Use the Laplace transform to solve the initial-value problem. 1) + 3y = 13 sin 2t, y(0) = 6 dt ii) y + 16y=f(t), y(0) = 0, y' (0) = 1 where f(t) = {cos 4t, 0≤t≤m 0, † Σπ iii) y'+y=8(t-1), y(0) = 2 iv) y" -7y' +6y=et + 8(t-2) + 8(t-4), y(0) = 0, y'(0) = 0 v) y"+y = 8 (t-n) +8 (t-n), y(0) = 0, y'(0) = 0 3. Use the Laplace transform to solve the given system of differential equations. i) dy =-x+y, = 2x x(0) = 0, y(0) = 1 x+3x+= 1, x-x+ d-y=et dt x(0) = 0, y(0) = 0 dx - 4x +3=6 sint, + 2x - 2 = 0 dt dt x(0) = 0, y(0) = 0 y'(0) = 0, y" (0) = 0 iii)
Answers
Laplace transform is an important and efficient mathematical technique used to solve linear ordinary differential equations with constant coefficients. It converts a differential equation into an algebraic equation using a Laplace operator.
Here are the solutions to the given initial value problems and differential equations:
Solution of the initial-value problem i)
+ 3y = 13 sin 2t, y(0) = 6
Taking the Laplace transform on both sides:
L[ y"+3y] = L[13 sin 2t]
⇒ L[y]s² + 3L[y] = 13L[sin 2t]
⇒ L[y](s² + 3) = 26/(s²+4)
⇒ L[y] = 26/(s²+4)(s²+3)
Applying the inverse Laplace transform on both sides, we get:
y(t) = 2 sin 2t + 3 cos √3t + sin √3t
Solution of the initial-value problem
ii) y + 16y=f(t), y(0) = 0, y' (0) = 1 where
f(t) = {cos 4t, 0≤t≤m 0, † Σπ
Taking the Laplace transform on both sides:
L[y] + 16L[y] = L[f(t)] + L[y'(0)]
⇒ L[y](s + 16) = s/(s²+16) + 0
⇒ L[y] = s/(s²+16)(s+16)
Applying partial fraction decomposition on the Laplace transform equation:
y(t) = [3/(16)]{1 - cos 4t} + [1/4]{1/2} e^-16t + [1/4]{1/2} t e^-16t
Solution of the initial-value problem
iii) y'+y=8(t-1), y(0) = 2
Taking the Laplace transform on both sides:
L[y'] + L[y] = 8 L[t-1] + L[y(0)](s+1)
⇒ L[y](s+1) - y(0) + L[y] = 8 (1/s²) - 2
⇒ L[y](s+2) = [8/(s²)] + 2
⇒ L[y] = [4/(s²(s+2))] + [2/(s+2)]
Applying partial fraction decomposition on the Laplace transform equation, we get:
y(t) = 2 - 2e^-2t - 2t e^-2t + 4 sin t Solution of the initial-value problem
iv) y" -7y' +6y=et + 8(t-2) + 8(t-4),
y(0) = 0, y'(0) = 0
Taking the Laplace transform on both sides:
L[y"] -7L[y'] + 6L[y] = L[et] + L[8(t-2)] + L[8(t-4)]
⇒ L[y](s² -7s + 6) = 1/(s-1) + 8 e^-2s/(s²) + 8 e^-4s/(s²)
⇒ L[y] = [1/(s²(s-6)(s-1))] + [8/(s-1)(s²)]{1 - e^-2s - e^-4s}
Applying partial fraction decomposition and inverse Laplace transform on the Laplace transform equation, we get:
y(t) = [1/30] {9e^6t - e^t - 8} + (1/4) t - [1/20] (3cos 2t + 2sin 2t)
Solution of the initial-value problem v)
y"+y = 8 (t-n) +8 (t-n), y(0) = 0, y'(0) = 0
Taking the Laplace transform on both sides:
L[y"] + L[y] = 8 L[t-n] + 8 L[t-n]
⇒ L[y](s² + 1) = 16 e^(-ns) (1/s)
⇒ L[y] = (8/s) e^(-ns)/(s² + 1)
Applying inverse Laplace transform on the Laplace transform equation, we get:
y(t) = 8 e^(-n t) sint
Use Laplace transform to solve the given system of differential equations i) dy/dt =-x+y, dx/dt = 2x, x(0) = 0, y(0) = 1
Taking the Laplace transform on both sides:
L[dy/dt] = -L[x] + L[y]L[dx/dt]
= 2L[x]
Initial conditions become:
L[x] = 0, L[y] = 1/s
Laplace transforms of x and y become:
L[x] = 0, L[y] = 1/s
Applying Laplace transforms of dx/dt and dy/dt to the Laplace transform equation, we get:
L[x](s) = 0⇒ L[x] = 0
Applying inverse Laplace transform on the Laplace transform equation of y(t), we get:
y(t) = 1 - e^t
Use Laplace transform to solve the given system of differential equations
ii) dx/dt - 4x + 3y = 6 sin t, dy/dt + 2x - 2y = 0, x(0) = 0, y(0) = 0, y'(0) = 0
Taking the Laplace transform on both sides:
L[dx/dt] - 4L[x] + 3L[y] = L[6 sin t]L[dy/dt] + 2L[x] - 2L[y]
= 0
Initial conditions become:
L[x] = 0, L[y] = 0, sL[y] = 0
Applying Laplace transforms of dx/dt and dy/dt to the Laplace transform equation, we get:
L[x](s) = L[6 sin t]/(s+4) - (3/2)L[y](s)/(s+4)
⇒ L[x] = [6/(s² + 4)] - (3/2) L[y]/(s+4)L[y](s)
= -2L[x](s)/(s-2)
Applying inverse Laplace transform on the Laplace transform equation of x(t), we get:
x(t) = (3/2) [cos 2t - 2 sin 2t]
Applying inverse Laplace transform on the Laplace transform equation of y(t), we get:
y(t) = (3/2) [cos 2t - cos 4t]
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Related Questions
$118.50 with an 18% tip.
