To solve the initial value problem (IVP) y" + 4y = f(t) with the given piecewise function f(t), we need to consider two cases: t < 1 and t > 1. Let's solve the IVP step by step.
Case 1: t < 1
In this case, the function f(t) is equal to t. To solve the differential equation, we assume a solution of the form y(t) = A(t) + B(t), where A(t) is the solution to the homogeneous equation y" + 4y = 0, and B(t) is a particular solution to the non-homogeneous equation.
The homogeneous equation y" + 4y = 0 has characteristic equation r^2 + 4 = 0, which yields the complex roots r = ±2i. Therefore, the homogeneous solution is A(t) = c1*cos(2t) + c2*sin(2t), where c1 and c2 are constants.
For the particular solution B(t), we assume B(t) = Ct, where C is a constant to be determined. Substituting B(t) into the differential equation, we get:
2C + 4Ct = t
6Ct + 2C = t
Comparing the coefficients, we have 6C = 0 and 2C = 1. Solving these equations, we find C = 0 and C = 1/2, respectively.
Therefore, the particular solution for t < 1 is B(t) = (1/2)t.
Combining the homogeneous and particular solutions, we have y(t) = A(t) + B(t) = c1*cos(2t) + c2*sin(2t) + (1/2)t.
To find the constants c1 and c2, we use the initial conditions y(0) = 2 and y'(0) = 0. Substituting t = 0 into the equation, we get:
y(0) = c1*cos(0) + c2*sin(0) + (1/2)*0 = c1 = 2
y'(0) = -2c1*sin(0) + 2c2*cos(0) + (1/2)*1 = 2c2 + (1/2) = 0
From the second equation, we find c2 = -1/4.
Thus, the solution for t < 1 is y(t) = 2*cos(2t) - (1/4)*sin(2t) + (1/2)t.
Case 2: t > 1
In this case, the function f(t) is equal to 11. The differential equation y" + 4y = 11 has a constant right-hand side, so we assume a particular solution of form B(t) = D, where D is a constant. Substituting B(t) into the equation, we have:
0 + 4D = 11
D = 11/4
Therefore, the particular solution for t > 1 is B(t) = 11/4.
The general solution for t > 1 is the homogeneous solution, which is the same as in Case 1, plus the particular solution B(t):
y(t) = A(t) + B(t) = c1*cos(2t) + c2*sin(2t) + 11/4
Since we have no additional initial conditions for t > 1, we can leave the constants c1 and c2 unspecified.
In conclusion, the solution to the IVP y" + 4y =
f(t) with y(0) = 2 and y'(0) = 0 is:
For t < 1: y(t) = 2*cos(2t) - (1/4)*sin(2t) + (1/2)t
For t > 1: y(t) = c1*cos(2t) + c2*sin(2t) + 11/4
Here, c1 and c2 are arbitrary constants, and the particular solutions take different forms depending on the value of t.
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HELP PLEASEEEE
ces Elizabeth Tailors Inc. has assets of $8,940,000 and turns over its assets 1.9 times per year. Return on assets is 13.5 percent. What is the firm's profit margin (returns on sales)? (Input your ans
The firm's profit margin is 7.11%.
Given data:
Asset Turnover (ATO) = 1.9
Return on Assets (ROA) = 13.5%
Assets (A) = $8,940,000
The formula of Return on assets (ROA) is:
ROA = Net Income / Total Assets
OR ROA = Profit Margin × Asset Turnover
OR Profit Margin = ROA / Asset Turnover
Profit Margin = ROA / Asset Turnover=
13.5% / 1.9= 0.0711 or 7.11%.
Therefore, the company's profit margin is 7.11%.
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A pet store receives 7 boxes of car food. Each box has 48 cans. The store wants to put the cans in equal stacks of 8 cans. Draw a bar model to help you find how many stacks can be formed
We can form a total of 42 stacks of 8 cans each from the 7 boxes of cat food received by the pet store.
