LQ - 10.4 Areas in Polar Coordinates Show all work and use proper notation for full credit. Find the area of the region enclosed by one loop of the curve. • Include a sketch of the entire curve. r = 4cos (20) LQ - 10.3 Polar Coordinates Show all work and use proper notation for full credit. Find the slope of the tangent line to the given polar curve at the point specified by the value of e. TT r = 1-2sine, =
Answers
The area of the region enclosed by one loop of the curve r = 4cos(θ) is 4 square units. The slope of the tangent line to the polar curve r=1 - 2sin(θ) at θ = π/4 is 2 + √2.
Area of region enclosed by one loop of the curve r = 4cos(2θ)
The curve r = 4cos(2θ) has two loops, and we need to find the area of one loop, which is from θ = 0 to θ = π/4.
To find the area, we use the formula for the area enclosed by a polar curve
A = (1/2) ∫[a,b] r^2 dθ
where r is the polar function, and a and b are the angles of the region we want to find the area for.
So, the area of one loop is
A = (1/2) ∫[0,π/4] (4cos(2θ))^2 dθ
= 8 ∫[0,π/4] cos^2(2θ) dθ
Using the identity cos(2θ) = (cos^2θ - sin^2θ), we can rewrite the integrand as
cos^2(2θ) = (cos^2θ - sin^2θ)^2
= cos^4θ - 2cos^2θsin^2θ + sin^4θ
= (1/2) (1 + cos(4θ)) - (1/2) sin^2(2θ)
So, the integral becomes
A = 8 ∫[0,π/4] [(1/2) (1 + cos(4θ)) - (1/2) sin^2(2θ)] dθ
= 4 [θ/2 + (1/8)sin(4θ) - (1/4)θ - (1/8)sin(2θ)]|[0,π/4]
= 1 + (2/π)
Therefore, the area of one loop of the curve r = 4cos(2θ) is 1 + (2/π).
Slope of tangent line to the polar curve r = 1-2sinθ at θ = π/4
To find the slope of the tangent line, we need to take the derivative of the polar function with respect to θ:
dr/dθ = -2cosθ
Then, we can use the formula for the slope of the tangent line in polar coordinates
dy/dx = (dy/dθ) / (dx/dθ) = (r sinθ) / (r cosθ) = tanθ + r dθ/dθ
At the point specified by θ = π/4, we have
r = 1 - 2sin(π/4) = 1 - √2/2 = (2 - √2)/2
dθ/dθ = 1
So, the slope of the tangent line is
dy/dx = tan(π/4) + r dθ/dθ
= 1 + (2 - √2)/2
= (4 + 2√2)/2
= 2 + √2
Therefore, the slope of the tangent line to the polar curve r = 1-2sinθ at θ = π/4 is 2 + √2.
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Related Questions
A large moving box has a volume of 45 cubic meters. The width of the box is 1. 5 meters. The length and height of the box are each whole number measurements that are greater than 2 meters. What could be the dimensions of the box? Give TWO possible answers
Answers
If A large moving box has a volume of 45 cubic meters then two possible sets of dimensions for the box are: 1.5 m x 5 m x 9 m, and 1.5 m x 3 m x 15 m.
One possible way to approach this problem is to use trial and error. Therefore, two possible sets of dimensions for the box are 1.5 m x 5 m x 9 m, and 1.5 m x 3 m x 15 m.
We know that the volume of the box is 45 cubic meters and that the width is 1.5 meters. We want to find two whole numbers for the length and height that work.
We can start by listing the factors of 45: 1, 3, 5, 9, 15, and 45. We can then try each of these factors as the length or height, and see if the other dimension is a whole number greater than 2.
For example, if we try length = 5, then the height would need to be 9 to get a volume of 45. However, the width would be 1.5, which is already less than 2, so this doesn't work. These are the dimensions.
Trying again, if we try length = 9, then the height would need to be 5 to get a volume of 45. In this case, the width would be height = 5, which is greater than 2, so this is a possible answer.
Continuing, if we try length = 3, then the height would need to be 15 to get a volume of 45. This gives us a width of 1.5, which is less than 2, so this doesn't work.
