Answers
The intervals in which the function is decreasing. [tex][-\pi , -\frac{\pi }{3} ], [\frac{\pi }{3}, \pi ][/tex]. Option 3
How do you find the interval in which the function is decreasing?We're given a function f(x) = 2 sin x - x, which describes a curve on a graph. We want to find the intervals where this curve is decreasing (going down) within the range of -π to π.
To find when the function is decreasing, we look at its slope. The slope tells us if the curve is going up or down. We find the slope by taking the first derivative of the function: f'(x) = 2 cos x - 1.
We now have an equation for the slope, f'(x) = 2 cos x - 1. A negative slope means the function is decreasing. So, we want to find where f'(x) is less than 0 (negative).
We set up the inequality: 2 cos x - 1 < 0. We solve it to find the x-values where the slope is negative. The solution is cos x < 1/2.
From the inequality cos x < 1/2, we find the intervals within the range of -π to π where the function is decreasing. These intervals are [-π, -π/3] and [π/3, π].
The above answer is in response to the question below as seen in the picture.
Determine the interval(s) in [tex][-\pi, \pi ][/tex] on
which f(x) = 2 sin x - x
is decreasing.
1. [tex][-\frac{\pi }{3}, \frac{\pi }{3} ][/tex]
2. [tex][-\frac{\pi }{6}, \frac{\pi }{6} ][/tex]
3. [tex][-\pi , -\frac{\pi }{3} ], [\frac{\pi }{3}, \pi ][/tex]
4. [tex][-\pi , -\frac{-2\pi }{3} ], [\frac{2\pi }{3}, \pi ][/tex]
5. [tex][-\pi , - \frac{5x}{6} ], [\frac{\pi }{6}, \pi ][/tex]
6. [tex][-\frac{\pi }{6}, \frac{5\pi }{6} ][/tex]
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Related Questions
K(10,..), L. (..,20), and M (30,.. .. are on that line too. Write their missing coordinates.
Answers
The coordinate of points K, L and M is the location of the points on a coordinate plane
Here, we have,
to determine the missing coordinates:
The given parameters are:
K = (10, )
L = ( ,10)
M = (30, )
The question has missing parameters.
So, I will assume that the line is a perfectly horizontal line.
This means that the y-coordinates of points K, L and M are equal.
The y-coordinate of point L is 10.
So, we have:
K = (10, 10)
L = ( ,10)
M = (30, 10)
Assume that point L is halfway points K and M, then we have:
K = (10, 10)
L = (20, 10)
M = (30, 10)
See attachment for the diagram of the coordinate plane showing the line
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Need help please
Geometry
Answers
Answer:
vertical because they are right across from each with the same degrees
Consider the number 3^6.
Rewrite the number as the product of two whole-number powers of 3.
Answers
We can rewrite 3^6 as the product of two whole-number powers of 3 as: [tex]3^4 . 3^2.[/tex]
How can we rewrite the number as the product?To rewrite 3^6 as the product of two whole-number powers of 3, we need to find two exponents that add up to 6.
A way to do this is to use the fact that 3^4 = 81, which is close to 3^6.
Specifically, we have:
3^6 = 3^4 * 3^2
We can check that this is true by using the rule for multiplying exponents that states that a^m * a^n = a^(m+n).
Applying this rule, we get:
= 3^4 * 3^2
= 3^(4+2)
= 3^6.
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Aisha and Malik are making fruit salads for a picnic. Aisha mixes 4 cups of melon and 1 cup of apple and Malik mixes 13 cups of melon and 3 cups of apple. Use Aisha and Malik’s percent of melon to determine whose fruit salad will taste more melony.
Answers
Answer:
Maliks fruit salad will taste more melony
Step-by-step explanation:
To determine whose fruit salad will taste more "melony," we can calculate the percentage of melon in each fruit salad for Aisha and Malik.