Now, We can use a bar model to represent the total number of cans and the number of stacks that can be formed.
First, let's find the total number of cans:
7 boxes x 48 cans/box = 336 cans
Now, let's use a bar model to represent this total:
| 336 cans of food |
Next, we want to find how many stacks of 8 cans we can form. We can use a separate bar model to represent the size of each stack:
| 42 stacks of 8 cans each |
Hence, The number of stacks that can be formed, we need to divide the total number of cans by the number of cans in each stack:
336 cans ÷ 8 cans/stack = 42 stacks
So the final bar model looks like this:
| 336 cans of food | = | 42 stacks of 8 cans each |
Therefore, we can form a total of 42 stacks of 8 cans each from the 7 boxes of cat food received by the pet store.
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Someone solve this please
Answer:
the pyramid is 80 ft² and the rectangular prism is 240 ft²
Step-by-step explanation:
Please help me with 7,8,9,10,11,12
Answer:
7. 9
8. 6.7
9. 12.2
10. 7.6
11. 7.3
12. 6.3
Step-by-step explanation:
7. 8^2+4^2=x^2
64+16=x^2
80=x^2
√80=√x^2
8.9=x
Round up
9=x
8. 6^2+3^2=x^2
36+9=x^2
45=x^2
√45=√x^2
6.7=x
9. 7^2+10^2=x^2
49+100=x^2
149=x^2
√149=√x^2
12.2=x
10. 3^2+7^2=x^2
9+49=x^2
58=x^2
√58=√x^2
7.6=x
11. 7^2+2^2=x^2
49+4=x^2
53=x^2
√53=√x^2
7.28=x
Round up
7.3=x
12. 6^2+2^2=x^2
36+4=x^2
40=x^2
√40=√x^2
6.3=x
Hope this helps
Answer:
7) X = a^2 + b^2 = c^2
X = (4^2) + (8^2) = c^2
X = 80^2 = c^2
X = √80 = 8.944
X = 8.9 = c^2
8) X = a^2 + b^2 = c^2
X = (6^2) + (3^2) = c^2
X = 45^2 = c^2
X = √45 = 6.708
X = 6.7 = c^2
9) X = a^2 + b^2 = c^2
X = (7^2) + (10^2) = c^2
X = 149^2 = c^2
X = √149 = 12.206
X = 12.2 = c^2
10) X = a^2 + b^2 = c^2
X = (3^2) + (7^2) = c^2
X = 58^2 = c^2
X = √58 = 7.615
X = 7.6 = c^2
11) X = a^2 + b^2 = c^2
X = (7^2) + (2^2) = c^2
X = 53^2 = c^2
X = √53 = 7.280
X = 7.3 = c^2
12) X = a^2 + b^2 = c^2
X = (2^2) + (6^2) = c^2
X = 40^2 = c^2
X = √40 = 6.324
X = 6.3 = c^2
Step-by-step explanation:
X = a^2 + b^2 = c^2
X and c^2 = missing side
a^2 and b^2 = the two legs of the triangle (the mesurments of the sides that are given)
c^2 = hipatanouse
The formula is called Pythagromtherom (a^2 + b^2 = c^2)
I need alittle help with this,
9x-5y+3x+2y-4y-y
Answer:
12x - 8y
Step-by-step explanation:
9x - 5y + 3x + 2y - 4y - y
9x + 3x - 5y + 2y - 4y - y
12x - 8y
Answer:
12x - 8y would be the simplified version of this equation if this is not the type of answer your looking for please lmk
Step-by-step explanation:
What is %6 tax added to $26?
A tax of 7.5 percent was added to the product to make it equal to 27.95.
Decide whether the normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions p_1 and p_2 at the given level of significance α using the given sample statistics. Assume the sample statistics are from independent random samples.
