The moment of inertia (I) is a measure of an object's resistance to rotational motion around an axis. The moment of inertia of a cylinder can be calculated using the following formula:
I = (1/2) × m × r²
where:
I is the moment of inertia of the cylinder
m is the mass of the cylinder
r is the radius of the cylinder.
Here are the steps to calculate the moment of inertia for a cylinder:
Determine the mass of the cylinder. This can be done by weighing the cylinder on a scale or by using its density and volume. The formula for the mass of a cylinder is:
m = density × volume
Measure the radius of the cylinder. This is the distance from the center of the cylinder to its outer edge.
Substitute the values of m and r into the moment of inertia formula:
I = (1/2) × m × r²
Calculate the moment of inertia using a calculator.
Note that the moment of inertia of a hollow cylinder is different from that of a solid cylinder. To calculate the moment of inertia of a hollow cylinder, you need to subtract the moment of inertia of the hollow space from the moment of inertia of the solid cylinder using the parallel axis theorem.
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if you borrow $500 for 4 years at an annual interest rate of 6 years how much will you pay altogether
The amount will be after 4 years at interest rate of 6%, $624.76
To calculate the total amount you will pay after borrowing $500 for 4 years at an annual interest rate of 6%, you can use the formula for compound interest:
A = P x (1 + r/n)^(nt)
where:
A is the total amount,
P is the principal amount (the initial amount borrowed),
r is the annual interest rate as a decimal,
n is the number of times interest is compounded per year,
t is the number of years the money is borrowed for.
In this case, P = $500, r = 6%, n = 1 and t = 4.
So, the formula becomes:
A = $500 x (1 + 0.06/1)^(1 x 4)
A = $500 x (1.06)⁴
A = $500 x 1.249536
A = $624.76
Therefore, you will pay a total of $624.76 after borrowing $500 for 4 years at an annual interest rate of 6%.
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Sarah’s dog ran into the woods. While in the woods,Sarah’s dog picked up 3 sticks,2 rocks, and an amount of leaves. Write an algebraic expression
The woods, Sarah’s dog picked up 3 sticks, 2 rocks, and an amount of leaves. An algebraic expression for this is, 3 + 2 + x
What are expressions?An expression is a sentence with at least two numbers or variables having mathematical operation. Math operations can be addition, subtraction, multiplication, division.
For example, 2x+3
The total number of items that Sarah's dog picked up in the woods can be represented by the algebraic expression:
3 + 2 + x
where x represents the number of leaves the dog picked up.
The expression 3 + 2 + x represents the sum of the 3 sticks, 2 rocks, and the unknown number of leaves that the dog picked up.
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A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 460 pounds of phosphoric acid, 960 pounds of nitrogen, and 220 pounds of potash. If x is the number of cubic yards of mix A used and y is the number of cubic yards of mix B used, write a system of linear inequalities that indicates appropriate restraints on x and y. Find the set of feasible solutions graphically for the amounts of mix A and mix B that can be used.
Graphically, the feasible solutions represent the points in the x-y plane that are inside the region defined by the three lines, and they can be found by plotting the three lines and shading the region below each line. The feasible solutions are the points in the shaded region that are common to all three.
Let x be the number of cubic yards of mix A used, and y be the number of cubic yards of mix B used. Then the total amount of phosphoric acid, nitrogen, and potash can be expressed as follows:
Phosphoric acid: 20x + 10y >= 460
Nitrogen: 30x + 30y >= 960
Potash: 5x + 10y >= 220
These inequalities represent the minimum monthly requirements for phosphoric acid, nitrogen, and potash. To find the feasible solutions, we need to find the values of x and y that satisfy all three inequalities simultaneously. This can be done graphically by plotting the three inequalities on the same coordinate plane and finding the region that is common to all three.
To plot the inequalities, we can choose convenient values for x and y, substitute them into the inequalities, and find the corresponding values of the right-hand side. For example, if we choose x = 0 and y = 0, then we have:
Phosphoric acid: 20 * 0 + 10 * 0 >= 460
Nitrogen: 30 * 0 + 30 * 0 >= 960
Potash: 5 * 0 + 10 * 0 >= 220
which gives us:
Phosphoric acid: 0 >= 460
Nitrogen: 0 >= 960
Potash: 0 >= 220
These values are all negative, which means that x = 0 and y = 0 do not satisfy the inequalities.