For Aisha's fruit salad:
Total cups of fruit = 4 cups of melon + 1 cup of apple = 5 cups
Percentage of melon = (4 cups of melon / 5 cups of total fruit) x 100%
For Malik's fruit salad:
Total cups of fruit = 13 cups of melon + 3 cups of apple = 16 cups
Percentage of melon = (13 cups of melon / 16 cups of total fruit) x 100%
Now we can calculate the percentages:
Aisha's percentage of melon:
(4 cups of melon / 5 cups of total fruit) x 100% = 80%
Malik's percentage of melon:
(13 cups of melon / 16 cups of total fruit) x 100% ≈ 81.25%
Based on the calculations, Malik's fruit salad has a higher percentage of melon (81.25%) compared to Aisha's fruit salad (80%). Therefore, Malik's fruit salad is likely to taste more "melony" compared to Aisha's fruit salad.
The graph of f(x) = e^x-1+5 is shown below. g(x) is a transformation of
f(x). How would you write the equation for the function g(x)?
Ax) = 8²1
+5
15
10
-10
-15
a.
g(x)
5
10 15 20 25
O A g(x) = 3e² - 1
B. g(x) = e²-¹ - 8
C. g(x) = e-5
D. g(x) = -1 -3
Answers
Answer- e^x-1-3
Step-by-step explanation:
8. You and your friend each collect coins and stamps. Your friend collects one fifth as many coins and twice as many stamps as you. You collect a total of 70 coins and stamps. Your friend collects a total of 95 coins and stamps.
How many coins and how many stamps do you each collect?
Answers
Answer:
you: 25 coins, 45 stampsfriend: 5 coins, 90 stampsStep-by-step explanation:
Given you have collected 70 coins and stamps, and your friend has collected 1/5 as many coins and twice as many stamps, for a total of 95 coins and stamps, you want the numbers of each.
SetupWe can let x represent the number of coins you have collected. Then 70 -x is the number of stamps you have collected. The corresponding numbers for your friend are x/5 coins and 2(70-x) stamps. Your friend's collection is ...
x/5 +2(70 -x) = 95
Solution-9/5x +140 = 95 . . . . . . . . simplify
45 = 9/5x . . . . . . . . . . . . . add 9/5x -95
25 = x . . . . . . . . . . . . . . . . multiply by 5/9
70 -x = 45
1/5x = 5
2(70 -x) = 90
You collect 25 coins and 45 stamps; your friend collects 5 coins and 90 stamps.
<95141404393>
help please math homework
Answers
Answer:
60.5 m
Step-by-step explanation:
We can use the arc length formula to calculate the length of highlighted arc. Let the numerical measure of the highlighted arc be known as C, the angle measure be known as [tex]\theta[/tex], and r as the radius of the circle.
Given variables:
C = Measure of the highlighted arcr = radius of the circle[tex]\theta[/tex] = angle length of the highlighted arcPlug in all the variables into the formula and solve for the letter C.
[tex]\boxed{\text{Formula: C = }{2\pi r\huge{\text(}\frac{\theta}{360}\huge\text{)}}}[/tex]
[tex]\implies C = }{2\pi (11)\huge{\text(}\dfrac{315}{360}\huge\text{)}}}[/tex] [tex]\implies C = }{22\pi \huge{\text(}\dfrac{315}{360}\huge\text{)}}} = 60.5 \ \text{m}[/tex]
Therefore, Option A is the correct option.
I need help on this im on a test pls
Pattern A follows the rule "add 2" and Pattern B follows the rule "subtract 2.
1, 3
1, 10
3, 6
5, 4
5, 6
7, 4
Select all correct answers
Answers
5,6
7,4
This is because those answers, following the pattern, work correctly together.
A party company is designing a new line of individualized bubbles for various celebrations such as birthday parties, weddings, and anniversaries. The bubbles will be sold in various-sized packs. The individual container of bubbles will be a plastic cylindrical tube with a diameter 2 cm and a height of 8 cm.
How many square centimeters of plastic are needed for one tube of bubbles?
Round your answer to the nearest whole number.