Finally, if we try length = 15, then the height would need to be 3 to get a volume of 45. This gives us a width of height = 3, which is greater than 2, so this is another possible answer.
Therefore, two possible sets of dimensions for the box are: 1.5 m x 5 m x 9 m, and 1.5 m x 3 m x 15 m.
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Jennifer had 7/8 of her pan of macoroni and chese left after supper. The next day she split what was left evenly between her five kids. What fraction of the total pan did each of them get
Answers
each of Jennifer's five kids received 1/40 of the total pan of macaroni and cheese.
What is equivalent ratio?
The concept of a ratio in mathematics is the divisional comparison of two quantities, the antecedent and consequent. As an illustration, each ingredient must be added according to a ratio during cooking. So, we may argue that a ratio is employed to represent one quantity as a portion of another. The ratio can be written as a fraction as well. If the ratio a:b is a fraction, its form is a/b. As a result, it is simple to compare two or more equivalent ratios expressed as equivalent fractions.
If Jennifer had 7/8 of her pan of macaroni and cheese left after supper, this means she had 1 - 7/8 = 1/8 of the pan remaining.
To split the remaining macaroni and cheese evenly between her five kids, we need to divide 1/8 by 5.
1/8 divided by 5 can be written as (1/8) ÷ 5 = (1/8) x (1/5) = 1/40.
Therefore, each of Jennifer's five kids received 1/40 of the total pan of macaroni and cheese.
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James and Padma are on opposite sides of a 100-ft-wide canyon. James sees a bear at an angle of depression of 45 degrees. Padma sees the same bear at an angle of depression of 65 degrees.
What is the approximate distance, in feet, between Padma and the bear?
A
21. 2ft
B
75. 2ft
C
96. 4ft
D
171. 6ft
Answers
The approximate distance between Padma and the bear is 21.2 ft, which corresponds to option A.
The approximate distance between Padma and the bear, we can use trigonometry. Since James and Padma are on opposite sides of the 100-ft-wide canyon,
we can form two right triangles with the bear's position as one of the vertices.
Step 1: Determine the horizontal distance from James to the bear.
Since the angle of depression from James to the bear is 45 degrees, the horizontal distance (x) and vertical distance (y) are equal due to the properties of a 45-45-90 right triangle. Therefore, x = y. Since the canyon is 100 ft wide, x + y = 100 ft. We can solve for x:
x + x = 100
2x = 100
x = 50 ft
Step 2: Determine the vertical distance from James to the bear.
Since x = y in the 45-45-90 right triangle, the vertical distance from James to the bear is also 50 ft.
Step 3: Determine the horizontal distance from Padma to the bear.
We can now use Padma's angle of depression, 65 degrees, to find the horizontal distance (p) from Padma to the bear. Using the tangent function:
tan(65) = vertical distance / horizontal distance
tan(65) = 50 ft / p
Solving for p:
p = 50 ft / tan(65) ≈ 21.2 ft
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Add the polynomials. (5x3+x)+(3x3+8) enter the answer in the box, in standard form (highest exponent to lowest).
Answers
The sum of the polynomials (5x³ + x) + (3x³ + 8) is 8x³ + x + 8
(5x³ + x) + (3x³ + 8) can be simplified by adding the coefficients of the like terms. The like terms are 5x³ and 3x³, which can be combined to give 8x³. The single terms are x and 8, which can be combined to give 8 + x. Therefore, the polynomials add up to
8x³ + x + 8
In standard form (highest exponent to lowest), the answer is:
8x³ + x + 8
Therefore, the sum of the polynomials (5x³ + x) + (3x³ + 8) is 8x³ + x + 8 in standard form (highest exponent to lowest)
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8 girls eat a total of 210 candies. after adding the number of candies eat by the ninth girl, the average number of candies eaten became 29. how many did the 9th girl eat?
Answers
The 9th girl ate 51 candies.
What is the number of candies eaten by the 9th girl, if the average number of candies eaten by 9 girls is 29 and the first 8 girls ate a total of 210 candies?Let the number of candies eaten by the 9th girl be x.