Next, we can choose x = 1 and y = 0, then we have:
Phosphoric acid: 20 * 1 + 10 * 0 >= 460
Nitrogen: 30 * 1 + 30 * 0 >= 960
Potash: 5 * 1 + 10 * 0 >= 220
which gives us:
Phosphoric acid: 20 >= 460
Nitrogen: 30 >= 960
Potash: 5 >= 220
These values are also negative, which means that x = 1 and y = 0 do not satisfy the inequalities either.
We can continue this process until we find values of x and y that satisfy all three inequalities. The feasible solutions are the values of x and y that satisfy all three inequalities, and they are represented by the region in the x-y plane that is common to all three. The feasible solutions can be found by shading the region below each of the lines representing the inequalities, and the set of feasible solutions is the region that is common to all three shaded regions.
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Kyle runs 18 miles in 112.5 minutes. If he continues at this rate, how long will it take him to complete the marathon (26.2 miles)?
What is -3x+y=6 on a graph?
Maileen has a stamp colection. Seventy percent of her is 300 stamps are philippine stamps. How many philipine stamps does Maileen have?
Maileen has 210 Philippine stamps in her collection. This equation can be read as 70 percent of 300, which is equal to 210.
Maileen has a stamp collection of 300 stamps. 70% of the stamps are Philippine stamps. To calculate the number of Philippine stamps Maileen has, we can use the following equation:
(70/100) x 300 = 210
Therefore, Maileen has 210 Philippine stamps in her collection. This equation can be read as 70 percent of 300, which is equal to 210. The equation can be expressed as a fraction, with the numerator being 70, and the denominator being 100. Multiplying the fraction by 300 gives us the answer. In this case, 70 percent of 300 is equal to 210.
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A scatter plot is shown on the coordinate plane. scatter plot with points plotted at 1 comma 7, 1 comma 10, 2 comma 6, 3 comma 7, 3 comma 8, 5 comma 8, 6 comma 6, 7 comma 4, 9 comma 2, and 10 comma 1 Which two points would a line of fit go through to best fit the data? (1, 10) and (10, 1) (1, 7) and (2, 6) (3, 7) and (7, 4) (3, 8) and (6, 6)
The two points that lie on the line of the best fit are (3, 7) and (7, 4).
What is the line of best fit?A straight line that minimizes the gap between it and certain data is called a line of best fit. In a scatter plot containing several data points, a relationship is expressed using the line of best fit. It is a result of regression analysis and a tool for forecasting indicators and price changes.
Given:
The data values of the scatterplot are given in the image below.
The line of the best fit and the calculation are given in the image below.
From the graph;
(3, 7) and (7, 4) are the two points that lie on the line of the best fit.
Therefore, (3, 7) and (7, 4) are the two points.
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T/F?The probability of the union of two events occurring can never be more than the probability of the intersection of two events occurring.
This statement is False as the probability of intersection of two events is always less than the probability of union.
This assertion is untrue. Contrary to popular belief, the likelihood of two occurrences coming together is always higher than or on pair with the likelihood of them colliding.
The chance of two events A and B occurring together can be calculated using the following formula to see why:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)According to this equation, the probability of A and B joining together is equal to the total of their individual probabilities less the likelihood of their colliding.
When we rewrite this calculation and focus only on the likelihood of the intersection, we obtain:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B)Now it is clear that the probability of the intersection is determined by subtracting the probability of their union from the sum of the probabilities of A and B.
P(A) and P(B) are both less than or equal to P(A B), since probabilities can never be negative. P(A + B) is therefore less than or equal to twice P(A + B) when they are added together. Thus, it follows:
P(A) + P(B) - P(A ∪ B) ≤ P(A ∪ B)This indicates that, contrary to what the original statement implied, the probability of the intersection is not greater than or equal to the probability of the union.
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PLEASEEEE HELP ASAP
Solve for x. Round your answer to the nearest tenth.
The value for x is x= [tex]sin^{-1[/tex] (0.5425).
What is Trigonometry?The area of mathematics that deals with particular angles' functions and how to use those functions in calculations. There are six popular trigonometric functions for an angle. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their respective names and acronyms (csc).