A. 50 cm^2
B. 57 cm^2
C. 63 cm^2
D. 53 cm^2
Answers
The correct answer is option B. That is the amount of plastic needed for making one plastic cylinder is 57 cm²
What is a cylinder?A cylinder is a three-dimensional geometric shape that has two congruent circular bases that lie parallel to each other and are connected by a curved surface. It can be thought of as a stack of circles or as a prism with a curved surface. Some common examples of cylinders include soda cans, water bottles, and pipes.
Given that the individual container of bubbles will be a plastic cylindrical tube with a diameter 2 cm and a height of 8 cm.
So radius = diameter/2 = 2/2 = 1 cm.
The surface area of the cylinder = 2πrh + 2πr²
SA = 2π × 1 ×8 + 2π × 1²
SA = 2π + 16π
SA = 18π
Substituting the value of π = [tex]\frac{22}{7}[/tex]
Hence, SA = 18π = 18 × [tex]\frac{22}{7}[/tex] ≈ 56.57
Rounding to the nearest integer, we get:
Answer: B. 57 cm^2.
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Calculate how far the village is north of O
Answers
Step-by-step explanation:
See image
A tourist wants to visit 7 cities in Israel. Driving distances, in kilometers, between the cities are shown below7 . Find a route for the person to follow, returning to the starting city:
a. Using Nearest Neighbor starting in Jerusalem
b. Using Repeated Nearest Neighbor
Answers
a. Using Nearest Neighbor starting in Jerusalem, the value will be 1000km.
b. Using Repeated Nearest Neighbor, the values are attached.
How to explain the informationUsing Nearest Neighbor starting in Jerusalem, the value will be from A to B to C to D to E to F to A.
This will be:
Total length = 58 + 95 + 35 + 29 + 233 + 241 + 309
= 1000km.
Therefore, Using Nearest Neighbor starting in Jerusalem, the value will be 1000km na using Repeated Nearest Neighbor, the values are attached.
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This is indirect measurement, I need some help on this one ):
Answers
The height of the building , given the shadow that it casts would be 93 meters tall .
How to find the height of the building ?Utilizing proportions can be an effective method to tackle this issue. By defining ' h ' as the height of the building responsible for casting a 39.75 meter shadow, we may establish the ensuing proportion :
h / 39. 75 = 3. 2 / 1. 37
Solving for h gives us :
h / 39. 75 = 2. 336
h = 2. 336 x 39. 75
h = 92. 8 56
h = 93 meters tall
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The data shows the number of years that a random sample of 20 employees worked for an insurance company before retirement. Employee Number Years Worked 1 8 2 13 3 15 4 3 5 13 6 28 7 4 8 12 9 4 10 26 11 29 12 3 13 10 14 3 15 17 16 13 17 15 18 15 19 23 20 13 The sample mean for the number of years worked is , and % of the employees in the sample worked for the company for at least 10 years. Round your answers to the nearest integer.
Answers
The sample mean for the number of years worked is 12.9, and 35% of the employees in the sample worked for the company for at least 10 years.
To find the sample mean, we need to add up all the years worked and divide by the total number of employees:
8 + 13 + 15 + 3 + 13 + 28 + 4 + 12 + 4 + 26 + 29 + 3 + 10 + 3 + 17 + 13 + 15 + 15 + 23 + 13 = 258
So, the sample mean is 258/20 = 12.9 (rounded to the nearest tenth).
To find the percentage of employees who worked for at least 10 years, we need to count the number of employees who worked for 10 years or more.
From the data, we see that 7 employees worked for at least 10 years (Employee Number: 2, 3, 5, 10, 11, 15, and 19). So, 7 out of 20 employees worked for at least 10 years, which is 35%. Therefore, the answer is 35% (rounded to the nearest integer).
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[tex]\sqrt[5]{128}[/tex]
Answers
this will be the square -
[tex]2 \times \sqrt[5]{4} [/tex]
or 2,63902
Un comerciante desea mezclar habichuelas rojas que cuestan $3.5 por libra, con habichuelas negras que cuestan $4 por libra, para obtener 20 libras de una mezcla que cueste $3.75 por libra. ¿Cuántas libras de cada variedad debe mezclar
Answers
The merchant needs to mix 10 pounds of red beans and 10 pounds of black beans to get a mix that costs $3.75 per pound.