The average number of candies eaten by 8 girls is given as (210/x+210)/8, which simplifies to 210/8 + x/8.
After the 9th girl eats x candies, the total number of candies eaten becomes 210 + x.
The new average is given as (210 + x)/9 = 29.
Solving for x, we get:
210 + x = 261
x = 51
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The height of three basketball players is 210 cm, 220 cm and 191 cm.
What is their average height?
Answers
Answer:
Average height = 207 cmStep-by-step explanation:
It's given that, The height of three basketball players is 210 cm, 220 cm and 191 cm.
h₁ = 210 cm
h₂ = 220 cm
h₃ = 191 cm
Total number of observations = 3.
Average = Sum of all observations/Total number of observations.
↠ Average = h₁ + h₂ + h₃/3
↠ Average = 210 +220 + 191 /3
↠ Average = 430 + 191/3
↠ Average = 621/3
↠ Average = 207
Therefore, The required average height is 207 cm.
The table shows the amount of money raised during a car wash for charity.
Number of Cars Washed Money Raised
3 $43.50
13 $279.50
18 $405.00
Which statement is true?
A. The group raised $14.50 per car.
B. The group raised $21.50 per car.
C. The group raised $22.50 per car.
D. The relationship is not a direct proportion.
Answers
Answer:
The correct answer is D. The relationship is not a direct proportion.
We can see that the money raised is not directly proportional to the number of cars washed. For example, when the number of cars washed is doubled from 3 to 13, the money raised is not doubled from $43.50 to $87.00. Instead, it is increased by a factor of 6.5, from $43.50 to $279.50. Similarly, when the number of cars washed is increased by 5 from 13 to 18, the money raised is increased by a factor of 1.4, from $279.50 to $405.00.
This suggests that the amount of money raised is not simply a linear function of the number of cars washed. Instead, it is likely a more complex function that takes into account other factors, such as the time of day, the weather, and the location of the car wash.:
A circle in the xy-coordinate plane has the equation
�
2
+
�
2
+
6
�
−
4
=
0
x
2
+y
2
+6y−4=0 . If the equation of the circle in written in the form
�
2
+
(
�
+
�
)
2
=
�
x
2
+(y+k)
2
=c , where k and c are constants, what is the value of k?
Answers
Answer:
Given the equation of the circle in the xy-coordinate plane as x^2 + y^2 + 6x - 4 = 0, we need to rewrite it in the form (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
Completing the square for x^2 + 6x, we get (x+3)^2 - 9. Thus, the equation becomes (x+3)^2 + y^2 - 9 = 4. Rearranging, we get (x+3)^2 + y^2 = 13.
Comparing with the required form, we get (x-h)^2 + (y-k)^2 = r^2, where h = -3, k = 0 and r^2 = 13. Thus, the value of k is 0.
A car dealership has 98 cars on its lot. Fifty-five of the cars are new. Of the new cars, 36 are domestic cars. There are 15 used foreign cars on the lot. Organize this information in a two-way table. Include the marginal frequencies
Answers
Here is a two-way table that summarizes the information:
The marginal frequencies (totals) are shown in the last row and last column. The dealership has a total of 98 cars on its lot, which is the sum of the new and used cars. There are 55 new cars and 15 used cars, which is the sum of the domestic and foreign cars in each category.
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Andrew went deep sea diving with some friends. If he descends at a rate of 4 feet per minute, what integer represents Andrews depth in ¼ of an hour?
Answers
The integer that represents Andrews depth in ¼ of an hour is 60 feet.
How to determine what integer represents Andrews depth in 1/4 of an hour?Word problems are sentences describing a 'real-life' situation where a problem needs to be solved by way of a mathematical calculation e.g. calculation of length and depth.
If Andrew descends at a rate of 4 feet per minute and we want to find his depth in ¼ of an hour.
1/4 of an hour = (1/4 * 60) minutes = 15 minutes
Thus, the integer that represents Andrews depth in ¼ of an hour will be:
(4 feet per minute) * (15 minutes) = 60 feet
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You saved $3 during Week 1, $6 during Week 2, $12 during Week 3, and $21 during Week 4. If the pattern continues, how much money will you save during Weeks 8 and 9 combined?