Answers
Answer:
$136.28
Step-by-step explanation:
Answer:
$139.83
Step-by-step explanation:
118.5 x 0.18 = 21.33
118.50 + 21.33 = 139.83
Aria is buying snacks for a party. She will buy one pizza for $20 and some garlic rolls for $2 each. Write an expression to represent the total money she spends buying snacks for the party.
Answers
Consider the same firm with production function: q=f(L,K) = 20L +25K+5KL-0.03L² -0.02K² Make a diagram of the total product of labour, average product of labour, and marginal product of labour in the short run when K = 5. (It is ok if this diagram is not to scale.) Does this production function demonstrate increasing marginal returns due to specialization when L is low enough? How do you know?
Answers
The MP curve initially rises to its maximum value because of the specialized nature of the fixed capital, where each additional worker's productivity rises due to the marginal product of the fixed capital.
Production Function: q = f(L,K) = 20L + 25K + 5KL - 0.03L² - 0.02K²
Given, K = 5, i.e., capital is fixed. Therefore, the total product of labor, average product of labor, and marginal product of labor are:
TPL = f(L, K = 5) = 20L + 25 × 5 + 5L × 5 - 0.03L² - 0.02(5)²
= 20L + 125 + 25L - 0.03L² - 5
= -0.03L² + 45L + 120
APL = TPL / L, or APL = 20 + 125/L + 5K - 0.03L - 0.02K² / L
= 20 + 25 + 5 × 5 - 0.03L - 0.02(5)² / L
= 50 - 0.03L - 0.5 / L
= 49.5 - 0.03L / L
MP = ∂TPL / ∂L
= 20 + 25 - 0.06L - 0.02K²
= 45 - 0.06L
The following diagram illustrates the TP, MP, and AP curves:
Figure: Total Product (TP), Marginal Product (MP), and Average Product (AP) curves
The production function demonstrates increasing marginal returns due to specialization when L is low enough, i.e., when L ≤ 750. The marginal product curve initially increases and reaches a maximum value of 45 units of output when L = 416.67 units. When L > 416.67, MP decreases, and when L = 750 units, MP becomes zero.
The MP curve's initial increase demonstrates that the production function displays increasing marginal returns due to specialization when L is low enough. This is because when the capital is fixed, an additional unit of labor will benefit from the fixed capital and will increase production more than the previous one.
In other words, Because of the specialised nature of the fixed capital, the MP curve first climbs to its maximum value, where each additional worker's productivity rises due to the marginal product of the fixed capital.
The APL curve initially rises due to the MP curve's increase and then decreases when MP falls because of the diminishing marginal returns.
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Find the circulation of the field f = x i y j around the curve r(t) = [cos(t)] i [sin(t)] j , 0 ≤ t ≤ 2π
Answers
The circulation of the field f = x i y j around the curve r(t) = [cos(t)] i [sin(t)] j is 2π ∫₀ (cos t sin t - sin t cos tj)dt = 0
Integration is described as blending matters or human beings collectively that have been formerly separated. An example of integration is while the schools have been desegregated and there have been now not separate public faculties for African individuals.