Claim: p_1 = p_2, α = 0.05
Sample statistics: x_1 = 32, n_1 = 119 and x_2 = 183, n_2 = 203
C. H_o: p_1 = p_2
H_a:p_1>p_2
D. H_o:p_1
H_a: p_1 = p_2
E. A normal sampling distribution cannot be used, so the claim cannot be tested.
Find the critical values. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The critical values are - z_o = - 1.96 and z_o = 1.96 (Round to two decimal places as needed.)
B. A normal sampling distribution cannot be used, so the claim cannot be tested.
Find the standardized test statistic. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. Z= ______(Round to two decimal places as needed.)
B. A normal sampling distribution cannot be used, so the claim cannot be tested.
The normal sampling distribution can be used
The critical values at α = 0.05 are z₀ = -1.96 and z₀ = 1.96
The standardized test statistic is -11.652
Deciding whether the normal sampling distribution can be usedFrom the question, we have the following parameters that can be used in our computation:
Claim: p₁ = p₂, α = 0.05Sample statistics: x₁ = 32, n₁ = 119 and x₂ = 183, n₂ = 203In the above we can see that the sample sizes are greater than 30 as required by the central limit theorem
This means that the normal sampling distribution can be used and the parameters are
H₀: p₁ = p₂
H₁: p₁ > p₂
Finding the critical valueIn (a), we have
α = 0.05
The critical values at α = 0.05 are z₀ = -1.96 and z₀ = 1.96
Finding the standardized test statistic.Start by calculating the pooled sample proportion using
p = (x₁ + x₂)/(n₁ + n₂)
So, we have
p = (32 + 183)/(119 + 203)
p = 0.67
So, we have
z = (x₁/n₁ - x₂/n₂)/√[p(1 - p)/n₁ + p(1 - p)/n₂)
substitute the known values in the above equation, so, we have the following representation
z = (32/119 - 183/203)/√[0.67(1 - 0.67)/119 + 0.67(1 - 0.67)/203]
Evaluate
z = -11.652
Hence, the standardized test statistic is -11.652
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A soccer ball on the ground is kicked with an initial velocity of 37 feet per second at an elevation angle of 70°. Which
parametric equations represent the path of the soccer ball?
Answer:
x(t) = 37cos(70o)t and y(t) = –16t2 + 37sin(70o)t
Step-by-step explanation:
*WILL GIVE BRAINLIEST FOR THE BEST ANSWER PLEASE HELP AND IF YOU DONT KNOW THE ANSWER DONT ANSWER!*How many people participated in the survey given the data in the histogram below?
Question 1 options:
7 people
12-15 people
20 people
0 people
Answer:
20 people participated in the survey.
Please tell me how to round 14,780
Answer:
for 100: 14,800
for 10: 14,780
for 1: 14,780
for 1000: 15,000
Step-by-step explanation:
ok so just go for like if its
18 its going to be 20 if its 11 its going to be 10
Hope this helps!!
PLEASE CROWN TYY
Answer:
Rounding Rules:
1. If the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up. Example: 38 rounded to the nearest ten is 40. ...
2. If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down. Example: 33 rounded to the nearest ten is 30.
Step-by-step explanation:
100: 14,800
10: 14,780
1: 14,780
1000: 15,000
Can someone please help me with 1-9
Answer: 1. No because there are no numbers to solve it. 2. Yes, has numbers. 3.Yes, has numbers. 4.No because there are no numbers to solve it. 5. not sure 6.not sure 7. not sure 8.not sure 9. not sure
Step-by-step explanation:
Find the area of the square. Round to one decimal place.
Answer:
309.8 mm
Step-by-step explanation:
17.6 x 17.6= 309.76
309.76 rounded is 309.8
Answer:
309.8
Step-by-step explanation:
The formula for area of square is one of its sides times another (or its side squared).
So if one of the side is 17.6, it would be 17.6^2 which is 309.76.