Answers
Felipe has several 2 liter bottles of lemonade. He wants to pour out 12 glasses for him and his friends each glass holds 500 milliliter of lemonade. How many two liter bottles will he need for all 12 glasses
Answers
Felipe needs a total of 6 liters of lemonade for all 12 glasses. Felipe will need a total of 3 two-liter bottles of lemonade to pour out 12 glasses for him and his friends.
Determining the total amount of lemonade needed:
12 glasses x 500 milliliters per glass = 6,000 milliliters.
Converting the total amount of lemonade needed to liters:
6,000 milliliters / 1,000 milliliters per liter = 6 liters.
Dividing the total amount of lemonade needed by the amount in each bottle:
6 liters / 2 liters per bottle = 3 bottles.
Therefore, Felipe will need 3 two-liter bottles of lemonade to pour out 12 glasses for him and his friends.
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HURRY PLEASE!!!!
The number of bacteria in a culture quadruples every hour. There were
65,536 bacteria in the culture at 8:00 A. M. The expression 65,536 4h models
the number of bacteria in the culture h hours after 8:00 A. M.
a. What is the value of the expression for h= -4?
b. What does the value of the expression in part (a) represent
Answers
The value of the expression 65,536 x 4h for h = -4 is 256. The value represents the number of bacteria in the culture 4 hours before 8:00 A.M.
To find the value of the expression for h = -4, we substitute -4 for h in the expression [tex]65536*4^{h}[/tex]. This gives us
65536*4⁻⁴
To simplify this, we need to evaluate the exponent first. Remember that a negative exponent means we take the reciprocal of the base raised to the positive version of the exponent. So 4⁻⁴ is the same as 1/(4⁴), or 1/256.
Substituting that back into the expression, we get
65536*(1/256) = 256
So the value of the expression for h = -4 is 256.
The value of the expression in part (a) represents the number of bacteria in the culture 4 hours before 8:00 A.M. Since h is negative, we are looking at a time before 8:00 A.M. Specifically, h = -4 means we are looking at 4 hours before 8:00 A.M., or 4:00 A.M.
So the expression 65536*4⁻⁴ tells us how many bacteria were in the culture at 4:00 A.M., assuming the bacteria quadrupled every hour from that point until 8:00 A.M.
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The cookies raina wants comes in packs of 6 she needs 23 of these for a party how many packs should she buy
Answers
Based on mathematical operations, since the cookies Raina wants for a party comes in packs of 6 and she needs 23 of these for the party, she should buy 4 packs.
What are the mathematical operations?The basic mathematical operations include addition, subtraction, multiplication, and division.
In this situation, we can determine the number of packs required by finding a number that when multiplied by 6 will be close to 23.
We know that 6 x 4 equals 24, which is the closest value to 23.
The number of cookies in each pack = 6
The total number of cookies Raina needs = 23
The number of packs to buy = 4 (6 x 4 = 24)
Thus, Raina needs to buy 4 packs of the cookies to satisfy her requirement for 23 pieces.
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Digitization positional errors may be less (say 2 meters) than the required positional accuracy of the data (say 5 meters) yet still prevent:
Answers
it is crucial to ensure that data collection methods and instruments meet the required accuracy standards.
How can Digitization positional errors can still be prevented?
Digitization positional errors can still prevent accurate analysis or decision making even if they are less than the required positional accuracy of the data. This is because the accuracy of the output or results depends on the accuracy of the input data.
For example, suppose a GPS receiver is used to collect data on the location of a pipeline, and the positional accuracy requirement is 5 meters. However, due to various factors such as signal interference or poor satellite coverage, the receiver only achieves an accuracy of 2 meters. Even though the positional error is less than the required accuracy, the resulting data may still be insufficient for the intended purpose, such as accurately identifying potential hazards or planning maintenance activities.
Inaccurate data can lead to wrong decisions, increased risks, and financial losses. Therefore, it is crucial to ensure that data collection methods and instruments meet the required accuracy standards.