Given:
H= 25
B = 21
Using Pythagoras theorem
P = √H² - B²
P = √25² - 21²
P = √184
P = 2√46
Using Trigonometry
sin x = P / H
sin x = 2√46 / 25
x= [tex]sin^{-1[/tex] (0.5425)
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The graph below shows the cost of college tuition per credit hour at a local community college.
(graph in image)
Write the equation in slope-intercept form for the cost of college tuition in terms of the number of credits. Then, complete the table of values for the function.
y = __x + __
How much money, y, will it cost to take 12 college credits, x?
It will cost $___.
Answer:
y = 350x + 225
It will cost $4425 for 12 credits
Step-by-step explanation:
From the two points shown in the graph, we can write the equation of the line.
(5, 1975), (8, 3025)
slope = m = (y_2 - y_1)/(x_2 - x_1) = (3025 - 1975)/(8 - 5) = 1050/3 = 350
y = mx + b
y = 350x + b
1975 = 350 × 5 + b
b = 225
The equation is
y = 350x + 225
For 12 credits, x = 12.
y = 350 × 12 + 225
y = 4425
Answer: It will cost $4425 for 12 credits
A freight elevator can hold a maximum weight of 2,500 pounds. A 180-pound person has a load of boxes to deliver. Some of the boxes weigh 25 pounds each, and some weigh 60 pounds each. Complete parts (a)-(c). Question content area bottom Part 1 a. Write and graph an inequality that represents the number of boxes the elevator can hold in one trip if the person is not in the elevator. Let x represent the number of 25-pound boxes, and let y represent the number of 60-pound boxes. Write an inequality that represents the number of boxes the elevator can hold in one trip if the person is not in the elevator.
The inequality is 25x + 60y + 180 ≤ 2500. x and y are variables.
What is inequality?
In mathematics, inequalities specify the relationship between two non-equal values. Equal does not imply inequality. Typically, we use the "not equal symbol (≠)" to indicate that two values are not equal. But various inequalities are used to compare the values, whether it is less than or greater than.
Given that a 180-pound person has a load of boxes to deliver. There are types of boxes. The weight of the first type of box is 25 pound and the second type of box is 60 pounds.
Let x represent the number of 25-pound boxes, and let y represent the number of 60-pound boxes.
The weight of x number of boxes is 25x
The weight of y number of boxes is 60y
The person carrying the box means he is also in the elevator.
The total weight of the boxes and the person is 25x + 60y + 180.
The elevator can hold a maximum weight of 2,500 pounds.
The wight of the boxes and the person must be less than or equal to 2500 pounds.
The inequality is
25x + 60y + 180 ≤ 2500
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the apothem is the perpendicular distance from the _____ of a regular polygon to any one of its sides.
The apothem is the perpendicular distance from the center of a regular polygon to any one of its sides.
A closed, two-dimensional, flat or planar structure with straight sides is referred to as a polygon. Its sides do not bend in any way. A polygon's sides are also referred to as its edges. A polygon's vertices (or corners) are the places where two sides converge. Polygons and polyhedrons are two names for geometric shapes having three or more sides. The names of a few polygons are listed below. In order to link and develop projects and blockchains that are compatible with Ethereum, Polygon (MATIC), a cryptocurrency and technological platform, was introduced. A fully closed polygon is a flat, two-dimensional form with straight sides. They must have straight sides. Any number of sides can be found in polygons.
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A random sample was taken to determine whether students from a certain grade prefer to wear Brand A, Brand B, or Brand C tennis shoes. There were eleven students in the sample. Two students preferred Brand A, four students preferred Brand B, and five students preferred Brand C.
If there are eighty-eight students in the population, about how many of them would you expect to prefer Brand C?
16
32
55
40
Answer:
40
Step-by-step explanation:
you can multiply 88 by 5/11 to get 40
{PLEASE HELP FOR 100 POINTS}
{DUE TODAY}
Select the correct answer.
Alex is surveying a car company. She decides to start by interviewing some people in her neighborhood. She prepares a questionnaire and emails it to 50 neighbors who own cars. Of those neighbors, 24 responded. Why is this sample likely to be biased?