You then round one decimal place and get 309.8
Have a good day
You have been looking to buy a pair of shoes and notice that on Saturday they are marked down 20% from the original price. They are still too expensive! On Tuesday the shoes are marked down with an additional 25% off the price from Saturday. They are now $63. (a) What was the original price? (b) How much did you save? (c) What fraction of the original price did you spend?
(a) The original price of the shoes was $100.
(b) You saved $37.
(c) You spent 63/100 = 0.63 or 63% of the original price.
Explain the relationship between evolution, natural selection and mutations.
You must write out a well thought out and cohesive explanation. Use the CER model for this answer.
Answer:
Natural selection is a mechanism, or cause, of evolution. Adaptations are physical or behavioral traits that make an organism better suited to its environment. Heritable variation comes from random mutations. Random mutations are the initial cause of new heritable traits.
Answer:
Darwin's theory of evolution
Charles Darwin developed a theory of evolution to explain the unity and diversity of life, based on the idea of shared common ancestors.
Natural selection
Darwin's theory was based on the mechanism of natural selection, which explains how populations can evolve in such a way that they become better suited to their environments over time.
Light colored mice are more easily seen by predators and are therefore preyed upon more. Dark mice are better adapted to their environment and better able to survive and reproduce.
Light colored mice are more easily seen by predators and are therefore preyed upon more. Dark mice are better adapted to their environment and better able to survive and reproduce.
Natural selection acting on mice population over time.
Individuals have variations within their heritable traits. Some variations make an individual better suited to survive and reproduce in their environment.
If this continues over generations, these favorable adaptations (the heritable features that aid survival and reproduction) will become more and more common in the population.
The population will not only evolve (change in its genetic makeup and inherited traits), but will evolve in such a way that it becomes adapted, or better-suited, to its environment.
Artificial selection
There are other types of selection, in addition to natural selection, that are out there in the world.
Artificial selection, also called "selective breeding”, is where humans select for desirable traits in agricultural products or animals, rather than leaving the species to evolve and change gradually without human interference, like in natural selection.
A timeline showing how dogs became domesticated over a long period of time due to artificial selection.
A timeline showing how dogs became domesticated over a long period of time due to artificial selection.
Dog breeding is a perfect example of how humans select for desirable or fashionable traits. Breeders deliberately mate parents with the hope of producing offspring with specific traits (such as color, size, ear shape, snout length, and so on).
Common mistakes and misconceptions
Evolution is not the same as adaptation or natural selection. Natural selection is a mechanism, or cause, of evolution. Adaptations are physical or behavioral traits that make an organism better suited to its environment.
Heritable variation comes from random mutations. Random mutations are the initial cause of new heritable traits. For example, a rabbit can't choose to have a different fur color. Rather, a genetic mutation causes a difference in fur color, which may help that rabbit hide better in its environment.
Natural selection acts on existing heritable variation. Natural selection needs some starting material, and that starting material is heritable variation. For natural selection to act on a feature, there must already be variation, and that variation must be able to be passed on to offspring.
Natural selection depends on the environment. Natural selection doesn't favor traits that are somehow inherently superior. Instead, it favors traits that are beneficial in a specific environment. Traits that are helpful in one environment might actually be harmful in another.
Step-by-step explanation:
Becky needs to decide how many cups of coffee and pieces of cake she will buy for a meeting with an office mate. She estimates her total utility for different quantities of coffee and cakes below. A cup of coffee is $3.00, and each piece of cake is $1.50. Becky has a total of S9 to spend. What is Becky's total utility if she buys 1 cup of coffee and 4 pieces of cake? Cups of Coffee 1 2 3 4 Utility from Consumption Total Utility from Pieces of Cups of Coffee Cake 35 1 61 2 84 3 Total Utility from Pieces of Cake 16 33 49 64 78 91 4 5 5 6 6 А 0 Total Utility of Different Consumption Bundles Consumption Bundle Cups of Coffee Pieces of Cake Total Utility 3 2 2 1 D B с 4 0 6
If Becky buys 1 cup of coffee and 4 pieces of cake, her total utility can be calculated by summing the individual utilities for each item consumed. According to the given estimates, the total utility from 1 cup of coffee is 35, and the total utility from 4 pieces of cake is 91. Therefore, Becky's total utility for this consumption bundle would be 35 + 91 = 126.