We have,
Let's assume that the merchant needs to mix x pounds of red beans and (20 - x) pounds of black beans.
To find the pounds of each variety, we can use the following equation:
= 3.5x + 4(20 - x)
= 3.75(20)
Simplifying the equation.
3.5x + 80 - 4x = 75
-0.5x = -5
x = 10
Therefore,
The merchant needs to mix 10 pounds of red beans and 10 pounds of black beans to get a mix that costs $3.75 per pound.
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The complete question.
A merchant wants to mix red beans that cost $3.5 per pound with black beans that cost $4 per pound to get 20 pounds of a mix that costs $3.75 per pound. How many pounds of each variety should he mix?
find the value of each trigonometric ratio in simplist form
Answers
Okay, let's go through this step-by-step:
1) sin(A) = opposite/adjacent = BC/AB = 18/24 = 3/4
2) cos(A) = adjacent/hypotenuse = AB/AC = 24/30 = 4/5
3) tan(A) = sin(A) / cos(A) = (3/4) / (4/5) = 3/4
4) sec(A) = 1/cos(A) = 1/4/5 = 5/4
5) csc(A) = 1/sin(A) = 1/3/4 = 12/3
In simple terms:
sin(A) = 3/4
cos(A) = 4/5
tan(A) = 3/4
sec(A) = 5/4
csc(A) = 12/3
Let me know if you need more details.
(11) Esi cut a rope into lengths which are exactly 18cm long, 30cm long or 40cm long. What is the shortest length that the rope can be?
Answers
The shortest length that rope can be, is 360 cm; such that the rope may be cut into lengths of exactly 18cm long, 30cm long or 40cm long.
A rope is cut into length which may be exactly 18cm long, 30cm long or 40cm long.
The shortest length that rope can be
Considering the individual lengths which are all integers, the shortest length is calculated as the LCM or the Least Common Multiple of the individual lengths that are considered
Factorize the numbers 18, 30 and 40 in the following way.
18 = 2 * 3 * 3
30 = 2* 3 *5
40 = 2* 2* 2* 5
The LCM of the numbers 18, 30 and 40 is
= 2* 2* 2* 3* 3* 5
= 360
So, the shortest length that rope can be, to make it cut into lengths of exactly 18cm long, 30cm long or 40cm long, is 360 cm.
Therefore, the required shortest length that rope can be, is 360 cm; such that the rope may be cut into lengths of exactly 18cm long, 30cm long or 40cm long.
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An element with mass 420 grams decays by 11.8% per minute. How much of the element is remaining after 16 minutes, to the nearest 10th of a gram?
Answers
After 16 minutes of decay, approximately 182.3 grams of the element remains.
To calculate how much of the element remains after 16 minutes of decay, we can use the formula for exponential decay:
A = A₀ * [tex]e^{(-kt)[/tex]
where A is the amount of the element remaining after time t, A₀ is the initial amount of the element, k is the decay constant, and t is the elapsed time.
In this case, we have A₀ = 420 grams, k = 0.118 (since the element decays by 11.8% per minute), and t = 16 minutes. Plugging these values into the formula, we get:
A = 420 * [tex]e^{(-0.118*16)[/tex]
A = 182.3 grams
We round our answer to the nearest 10th of a gram, as requested in the problem.
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To have a driver's license you must be at least 16 years old, and no more than 85 years old.
a. Represent the ages of people who can get a driver's license in three forms.
b. Represent the ages of people who cannot get a driver's license in three forms.