A.
Alex sent a questionnaire by email instead of going door to door.
B.
Alex sent the questionnaire only to those who live near her.
C.
Not everyone who got the questionnaire responded.
D.
Those who got the questionnaire may not have responded on time.
Answer: B Alex sent the questionnaire only to those who live near her.
Step-by-step explanation:
it is because it were her nieghbors
1 kilograms of pears cost as much as 1. 5 kilograms of apples. Alex paid $34. 20 for 4 kilograms of pears and 3 kilograms for apples. How much did 1 kilogram of pears cost?
The cost of 1 kilogram of pears is $5.70.
How determine the costThe cost of 1 kilogram of pears can be found by using the given information and setting up an equation. First, let's repeat the question: "1 kilogram of pears cost as much as 1.5 kilograms of apples. Alex paid $34.20 for 4 kilograms of pears and 3 kilograms for apples. How much did 1 kilogram of pears cost?"
Let's assign variables to the unknowns:
- P = cost of 1 kilogram of pears
- A = cost of 1 kilogram of apples
From the question, we know that:
- 1P = 1.5A - 4P + 3A = 34.20
Now, we can use substitution to solve for P.
1. Substitute 1.5A for P in the second equation: 4(1.5A) + 3A = 34.20
2. Simplify: 6A + 3A = 34.20
3. Combine like terms: 9A = 34.20
4. Solve for A: A = 34.20/9 A = 3.8
5. Substitute 3.8 for A in the first equation: 1P = 1.5(3.8)
6. Solve for P: P = 5.7
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5/18+4/9 I need help pls NPC
Answer:
13/18
Step-by-step explanation:
Answer:
13/18
Step-by-step explanation:
5/18 + 4/9 .Steps for adding fractions. Find the least common denominator or LCM of the two denominators
15) If a square prism has a volume of 104 cubic centimeters and a height of eight centimeters, explain how to
find the area of the side that is a square.
Answer:
141 cm^2
Step-by-step explanation:
Given: V = 104 cm3, h = 8 cm
Since V = Bh, where B is the area of the base, and h is the height of a prism, then
B = V/h substitute the given values:
B = 104 cm3/8 cm
B = 13 cm2, which tells us that the long side of the square base equals the square root of 13.
The surface area of a prism = L.A. + 2B, where L.A. is the lateral area of the prism.
L.A. = pH, where p is the perimeter of the base.
Since p = 4(square root of 13) cm, h = 8cm, and B = 13 cm2, we have:
Surface Area = pH + 2B = [4(square root of 13) cm](8 cm) + 2(13 cm2) = 115.38 cm2 + 26 cm2 = 141 cm2
Thus, the surface area of the prism is approximately 141 cm2.
What is the value of 30 ml in teaspoon?
The conversion from ml to teaspoon is 1 ml = 0.202884 tsp. This means that 30 ml is equal to 6.08652 teaspoons.
This conversion can be calculated by multiplying the volume in ml with 0.202884. The formula for this conversion is Volume (tsp) = Volume (ml) x 0.202884. For example, if the volume is 30 ml, the calculation would be 30 ml x 0.202884 = 6.08652 tsp. This means that 30 ml is equal to 6.08652 teaspoons.To convert between other units of volume, such as liters and gallons, the conversion factor must be changed. For example, to convert from liters to teaspoons, the conversion factor would be 0.0147867. To convert from gallons to teaspoons, the conversion factor would be 76.7526. It is important to remember that the conversion factor must be changed depending on the units being converted.In conclusion, 30 ml is equal to 6.08652 teaspoons. The conversion from ml to teaspoon is done by multiplying the volume in ml with 0.202884. To convert between other units of volume, the conversion factor must be changed.
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The figure is cut into 8 equal pieces. Shade 1/2 of the figure.
PLEASE HELP ASAP
a human factors expert recommends that there be at least 12 square feet of floor space in a college classroom for every student in the class due to lack of space a university converts a 24 by 27 foot conference room into a classroom find the max number of students the room can accommodate
Answer:
The room can accommodate 4.5 students.
Step-by-step explanation:
12 square feet = 144 square inches of floor space
24 x 27 = 648 square feet of floor space
648 / 144 = 4.5
The room can accommodate 4.5 students.