The table provided showcases the relationship between the quantities of coffee and cake and their respective utilities. Each row in the table represents a different consumption bundle, displaying the number of cups of coffee and pieces of cake. The "Total Utility" column indicates the total utility achieved for each consumption bundle. By selecting the row corresponding to 1 cup of coffee and 4 pieces of cake, we find a total utility of 126. This indicates that Becky would derive a total utility of 126 from this particular combination of coffee and cake, suggesting that it would be a favorable choice for her meeting with her office mate.
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PLS HURRY! ANSWER FOR BRAINLY.
question:
Which statement best describes the equation
x = 3?
A. The equation can not be graphed.
B. The equation represents a function, but not a linear function.
C. The equation represents a linear function.
D. The equation represents a line, but not a linear function.
Answer:
D
Step-by-step explanation:
A: Not A. It can be graphed. It is a vertical line where x = 3. Every point must have an x value of 3. (3,4), (3,0), (3.- 6) are all on x = 4. It can be graphed, but it is not a function.
B: Not B. It is not a function at all. To be a function, it can only have 1 y value associated with it.
C: It's not a linear function because it is not a function.
D is your answer
What is the equation of the graph below?
y = sec(x – 4)
y = sec(x) – 4
y = sec(x + 4) + 4
y = sec(x + 4) – 4
Answer:
It's A don't listen to the other guy
Step-by-step explanation:
Answer:
A
Step-by-step explanation:
edg 22
Find all solutions to the equation 34 – 38 = 3 where a, ß are positive integers (or show there are none). (6) Find all solutions where x and y are integers to the diophantine equation 1/x + 1/y =1/5
a) There are no solutions to the equation.
To find all solutions to the equation 34 - 38 = 3, we can simplify the equation:
34 - 38 = 3
-4 = 3
However, this equation is not true, as -4 is not equal to 3. Therefore, there are no solutions to the equation.
b) Now, let's consider the diophantine equation 1/x + 1/y = 1/5, where x and y are integers. To find all solutions, we can follow these steps:
1: Multiply both sides of the equation by 5xy to eliminate the denominators:
5y + 5x = xy
2: Rearrange the equation:
xy - 5x - 5y = 0
3: Apply Simon's Favorite Factoring Trick:
Add 25 to both sides to complete the square:
xy - 5x - 5y + 25 = 25
xy - 5x - 5y + 25 = 25
(xy - 5x - 5y + 25) + 25 = 25 + 25
(x - 5)(y - 5) = 50
4: List all possible factor pairs of 50:
(1, 50), (2, 25), (5, 10), (-1, -50), (-2, -25), (-5, -10)
5: Solve for x and y in each factor pair:
For (x - 5)(y - 5) = 50:
If x - 5 = 1 and y - 5 = 50, then x = 6 and y = 55.
If x - 5 = 2 and y - 5 = 25, then x = 7 and y = 30.
If x - 5 = 5 and y - 5 = 10, then x = 10 and y = 15.
If x - 5 = -1 and y - 5 = -50, then x = 4 and y = -45.
If x - 5 = -2 and y - 5 = -25, then x = 3 and y = -20.
If x - 5 = -5 and y - 5 = -10, then x = 0 and y = -5.
Therefore, the solutions to the diophantine equation 1/x + 1/y = 1/5, where x and y are integers, are:
(x, y) = (6, 55), (7, 30), (10, 15), (4, -45), (3, -20), (0, -5).