*In Symbolic/Inequality and number line*
Answers
The ages of people who can get a driver's license can be represented in three forms as follows;
x ≥ 16
x ≤ 85
16 ≤ x ≤ 85
The ages of people who cannot get a driver's license can be represented in three forms as follows;
x ≤ 16
x ≥ 85
16 ≥ x ≥ 85
What is an inequality?In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;
Greater than (>).Less than (<).Greater than or equal to (≥).Less than or equal to (≤).Assuming the variable x denotes the ages of people who wishes to obtain a driver's license, a set of inequality which can be used to represent those who are eligible in three forms include;
x ≥ 16
x ≤ 85
16 ≤ x ≤ 85
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9. data was collected on houses currently for sale. based on a cubic modeling equation, which prediction about this data is the most reliable? a.
a $70k house would have 1,354 square ft
b. a $70k house would have about 1,460 square ft
c. a $400k house would have 492 square ft
d. a $400k house would have 4,328 square ft
10. suppose a scatterplot is created from the points in the following table. when x=7, what is the second coordinate in a scatterplot of the linearized data? round the answer to the tenth place.
a. 0.8
b. 2.1
c. 54.3
d. 112.8
11. the equation that best models the linearized data for a particular data set is log y= 0.75821x+0.03442. find the approximate value x=4. round your answer to the nearest whole number.
a. 3
b. 25
c. 1,168
d. 62,034
Answers
question 9.
Based on a cubic modeling equation, the prediction about this data that is most reliable is a $70k house would have 1,354 square ft.
Option A is correct.
Question 10.
The second coordinate in a scatterplot of the linearized data and rounding the answer to the tenth place is 0.8
Option A is correct.
Question 11.
The approximate value when x=4 and rounding the answer to the nearest whole number is 1,168
Option C is correct.
How do we calculate?The equation is given as
Log y= 0.75821x+0.03442
We substitute x = 4 into the given equation and solve for y:
log y = 0.75821(4) + 0.03442
log y = 3.07286 + 0.03442
log y = 3.10728
We take the antilogarithm of both sides,
y = 10^(3.10728)
y = 1,168.66
Rounding off, the closest answer is option (c) 1,168.
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Two containers designed to hold water are side by side, both in the shape of a cylinder. Container A has a diameter of 14 feet and a height of 11 feet. Container B has a diameter of 10 feet and a height of 19 feet. Container A is full of water and the water is pumped into Container B until Container B is completely full.
Answers
After filling Container B completely, there are approximately 209.56 cubic feet of water left in Container A.
How to solveContainer A:
Diameter = 14 feet
Radius = Diameter / 2 = 14 / 2 = 7 feet
Height = 11 feet
Volume of Container A = π * (7^2) * 11 ≈ 1,696.46 cubic feet
Container B:
Diameter = 10 feet
Radius = Diameter / 2 = 10 / 2 = 5 feet
Height = 19 feet
The volume of Container B = [tex]π * (5^2) * 19[/tex] ≈ 1,486.90 cubic feet
Now, let's find out how much water is left in Container A after filling Container B completely.
Water left in Container A = Volume of Container A - Volume of Container B
Water left in Container A ≈ 1,696.46 - 1,486.90 ≈ 209.56 cubic feet
So, after filling Container B completely, there are approximately 209.56 cubic feet of water left in Container A.
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Suppose a truck rental costs $20 plus $0.30 for each mile
driven. If your total cost of a rental was $39. 80, how many miles
did you drive?
Answers
If the total cost is $39.80, then the number of miles driven is 66 miles.
If your total cost of a rental was $39. 80, how many miles did you drive?Here we know that a truck rental costs $20 plus $0.30 for each mile driven.
Then we can model this with a linear equation where the y-intercept is 20, and the slope is 0.30 (where the independent variable will be the number of miles driven).
Then the total cost is modeled by the linear equation:
y = 20 + 0.3*x
If the total cost is $39.80, then we can solve:
39.8 = 20 + 0.3*x
39.8 - 20 = 0.3*x
19.8/0.3 = x
66 =x
There are 66 miles.
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Ms. Estrada, the art teacher at Riverview Elementary School, is painting a mural in the school library. The mural is 20 feet long, and Ms. Estrada splits the mural into 8-inch sections to paint. It takes her about 50 minutes to paint each section. About how many hours will it take Ms. Estrada to complete the mural?
Answers
It will take Ms. Estrada about 25 hours to complete the mural.