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Digitization positional errors may be less (say 2 meters) than the required positional accuracy of the data (say 5 meters) yet still prevent achieving the desired level of accuracy for the intended use of the data.
Find the surface area of this cone please help
Answers
The calculated value of the surface area of the cone is 36π
Calculating the surface area of the cone
From the question, we have the following parameters that can be used in our computation:
Radius, r = 4 meters
Slant height, l = 5 meters
using the above as a guide, we have the following:
SA = πr(r + l)
Substitute the known values in the above equation, so, we have the following representation
SA = π * 4 * (4 + 5)
Evaluate
SA = 36π
Hence, the surface area of the cone is 36π
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A prism 5 feet tall whose base is a right triangle with leg lengths 6 feet and 7 feet
what is the volume in cubic feet?
Answers
The volume of the prism is 21 * 5 = 105 cubic feet.
To find the volume of a prism with a triangular base, you need to follow these steps:
1. Determine the area of the triangular base: Since the base is a right triangle with leg lengths of 6 feet and 7 feet, you can use the formula for the area of a right triangle: (1/2) * base * height. In this case, the area would be (1/2) * 6 * 7 = 21 square feet.
2. Multiply the area of the triangular base by the height of the prism: The prism is 5 feet tall, so the volume can be calculated by multiplying the area of the base (21 square feet) by the height (5 feet).
Thus, the volume of the prism is 21 * 5 = 105 cubic feet.
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7 NEXT QUESTION e = READ NEXT SECTION G # O ASK FOR HELP 2 e By what percentage did the median earnings of college degreed exceed that of high school degreed for 1973 for men (to the nearest tenth)? 2 3 TURN IT IN
Answers
The percentage by which the median earnings of college degree exceed that of high school degreed for 1973 for men is 17.9%
Why is this so?College: Women= 4400
H.School: Women = 3300
Solving we have
The base number is the high school women.
The difference is 4400 - 3300 = 1100
So the % = (1100/3300) * 100% = 33.3%
1973
The base number is again high school 5600
Difference: 6600 - 5600 = 1000
% = (1000/5600) * 100% = 17.9
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Full Question:
Although part of your question is missing, you might be referring to this full question:
See the aattached image.
Question 5
Not yet answered
An ice-cream parlor used a scatterplot to record their total sales, in dollars, each day (s) and
the corresponding average temperature, in ºf, on that day (t). They then found a trend line of
this data to be s = 12. 75t + 32. What is the predicted total sales the ice-cream parlor makes
if the average temperature of the day is 72°f?
Marked out of
1. 00
P Flag question
O a.
$950. 00
O b. $820. 00
O c. $1,275. 00
O d. $740. 00
Answers
The predicted total sales for the ice-cream parlor when the average temperature is 72°F is $950.00.
You are asked to find the predicted total sales (s) for the ice-cream parlor when the average temperature (t) is 72°F, using the trend line equation s = 12.75t + 32.
Step 1: Plug the given temperature (72°F) into the trend line equation:
s = 12.75(72) + 32
Step 2: Calculate the value of 12.75(72):
12.75 * 72 = 918
Step 3: Add 32 to the result from Step 2:
918 + 32 = 950
So, the predicted total sales for the ice-cream parlor when the average temperature is 72°F is $950.00. Therefore, the correct answer is (a) $950.00.
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Help with problem in photo
Answers
Answer:
(x-6)²+(y+3)²=17²
Step-by-step explanation:
The equation of a circle with center (a,b) and radius r is given by the formula:
(x - a)² + (y - b)² = r²
In this case, the center of the circle is (6,-3), and it passes through the point (-9,5). To find the radius of the circle, we need to calculate the distance between the center and the point on the circle. Using the distance formula, we get:
r = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(-9 - 6)² + (5 - (-3))²]
= √[225 + 64]
= √289
= 17
So, the radius of the circle is 17. Now we can plug in the values for the center and radius into the equation of a circle:
(x - 6)² + (y + 3)² = 17²
The following two-way table shows the distribution of daily traffic and weather issues in a certain large city.
A 4-column table with 3 rows. Column 1 has entries bad weather, good weather, total. Column 2 is labeled heavy traffic with entries 25, 55, 80. Column 3 is labeled Light traffic with entries 5, 15, 20. Column 4 is labeled Total with entries 30, 75, 100. The columns are titled daily traffic and the rows are titled weather conditions.
Suppose a day from this city is selected at random. Let event A = heavy traffic and event B = bad weather. Are events A and B independent?
No, P(A) = P(B|A).
No, P(A) ≠ P(A|B).
Yes, P(A) = P(A|B).
Yes, P(A) ≠ P(B|A)
Answers
The correct answer is "No, P(A) ≠ P(A|B)".
How to solveIf the occurrence of one event has no bearing on the likelihood that the other will also occur, then the two events are independent.
The likelihood that both occurrences A and B will occur is defined mathematically as being equal to the product of the probabilities of A and B (i.e., P) (A and B).
Let event A = heavy traffic and event B = bad weather.
P(A)= 25/30 and P(B)=25/30
Two events are independent if the occurrence of one does not change the probability of the occurrence of the other.
This means P(A) = P(A|B) or P(B) = P(B|A).
For event A (heavy traffic), P(A) = 80/100 = 0.8.
For event B (bad weather), P(A|B) = 25/30 = 0.833.
Since P(A) ≠ P(A|B), events A and B are not independent.
So, the correct answer is "No, P(A) ≠ P(A|B)".
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Which is the most precise measurement?
4 yd., 9 ft., 638
6
3
8
ft., or 1518
15
1
8
yd.
Answers
The most precise measurement is the one with the smallest value, which is 9 ft.
To determine which is the most precise measurement among 4 yd., 9 ft., 638638 ft., or 15181518 yd., we first need to convert all the measurements to a common unit. Let's convert everything to feet:
1 yard = 3 feet
- 4 yd. = 4 (3 ft). = 12 ft.
- 9 ft. = 9 ft. (no conversion needed)
- 638638 ft. = 638638 ft. (no conversion needed)
- 15181518 yd. = 15181518 (3 ft.) = 45544554 ft.
Now that we have all the measurements in feet, we can compare them:
- 12 ft.
- 9 ft.
- 638638 ft.
- 45544554 ft.
The most precise measurement is the one with the smallest value, which is 9 ft.
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2n-1/3=n+2/2 please help me
Answers
[tex] \sf \longrightarrow \: \frac{2n - 1}{3} = \frac{n + 2}{2} \\ [/tex]
[tex] \sf \longrightarrow \: 2( 2n - 1) = 3(n + 2) \\ [/tex]
[tex] \sf \longrightarrow \: 4n - 2 = 3n +6 \\ [/tex]
[tex] \sf \longrightarrow \: 4n = 3n +6 + 2\\ [/tex]
[tex] \sf \longrightarrow \: 4n - 3n= 6 + 2\\ [/tex]
[tex] \sf \longrightarrow \: 1n= 6 + 2\\ [/tex]
[tex] \sf \longrightarrow \: n= 8\\ [/tex]
[tex] \longrightarrow { \underline{ \overline{ \boxed{ \sf{\: \: \: n= 8 \: \: \: }}}}} \: \: \bigstar\\ [/tex]
Help me please first correct answer get branliest and please no essay
Answers
Answer:
6cm
Step-by-step explanation:
all the sides of a square are identical so we can assume the missing side as x
since all four sides are the same and the perimeter is the sum of all sides, we can write
x+x+x+x=24cm
4x=24cm
x=24/4
x=6cm
UNO is card game: standard UNO deck consist of 108 cards four each of Wild and Wild Draw Four; and 25 each of four different colors (red yellow; green, blue): Each color consists of number cards (one zero, two each of through 9) and two cards each of Skip, Draw Two and Reverse_ These last three types are known as action cards. To begin the game (using well shufiled deck) seven card hand is dealt to each player (without replacement) . Consider an outcome space without order Without worrying about how many players are there, find the probability of a player starting of with: a whole hand of action cards_ b a hand of cards with out any Wild (Wild or Wild Draw Four) cards. Explain your answer by giving clear description of an equally-likely outcomes model on which it is based_ In other words, tell me what 0 and events you are using; how many elements they have?
Answers
Therefore, a person starts with a hand of cards without any Wild (Wild or Wild draw four) is 0.5741
and a number of outcomes of event 'B' is 16007560800.
How to solvetotal number of cards = N = 108
wild cards = 4
wild draw four = 4
we have 4 different colors and each color has 25 cards with break down as
zero numbered card = 1
numbered from 1 to 9 = 2 each (total 18)
skip = 2
draw two = 2
reverse = 2
skip, draw two and reverse are action cards
therefore, in a deck of UNO we have 3*2*4 = 24 action cards
a seven-card hand is dealt to each player from a well shuffled pack of card.
therefore, as each card is equally likely, we have total possible outcomes as 108C7
Ω is a event of every possible outcome and it has 108C7 = 27883218168 elements
an equally likely model is a model where equal weights are attached to every outcome and thus the probability of each outcomes become equal.
1) let 'A' be the event a whole hand is of action cards.
we have a total of 24 action cards in a UNO deck of 108 cards.and we have to draw a hand of 7 cards
total number of outcomes for event 'A' is 24C7
therefore, the probability that a player starting with a whole hand of action cards is
= [tex]24C7\n108C7\n[/tex]
= 0.0000124
therefore, a person starts with a whole hand of action card is 0.0000124
and the number of outcomes of event 'A' is 346104
2) let 'B' be the event that a hand of card is without any wild card
therefore we have a total of 100 cards that do not have any wild ( wild or wild draw four) cards
therefore a hand of 7 out of 100 cards for event 'B' can be drawn in 100C7 total ways
therefore, the probability that a player starts with a hand of cards without any Wild (Wild or Wild draw four) cards is
= [tex]100C7\n108C7\n[/tex]
= 0.5741
therefore, a person starts with a hand of cards without any Wild (Wild or Wild draw four) is 0.5741
and the number of outcomes of event 'B' is 16007560800.
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Select allá transformations that Will map a pentágon onto itself
Answers
There are several transformations that can be applied to a pentagon in order to map it onto itself. One such transformation is a rotation of 72 degrees, which can be performed by rotating the pentagon about its center point by 72 degrees clockwise. This will result in the pentagon appearing exactly as it did before the rotation, but in a different orientation.
Another transformation that will map a pentagon onto itself is a reflection along one of its symmetry lines. A pentagon has five symmetry lines, which are lines that divide the shape into two congruent halves. Reflecting the pentagon along any of these lines will result in the same shape being produced, but in a mirror image orientation.
Finally, a translation can also be used to map a pentagon onto itself. This involves moving the pentagon a certain distance in a particular direction, such as shifting it 2 units to the right or 3 units upwards. As long as the distance and direction of the translation are such that the pentagon ends up exactly where it started, it will be a valid transformation.
Overall, there are several transformations that can be applied to a pentagon in order to map it onto itself, including rotations, reflections, and translations.
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Find the derivative of the given function.
y= (4x² – 9x) e⁻⁴
ˣy' = ... (Type an exact answer.)
Answers
The derivative of the given function is:
y' = (8x – 9) e⁻⁴ - 16x (4x - 9) e⁻⁴
Find the derivative?
To find the derivative of the given function y= (4x² – 9x) e⁻⁴, we need to use the product rule of differentiation. The formula for the product rule is:
(fg)' = f'g + fg'
Where f and g are two differentiable functions. Applying this formula, we get:
y' = (4x² – 9x)' e⁻⁴ + (4x² – 9x) (e⁻⁴)'
The first term on the right-hand side can be simplified using the power rule and the constant multiple rule of differentiation:
(4x² – 9x)' = 8x – 9
The second term on the right-hand side requires the chain rule of differentiation. Let u = -4x, then we have:
(e⁻⁴)' = (e^u)' = e^u (-4) = -4e⁻⁴x
Substituting these results back into the expression for y', we get:
y' = (8x – 9) e⁻⁴ + (4x² – 9x) (-4e⁻⁴x)
Simplifying this expression, we get:
y' = (8x – 9) e⁻⁴ - 16x (4x - 9) e⁻⁴
Therefore, the derivative of the given function is:
y' = (8x – 9) e⁻⁴ - 16x (4x - 9) e⁻⁴
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Suppose there is a simple index of two stocks, stock A and stock B. Stock A
opens on Monday with 10,000 shares at $5. 50 per share. Stock B opens on
Monday with 8000 shares at $6. 25 per share. Stock A opens on Tuesday at
$5. 80 per share, and stock B opens on Tuesday at $6. 65 per share. Both
stocks have the same number of shares that they opened with on Monday.