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From her eye, which stands 1.75 meters above the ground, Myesha measures the angle of elevation to the top of a prominent skyscraper to be 19 degrees
. If she is standing at a horizontal distance of 337 meters from the base of the skyscraper, what is the height of the skyscraper? Round your answer to the nearest hundredth of a meter if necessary.
The height of the skyscraper is approximately 115.25 meters (rounded to the nearest hundredth of a meter).
To find the height of the skyscraper, we can use trigonometry and the information provided about the angle of elevation and the horizontal distance.
Let's denote the height of the skyscraper as h. We are given that Myesha's eye height above the ground is 1.75 meters, and she measures the angle of elevation to be 19 degrees.
In a right triangle formed by Myesha's eye, the top of the skyscraper, and a point on the ground directly below the top of the skyscraper, the angle of elevation (θ) is the angle between the line of sight from Myesha's eye to the top of the skyscraper and the horizontal ground.
The opposite side of the triangle is the height of the skyscraper (h), and the adjacent side is the horizontal distance from Myesha to the base of the skyscraper (337 meters).
Using the trigonometric function tangent (tan), we can set up the following equation:
tan(θ) = h / 337
Since we know the value of the angle of elevation (θ = 19 degrees), we can substitute it into the equation:
tan(19 degrees) = h / 337
Now we can solve for h:
h = tan(19 degrees) * 337
Using a calculator or trigonometric tables, we find that tan(19 degrees) is approximately 0.34202. Substituting this value into the equation:
h = 0.34202 * 337
h ≈ 115.25
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To figure out the distance for a trip, you use a ruler to measure the distance from Orlando to Gainesville on the map. You measure 2.3 cm. Find the actual mileage
between the two cities, rounded to the nearest mile.
will give u brainlist
Answer:
The answer is 71
Step-by-step explanation:
Why because
1 cm /31 mi = 2.3 cm / m
1 x m = 31 x 2.3
n = 71.3
71.3 rounded to nearest mile is 71.
Question 5
Find MZA
(5x - 1)
(4x + 17)°
B
A
(3y - 20°
(2y + 1°
Need help with this question?
Answer:
m∠A is 89°
Step-by-step explanation:
By circle theorem, two or more inscribed angles subtended by the same arc are equal
Therefore, from the diagram, we have;
Angle (5·x - 1)° and angle (4·x + 17)° subtends the same arc DC
∴ (5·x - 1)° = (4·x + 17)°
We can write the angles as follows since they are both in degrees
(5·x)° - 1° = (4·x)° + 17°
(5·x - 4·x)° = 17° + 1°
x = 18°
m∠A = (5·x - 1)°
∴ m∠A = (5 × 18 - 1)° = (90 - 1)° = 89°
m∠A = 89°.
In a group of 700 people, must there be 2 who have matching first and last initials? Why? (Assume each person has a first and last name.)
---Select--- Yes No . Let A be the set of 700 distinct people and let B be the different unique combinations of first and last initials. If we construct a function from A to B, then by the ---Select--- pigeonhole zero product mathematical induction principle, the function must be ---Select--- onto, one-to-one, not a one-to-one correspondence . Therefore, in a group of 700 people, it is ---Select--- possible, impossible, that no two people have matching first and last
Yes, in a group of 700 people, there must be two people with the identical first and last initials.
Because there are 26 distinct letters in the English alphabet, there are 26 × 26 = 676 potential combinations of two initials. According to the pigeonhole principle, at least two people in a group of 700 share the same initials. (Here: persons are pigeons, combinations of two initials are pigeonholes).
Note that in alphabets with 27 or more letters, this is no longer valid because there are 27 × 27 = 729 > 700 options for the initials.
If we try to assign a distinct combination to each individual in the group, we can only do so for the first 676 persons, leaving the remaining 24 with the same first and last initial. As a result, the answer is yes, there must be two people with the same first and last initials.
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Correct question:
In a group of 700 people, must there be 2 who have the same first and last initials? Why?