Describe Algebra?Algebra is a branch of mathematics that deals with the study of mathematical symbols and their manipulation. It involves the use of letters, symbols, and equations to represent and solve mathematical problems.
In algebra, we use letters and symbols to represent unknown quantities and then use mathematical operations such as addition, subtraction, multiplication, division, and exponentiation to manipulate those quantities and solve equations. We can use algebra to model and solve real-world problems in various fields such as science, engineering, economics, and finance.
Some common topics in algebra include:
Solving equations and inequalities
Simplifying expressions
Factoring and expanding expressions
Graphing linear and quadratic functions
Using logarithms and exponents
Working with matrices and determinants
First, we need to convert the length of the mural from feet to inches to match the unit of the sections.
20 feet x 12 inches/foot = 240 inches
Then, we need to determine the number of sections in the mural:
240 inches / 8 inches/section = 30 sections
Now, we can find the total time it will take Ms. Estrada to complete the mural:
30 sections x 50 minutes/section = 1500 minutes
Finally, we can convert the time to hours:
1500 minutes / 60 minutes/hour = 25 hours
Therefore, it will take Ms. Estrada about 25 hours to complete the mural.
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Which graph represents this equation? y = 3 2 x 2 − 6 x A. The graph shows an upward parabola with vertex (3, minus 4.5) and passes through (minus 1, 3.5), (0, 0), (6, 0), and (7, 3.5) B. The graph shows an upward parabola with vertex (2, minus 6) and passes through (minus 1, 7), (0, 0), (4, 0), and (5, 7) C. The graph shows an upward parabola with vertex (minus 3, minus 4.5) and passes through (minus 7, 3.5), (minus 6, 0), (0, 0), and (1, 3.5) D. The graph shows an upward parabola with vertex (minus 2, minus 6) and passes through (minus 5, 7), (minus 4, 0), (0, 0), and (1, 7)
Answers
The graph that represents the equation is:
The graph shows an upward parabola with vertex (2, -6) and passes through (-1, 7), (0, 0), (4, 0), and (5, 7).
Option B is the correct answer.
We have,
The equation y = (3/2)x² - 6x represents an upward parabola since the coefficient of x² is positive.
Now,
The vertex of the parabola.
x = -b/2a,
where a and b are the coefficients of x² and x, respectively.
So,
a = 3/2 and b = -6
x = -(-6)/(2(3/2)) = 6/3 = 2.
Plugging x = 2 into the equation,
We get y = 3/2(2)² - 6(2) = -6,
so the vertex is (2, -6).
We can eliminate options A and C since their vertices are not (2, -6).
Now,
To check which of the remaining options fits the equation, we can plug in the given points and see if they satisfy the equation.
Option B gives:
When x = -1, y = 3/2(-1)² - 6(-1) = 7
When x = 0, y = 3/2(0)² - 6(0) = 0
When x = 4, y = 3/2(4)² - 6(4) = 0
When x = 5, y = 3/2(5)² - 6(5) = 7.5
So option B fits the equation, and is the graph that represents
y = (3/2)x² - 6x.
Therefore,
The graph that represents the equation is:
The graph shows an upward parabola with vertex (2, -6) and passes through (-1, 7), (0, 0), (4, 0), and (5, 7).
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Why do we receive two answers in a projectile motion formula , a negative and a positive?
Answers
Answer:
Both solutions are valid, and depending on the context of the problem, one or both solutions may be useful in solving for different parameters, such as the maximum height or the time of flight of the projectile.
Step-by-step explanation:
In a projectile motion formula, such as the one that describes the height of an object in motion as a function of time, there are two solutions because the object can reach the same height at two different times during its trajectory.
The negative solution represents the time it takes for the object to reach its maximum height and then start falling back down to the ground. The positive solution represents the time it takes for the object to reach the same height on its way up.
So, both solutions are valid, and depending on the context of the problem, one or both solutions may be useful in solving for different parameters, such as the maximum height or the time of flight of the projectile.
On Team B 80% of the football players weigh more than 200 pounds. If 41 men are on the team, how many weigh more than 200 pounds?