Solve for x provide proofs. Helppp
Answer:
36 degrees
Step-by-step explanation:
since your trying to find x you know that a wright angle is 90 degrees you minus 54 from 90
The table below shows the value in dollars of a car at the end of x years
The exponential function that models this situation is y = 11,000 (0.85)ˣ. The correct answer would be an option (B).
What is an exponential function?An exponential function is defined as a function whose value is a constant raised to the power of an argument is called an exponential function.
The table below shows the value in dollars of a car at the end of x years.
Number of Years, xValue, v(x) (dollars)
0 11,000
1 9,350
2 7,948
3 6,755
Let the formula for an exponential function would be as
⇒ v(x) = abˣ
Substitute the value of x = 0 and v(x) = 11,000 in the above equation,
⇒ 11,000 = ab⁰
⇒ a = 11,000
Now, substitute the value of x = 1 and v(x) = 9,350, and we get
⇒ 9,350 = ab¹
⇒ 9,350 = (11,000)b
⇒ b = 9,350/11,000
⇒ b = 0.85
Substitute the value of a = 11,000 and b = 0.85 in v(x) = abˣ,
⇒ y = 11,000 (0.85)ˣ
Thus, the exponential function is y = 11,000 (0.85)ˣ.
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Write a function that models y in terms of x. Round any coefficients to the nearest tenth.
X 0 5 10 15 20
y 0 4.02 5.69 6.97 8.05
y= ?
The exponential equation representing the given table is -
y = 2.84(1.072)ˣ.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is a function table as shown in below -
{X} 0 5 10 15 20
{Y} 0 4.02 5.69 6.97 8.05
Let the expression be -
y = A(B)ˣ
At {x} = 5, y = 4.02.
We can write -
4.02 = A(B)⁵ ... Eq { 1 }
At {x} = 10, {y} = 5.69
We can write -
5.69 = A(B)¹⁰ ... Eq { 2 }
Refer to the graph. The point of intersection represents -
A = 2.84
B = 1.072
The equation would be -
y = 2.84(1.072)ˣ ... Eq { 3 }
Therefore, the exponential equation representing the given table is -
y = 2.84(1.072)ˣ.
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The equation, 3x2 + 6x + 3y2 + 7y + 4 = 0, represents a conic. The conic represented by the equation is a/an because.
The ratio of the variables makes the equation a circle [tex]x^2\: and \:y^2[/tex] must be equal,
A circle is a particular type of ellipse in mathematics or geometry where the eccentricity is zero and the two foci are congruent. A circle is also known as the location of points that are evenly spaced out from the centre. A circle's radius is calculated by measuring it from the centre to the edge.The line that splits a circle into two identical halves is its diameter, which is also twice as wide as its radius.A circle's radius may be measured, making it a basic 2D shape.The inner and exterior parts of the aircraft are separated by the circles.
Equation of a circle
Circle's standard equation is written as
[tex]x^2+y^2+2gx+2fy+C=0[/tex]
Given the equation of the circle
[tex]3x^2 + 6x + 3y^2 + 7y + 4=0[/tex]
Rearrange the equation
[tex]3x^2 + 6x + 3y^2 + 7y + 4=0\\3x^2 + 3y^2+ 6x + 7y + 4=0\\[/tex]
The equation is a circle because the ratio of [tex]x^2\: and \:y^2[/tex] must be equal,
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Answer:
hope this helps.
Step-by-step explanation:
Solve the system by substitution.
Answer:
x=3
y=-34
this should be correct
Assume the distance from San Francisco to Big Sur California is 240 kilometers. Han Fei leaves Big Sur on her bicycle heading for San Francisco at 20 kilometers per hour; at the same time Mohammed leaves San Francisco in his car heading to Big Sur on the same route averaging 40 miles per hour. How far from San Francisco will Han Fei be when Mohammed meets her in his car?
Answer: When Mohammed meets Han Fei, she will be 74.6 kilometers away from San Francisco.
Step-by-step explanation:
First, we need to convert Mohammed's speed from miles per hour to kilometers per hour, as Han Fei's speed is given in kilometers per hour.