Which expression is already in its simplified form?
a.) 3t+ (s +t)
b.) 6h + 2(h+ 3k)
c.) 5m + 7mn + 4n
d.) 8 + y3 - 5z2 + 2y3
In a local, recreational soccer league there are 252 men and 90 women signed up to play. The organizers need to make sure that everyone is on a team, the teams are the same size, each team has the same number of males; and each team has the same number of females. a. What is the largest possible number of teams? b. How many people will be on each team?
(a) There can be a maximum of 18 teams.
(b) The largest possible number of teams is 18, and there will be a total of 19 players on each team.
Given information:
In a local, recreational soccer league there are 252 men and 90 women signed up to play. The organizers need to make sure that everyone is on a team, the teams are the same size, each team has the same number of males; and each team has the same number of females.
(a) To find the largest possible number of teams, we need to divide the players (252 men and 90 women) into teams, where each team has the same number of males and females. Let's find the GCD (greatest common divisor) of 252 and 90:
GCD of 252 and 90 = GCD (252,90) = 18.
Therefore, there can be a maximum of 18 teams.
(b) Since the number of teams is 18, and we have 252 men and 90 women, there will be a total of 342 players.
Now, we need to divide these 342 players into 18 teams. Each team must have an equal number of males and females. Number of males = 252, Number of females = 90.
To divide the players equally into teams, we need to find the number of males and females that can be in one team.
Number of males in one team = 252/18 = 14
Number of females in one team = 90/18 = 5
Therefore, each team will have 14 males and 5 females, and there will be a total of 19 players on each team (14 + 5).
Thus, the largest possible number of teams is 18, and there will be a total of 19 players on each team.
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Let a/m and b/n be rational numbers expressed in lowest terms. Prove that (an + bm)/(mn) is in lowest terms if and only if m and n are relatively prime.
If m and n are relatively prime, the numerator (an + bm) and denominator (mn) have no common factors other than 1, and therefore, (an + bm)/(mn) is in lowest terms.
To prove that (an + bm)/(mn) is in lowest terms if and only if m and n are relatively prime, we will need to establish two separate claims:
Claim 1: If (an + bm)/(mn) is in lowest terms, then m and n are relatively prime.
Claim 2: If m and n are relatively prime, then (an + bm)/(mn) is in lowest terms.
Proof of Claim 1:
Let's assume that (an + bm)/(mn) is in lowest terms. This means that the numerator (an + bm) and the denominator (mn) have no common factors other than 1.
Suppose m and n are not relatively prime, which means they have a common factor greater than 1.
Let's say that factor is d, where d > 1. Then we can express m and n as follows: m = dx and n = dy, where x and y are integers.
Now, we rewrite the numerator (an + bm) as follows:
an + bm = an + b(dx) = n(a + bx/d)
Since m = dx and n = dy, we have (an + bm)/(mn) = (n(a + bx/d))/(dx * dy) = (a + bx/d)/(xy).
We can see that both the numerator (a + bx/d) and the denominator (xy) have the factor d.
This contradicts our assumption that (an + bm)/(mn) is in lowest terms, as they have a common factor greater than 1.
Therefore, m and n must be relatively prime.
Proof of Claim 2:
Now, let's assume that m and n are relatively prime, which means they have no common factors other than 1.
To show that (an + bm)/(mn) is in lowest terms, we need to demonstrate that its numerator and denominator have no common factors other than 1.
Let's suppose that the numerator (an + bm) and the denominator (mn) have a common factor d, where d > 1.
This implies that d divides both an + bm and mn.
Since d divides mn, we can express m and n as follows: m = dx and n = dy, where x and y are integers.
Now, let's rewrite the numerator (an + bm) as follows:
an + bm = an + b(dx) = n(a + bx/d)
We can see that d divides both n and the expression (a + bx/d). Therefore, d also divides the numerator (an + bm).
However, we initially assumed that (an + bm)/(mn) is in lowest terms, which means the numerator and denominator have no common factors other than 1. This contradicts the assumption that they have a common factor d > 1.