Answers
Team B has 33 players that weigh more than 200 pounds.
How to calculate the percentage?To calculate the percentage, divide the amount by the total value and multiply the result by 100. The percentage is calculated using the formula: (value/total value)100%.
When 80% of the players weigh more than 200 pounds, 20% of the players weigh less than 200 pounds.
We can start by locating 20% of the players:
20% of 41 = 0.20 x 41 = 8.2
So we can guess that there are about 8 players weighing 200 pounds or fewer.
We may subtract this estimate from the total number of participants to obtain the number of players who weigh more than 200 pounds:
41 - 8 = 33
As a result, Team B has 33 players that weigh more than 200 pounds.
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Help me with this math if you may.
Answers
Answer:
4/5
Step-by-step explanation:
2 : 5/2
4/2 : 5/2
4 : 5
The scale factor is:
4/5
Plot the numbers -2 3/4 and 5/2 on the number line below.
Answers
Answer: see attached image
Step-by-step explanation:
since the distance between each number is split into fourths, that is the interval it is increasing.
5/2 simplified is 2 and 1/2. so you go 2 tick marks after the 2.
-2 and 3/4 goes 3/4 (3 tickmarks) after the -2.
Help me on this please
Answers
Answer:
infinitely manyno solutionone solutionStep-by-step explanation:
You want to determine the number of solutions to three different systems of equations.
Standard formA linear equation is written in standard form when the coefficients are mutually prime, and the leading coefficient is positive:
ax +by = c . . . . . . GCF(a, b, c) = 1, a > 0
The equations are easiest to compare when they are all written in standard form.
Numbers of solutionsA system will have an infinite number of solutions when the equations are identical.
A system will have zero solutions when it reduces to ...
(non-zero constant) = 0
A system will have one solution when the equations are different.
System 1A factor of 2 can be removed from the first equation:
2x -3y = 5
A factor of 3 can be removed from the second equation:
2x -3y = 5
These equations are identical, so have infinitely many solutions.
System 2Multiplying the first equation by 2 gives ...
2y = -3x +6
Adding 3x, we have ...
3x +2y = 6
When we subtract the second equation from this, we get ...
(3x +2y) -(3x +2y) = (6) -(3)
0 = 3
These equations have no solution.
System 3These equations are already in standard form, and are different. This system has one solution.
(The exact solution is (x, y) = (0.48, 3.36).)
a. Use the appropriate formula to determine the periodic deposit.
b. How much of the financial goal comes from deposits and how much comes from interest?
Periodic Deposit
Rate
$? at the end of each month 6.75% compounded monthly
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Time
45 years
Financial Goal
$1,000,000
***
a. The periodic deposit is $.
(Do not round until the final answer. Then round up to the nearest dollar as needed.)
Answers
The periodic deposit required to reach the financial goal of $1,000,000 in 45 years with a 6.75% annual interest rate compounded monthly is approximately $541.05.
And, Amount $292,593.00 comes from deposits and $707,407.00 comes from interest.
For the periodic deposit, we can use the formula:
P = (FV × r) / ((1 + r)ⁿ - 1)
where P is the periodic deposit, FV is the financial goal, r is the interest rate per period, and n is the total number of periods.
Using the given values, we get:
P = ($1,000,000 0.0675) / ((1 + 0.0675/12)^(45x12) - 1)
P ≈ $541.05
So, the periodic deposit required to reach the financial goal of $1,000,000 in 45 years with a 6.75% annual interest rate compounded monthly is approximately $541.05.
And, To determine how much of the financial goal comes from deposits and how much comes from interest, we can calculate the total amount of deposits made over the 45-year period:
Hence, Total deposits is,
P n = $541.05 (45 x 12)
≈ $292,593.00
Then we can subtract this amount from the financial goal to get the amount that comes from interest:
Amount from interest = FV - Total deposits
= $1,000,000 - $292,593.00
≈ $707,407.00
So , Amount $292,593.00 comes from deposits and $707,407.00 comes from interest.
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