What is the rate of change of this simple index over 1 day?
I
Answers
To calculate the rate of change of the simple index over 1 day, we need to first calculate the index value for Monday and Tuesday.
On Monday, the value of stock A is 10,000 x $5.50 = $55,000, and the value of stock B is 8,000 x $6.25 = $50,000. The total value of the index on Monday is $55,000 + $50,000 = $105,000.
On Tuesday, the value of stock A is 10,000 x $5.80 = $58,000, and the value of stock B is 8,000 x $6.65 = $53,200. The total value of the index on Tuesday is $58,000 + $53,200 = $111,200.
To calculate the rate of change, we can use the formula:
(rate of change) = (new value - old value) / old value x 100%
Using this formula, we get:
(rate of change) = ($111,200 - $105,000) / $105,000 x 100% = 5.90%
Therefore, the rate of change of this simple index over 1 day is 5.90%.
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You are at the beach with your friends. You have brought some supplies to make sand castles. These supplies include a pail that has a base with a circumference of 87 inches, is 12 inches tall, and has an opening on top that is twice the diameter of the base. You also have a plastic pyramid mold that has a square base with an edge that measures 6 inches and is 7 inches tall, and an empty soup can with a diameter of 5. 25 inches and is 6. 5 inches tall
Answers
The opening of the pail has a diameter of approximately 55.4 inches (since it's twice the diameter of the base).
Based on the information you provided, it sounds like you have some great supplies for making sand castles at the beach with your friends.
Firstly, let's take a look at the pail. You mentioned that it has a base with a circumference of 87 inches, which means that the diameter of the base is approximately 27.7 inches (since circumference = pi x diameter). The pail is also 12 inches tall and has an opening on top that is twice the diameter of the base. Therefore, the opening has a diameter of approximately 55.4 inches (since it's twice the diameter of the base). With these measurements, you can use the pail to make some pretty big sand castles!
Next, you mentioned a plastic pyramid mold that has a square base with an edge that measures 6 inches and is 7 inches tall. This mold should be perfect for making pyramid-shaped sand castles. Just fill it with sand, pack it down, and carefully remove the mold to reveal your pyramid!
Finally, you mentioned an empty soup can with a diameter of 5.25 inches and is 6.5 inches tall. This can could be used to make cylindrical shapes in your sand castles. Simply pack sand around the can, press down firmly, and carefully remove the can to reveal your cylinder.
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What is the volume of the rectangular prism?a prism has a length of 8 inches, width of 2 inches, and height of 12 and one-half inches.16 in.322 and one-half in.3200 in.3 212 and one-half in.3
Answers
The volume of the rectangular prism is 200 in³.
The volume of a rectangular prism is found by multiplying its length, width, and height. In this case, the prism has a length of 8 inches, width of 2 inches, and height of 12 and one-half inches. To calculate the volume, you would use the following formula:
Volume = Length × Width × Height
Now, plug in the given dimensions:
Volume = 8 in × 2 in × 12.5 in
Perform the calculations:
Volume = 16 in² × 12.5 in
Volume = 200 in³
So, the rectangular prism has a volume of 200 cubic inches. This means that the space occupied by the prism is equal to 200 cubic inches. The other options provided, such as 16 in³, 22 and one-half in³, and 212 and one-half in³, are not correct because they do not represent the product of the length, width, and height of the given prism. In conclusion, the correct answer for the volume of this rectangular prism is 200 in³.
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