Thus, if m and n are relatively prime, the numerator (an + bm) and denominator (mn) have no common factors other than 1, and therefore, (an + bm)/(mn) is in lowest terms.
By proving both Claim 1 and Claim 2, we have established that (an + bm)/(mn) is in lowest terms if and only if m and n are relatively prime.
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You have a green house. The green house is made completely of glass, except
for the door. The entire building is 15 feet tall. The height of the vertical walls is 10 ft. The
green house is 20 ft long (on side with door) and 16 feet wide. The triangles that make up the
roof are isosceles triangles (both sides are equal and height is measured at the middle of the base).
The door is 8 feet wide and 7 feet tall. Answer each of the following questions about your greenhouse.
1. How much glass is used to make this greenhouse?
The height of the vertical walls is 10 ft=480 sq ft. The triangles that make up the roof are isosceles triangle 80 sq ft. The greenhouse uses 504 square feet of glass to construct.
To calculate the amount of glass used to make the greenhouse, to determine the total surface area of the glass panels.
The greenhouse consists of four vertical rectangular walls and two triangular roof sections.
The total surface area of the vertical walls calculated by adding the areas of each wall. Since the greenhouse is rectangular, the height of the walls is 10 ft, and the total width of the walls (including the door) is 16 ft + 8 ft = 24 ft.
Area of each wall = height x width
Area of vertical walls = 2 x (height x width) = 2 x (10 ft x 24 ft) = 480 sq ft
Each roof section forms an isosceles triangle. The base of each triangle is the width of the greenhouse, which is 16 ft. The height of the triangle is the difference between the total height of the greenhouse (15 ft) and the height of the vertical walls (10 ft), which is 5 ft.
Area of each triangular roof section = 0.5 x base x height
Area of triangular roof = 2 x (0.5 x 16 ft x 5 ft) = 80 sq ft
The door is not made of glass, so we need to subtract its area from the total.
Area of the door = width x height = 8 ft x 7 ft = 56 sq ft
calculate the total glass surface area by subtracting the area of the door from the sum of the vertical walls and triangular roof sections.
Total glass surface area = (Area of vertical walls + Area of triangular roof) - Area of the door
Total glass surface area = (480 sq ft + 80 sq ft) - 56 sq ft
Total glass surface area = 504 sq ft
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Of the 100 gymnasts at a meet, 34 are boys. Write a decimal for the part of the
gymnasts that are girls.
Answer:
0.66
Step-by-step explanation:
1.00 - 0.34 = 0.66
a camera has a listed price of $634.95 before tax. if the sales tax rate is 8.75%, find the total cost of the camera with sales tax included. round your answer to nearest cent, as necessary
Answer:
690.50
Step-by-step explanation:
3. A system of functions is given. Select
ALL values where f(x) = g(x). Round all
answers to the nearest tenth
f(x) = - x - 2x + 6
g(x) = 2x2 + 5x + 3
a) -2.7
b) -1.8
c) -0.6
d) 0.4
e) 4.1
f) 5.1
g)7.0
Answer:
Step-by-step explanation:
The easiest way is to graph both functions and see where they intersect. The intersection is where f(x) = g(x).
f(x) = -x² - 3x + 6
g(x) = 2x² + 5x + 3
The distance y (in miles) that a truck travels on x gallons of gasoline is represented by the equation y=18x The graph shows the distance that a car travels. Which vehicle gets the better gas mileage?
Answer:
Car M:
50.4/2 = 25.2
car M uses up 1 gallon every 25.2 miles
Car P:
Just from the graph, you can see that it uses up 1 gallon every 30 miles
The two graphs vary the /miles slightly but it is around their zones of 25.2 and 30. It varies slightly because the cars may be traveling at a fast speed or slower speed thus using up more or less fuel by the time they've reached the recorded distances on the graphs.
Answer:
Step-by-step explanation: