters that can be used in our computation:

10b³ + 5b² + 4

The terms of the above expressions are

10b³, 5b² and 4

The terms of the above polynomial have no common factor other than 1.

Hence, the polynomial cannot be factored

Answers

The standard form is y= 3x² - 16x + 132.

We have Equation,

f(x) = 3(x-8)² - 160

We know the standard form is

y= ax² + bx + c

Now, converting vertex form to standard form

y = 3(x-8)² - 160

y = 3 (x² + 64 - 16x )-60

y= 3x² + 192 - 16x -60

y= 3x² - 16x + 132

Thus, the standard form is y= 3x² - 16x + 132.

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Please help me with this homework only the answer

Answers

Answer:

[tex]\frac{5}{8}[/tex]

Step-by-step explanation:

[tex]\frac{-8-(-3)}{-12-(-4)}[/tex] = [tex]\frac{-8+3}{-12+4}[/tex] = [tex]\frac{-5}{-8}[/tex] = [tex]\frac{5}{8}[/tex]

Helping in the name of Jesus.

Find the area of the shaded region.
80⁰
5 cm
A≈ [?] cm²
Enter a decimal rounded to the nearest tenth.

Answers

The area of the shaded region for the circle is derived to be equal to 73.4 square centimeters.

How to evaluate for the area of shaded region

The shaded region is the segment area in the circle, so it is derived by subtracting the area of the segment from the area of the circle as follows:

area of the circle = 22/7 × 5 cm × 5 cm

area of the circle = 78.57 cm²

area of sector = 80/360 × 22/7 × 5 cm × 5 cm

area of sector = 17.46 cm²

area of triangle = 5 cm × sin40° × 5 cm × cos40°

area of triangle = 12.31 cm²

area of the segment = 17.46 cm² - 12.31 cm²

area of the segment = 5.15 cm²

area of the shaded region = 78.57 cm² - 5.15 cm²

area of the shaded region = 73.42 cm²

Therefore, the area of the shaded region for the circle is derived to be equal to 73.4 square centimeters.

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Use the image to determine the direction and angle of rotation

Answers

Answer:

180

Step-by-step explanation:

rotate it 90 degrees 2 times

List the sample space for rolling a fair seven-sided die.

S = {1, 2, 3, 4, 5, 6, 7}
S = {1, 2, 3, 4, 5, 6, 7, 8}
S = {1}
S = {7}

Answers

The sample space for rolling a fair seven-sided die is S = {1, 2, 3, 4, 5, 6, 7}.

Given that,

A fair seven sided die is rolled.

We have to find the sample space of the rolling.

A sample space is a set of all the possible outcomes in a random experiment. It is usually denoted by the letter, S.

The subset of the sample space are events.

The die has the numbers marked from 1 to 7.

So when we roll the die,

The possible numbers are 1, 2, 3, 4, 5, 6, 7

Sample space = {1, 2, 3, 4, 5, 6, 7}

Hence the sample space is {1, 2, 3, 4, 5, 6, 7}.

Learn more about Sample Space here :

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what is the area of a equilateral triangle with radius 8√ 3

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Answer:

Answer and Explanation: The side length of the equilateral triangle is 8√3 . Therefore, the area of the equilateral triangle is 48√3 square units 48 3 square units .

Step-by-step explanation:

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