40 miles per hour = 64.37 kilometers per hour (approximately)
Now, let's determine how long it will take Mohammed to reach the point where he meets Han Fei:
time = distance ÷ speed
time = 240 km ÷ 64.37 km/h
time ≈ 3.73 hours
So, after 3.73 hours, Mohammed will meet Han Fei.
In that time, Han Fei will have traveled:
distance = speed × time
distance = 20 km/h × 3.73 h
distance ≈ 74.6 km
A restaurant charges $4. 50 for a slice of pizza that is 1/4 of a pizza and $4. 00 for a slice that is 1/6 of a pizza. One day they cooked 6 pizzas that were all the same size. If they sell all the slices, would they make more money by slicing each pizza into fourths or sixths? How much more? Explain
The restaurant would make more money by slicing each pizza into sixths. The restaurant would make $36 more than if they sliced each pizza into fourths.
To determine if the restaurant would make more money by slicing each pizza into fourths or sixths, we need to calculate the total amount of money they would make for each option.
Option 1: Slicing each pizza into fourths
Option 2: Slicing each pizza into sixths
Each pizza would have 6 slices, so 6 pizzas would have a total of 36 slices.Each slice would be sold for $4.00, so the total amount of money made would be 36 slices × $4.00/slice = $144.Comparing the two options, we can see that the restaurant would make more money by slicing each pizza into sixths. They would make $144 - $108 = $36 more by slicing each pizza into sixths instead of fourths.
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The height of an object at any time can be found with the polynomial - 16t ^ 2 + 550 where t is the time in seconds. How high is the object after 3.5 seconds?
Answer:
354 m
Step-by-step explanation:
to find the height substitute t = 3.5 into the height polynomial
height = - 16(3.5)² + 550 = - 16(12.25) + 550 = - 196 + 550 = 354
Sophia and Joseph are making fruit salads for a picnic. Sophia mixes 5 cups of melon and 2 cups of apple and Joseph mixes 8 cups of melon and 5 cups of apple. Use Sophia and Joseph's percent of melon to determine whose fruit salad will taste more melony.
Based on the percentage of melons, Sophia's fruit salad will taste more Melony than Joseph's.
What is the percentage?The percentage is the quotient of the numerator and denominator.
Like ratios, percentages show the relative size of a value compared to another.
Percentages are expressed as the quotient multiplied by 100.
Sophia Joseph
Cups of melon 5 8
Cups of apple 2 5
Ratios 5:2 8:5
The sum of ratios 7 13
Percentages of melon to apple:
Sophia Joseph
71.43% (5/7 x 100) 61.54% (8/13 x 100)
Thus, using the percentage of melon to apple, we can conclude that Sophia's fruit salads tastes 71.43% more Melony than Joseph's.
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Find the area inside the loop of the following limacon: r=7−14sinθ
The area inside the loop of the limacon given by the polar equation r = 7 - 14sin(θ) is 294π square units.
To find the area inside the loop of the limacon given by the polar equation r = 7 - 14sin(θ), we need to evaluate the integral for half of the area and then double the result.
The curve of the limacon has a loop when 0 ≤ θ ≤ π, so we integrate from 0 to π. The area can be calculated as follows:
A = 2 ∫₀^π 1/2 r² dθ
Using the equation for r given, we can substitute and simplify:
A = 2 ∫₀^π 1/2 (7 - 14sin(θ))² dθ
A = 2 ∫₀^π 1/2 (49 - 196sin(θ) + 196sin²(θ)) dθ
A = ∫₀^π (98 - 392sin(θ) + 392sin²(θ)) dθ
Using the trigonometric identity 1 - cos(2θ) = 2sin²(θ), we can simplify further:
A = ∫₀^π (98 - 392sin(θ) + 196 - 196cos(2θ)) dθ
A = ∫₀^π (294 - 392sin(θ) - 196cos(2θ)) dθ
Integrating with respect to θ:
A = [294θ + 392cos(θ)sin(θ) + 98sin(θ)]₀^π
Evaluating at the limits of integration, we get:
A = [294π + 0 + 0] - [0 + 392cos(0)sin(0) + 98sin(0)]
A = 294π - 0 - 0 = 294π
Therefore, the area inside the loop of the limacon r = 7 - 14sin(θ) is 294π square units.
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