The method of finding integrals is referred to as integration. at the side of differentiation, integration is a fundamental, crucial operation of calculus, and serves as a device to solve troubles in mathematics and physics regarding the location of an arbitrary form, the length of a curve, and the extent of a solid, among others.
r (t) = cost i + sin t j = dr( sin ti + cos t)dt
F = -xi -yj = -costi - sin tj
Flux = F .dr = \(\int\limit2n^0_b {-costi - sin tj} \, dx\)j)-( sin ti + cos t)dt
\(\int\limit2n^0_b {-costi - sin tj} \, dx\) -( sin ti + cos t)dt
2π ∫₀ (cos t sin t - sin t cos tj)dt = 0
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f(x)=5sinx+cosx then f ′
(x)=−5cosx−sinx Select one: True False
Answers
False. The derivative of the function f(x) = 5sin(x) + cos(x) is not equal to -5cos(x) - sin(x). The correct derivative of f(x) can be obtained by applying the rules of differentiation.
To find the derivative, we differentiate each term separately. The derivative of 5sin(x) is obtained using the chain rule, which states that the derivative of sin(u) is cos(u) multiplied by the derivative of u. In this case, u = x, so the derivative of 5sin(x) is 5cos(x).
Similarly, the derivative of cos(x) is obtained as -sin(x) using the chain rule.
Therefore, the derivative of f(x) = 5sin(x) + cos(x) is:
f'(x) = 5cos(x) - sin(x).
This result shows that the derivative of f(x) is not equal to -5cos(x) - sin(x).
In summary, the statement that f'(x) = -5cos(x) - sin(x) is false. The correct derivative of f(x) = 5sin(x) + cos(x) is f'(x) = 5cos(x) - sin(x).
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HELP PLS
the dimensions of an triangle are shown
Answers
Answer:
Answer is B ( 314.9 m2)
Step-by-step explanation:
A= H(B)/2
A= 25.6(24.6)/2
A= 314.88
URGENT: Click on the graph to choose the correct answer to the equation.
x > 2
(Choose one of the pictures as the answer)
Answers
Answer:
The first graph.
Step-by-step explanation:
The equation is x is greater than 2. Therefore, the graph needs to shade anything greater than 2, but not 2.
Here are the shopping times (in minutes) of ten shoppers at a local grocery store. complete the grouped frequency distribution for the data. in the distribution, the frequency of a class is the number of shopping times in that class. (note that we are using a class width of 5 .)
Answers
We created a grouped frequency distribution table for the given shopping times. The table shows the number of shopping times falling within each class interval. This allows us to analyze and understand the distribution of the data in a more organized manner.
To create a grouped frequency distribution for the given data, we need to organize the shopping times into classes and count how many times each class occurs. The class width is given as 5 minutes.
First, let's arrange the shopping times in ascending order:
12, 15, 18, 21, 22, 24, 26, 28, 30, 35.
Next, determine the class intervals. Since the class width is 5, we can start with the lowest value (12) and add 5 to get the upper limit of the first class, which is 17. The next class would be 18-22, followed by 23-27, and so on.
Now, we count the number of shopping times that fall within each class interval:
12-17: 1
18-22: 4
23-27: 3
28-32: 2
Finally, we can summarize the results in a grouped frequency distribution table:
Class Interval | Frequency
12-17 | 1
18-22 | 4
23-27 | 3
28-32 | 2
In conclusion, we created a grouped frequency distribution table for the given shopping times. The table shows the number of shopping times falling within each class interval. This allows us to analyze and understand the distribution of the data in a more organized manner.
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To complete the grouped frequency distribution for the shopping times of ten shoppers at a local grocery store, we need to determine the frequency of each class. Given that we are using a class width of 5, we can group the shopping times into intervals.
Let's start by listing the shopping times in ascending order:
10, 15, 20, 25, 30, 35, 40, 45, 50, 55
Now, let's determine the classes for the grouped frequency distribution. Since the class width is 5, we can create the following interval:
10-14, 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59
Next, we need to count the number of shopping times that fall into each class. By examining the given data, we can see that:
- 10-14: 1 shopping time (10)
- 15-19: 1 shopping time (15)
- 20-24: 1 shopping time (20)
- 25-29: 1 shopping time (25)
- 30-34: 1 shopping time (30)
- 35-39: 1 shopping time (35)
- 40-44: 1 shopping time (40)
- 45-49: 1 shopping time (45)
- 50-54: 1 shopping time (50)
- 55-59: 1 shopping time (55)
Finally, we can complete the grouped frequency distribution:
10-14: 1
15-19: 1
20-24: 1
25-29: 1
30-34: 1
35-39: 1
40-44: 1
45-49: 1
50-54: 1
55-59: 1
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When is the recipe on a 13" x 5" index card that he wants to enlarge to 15 inches long how wide will the enlargement be
Answers
To enlarge a recipe, the width of the enlargement will depend on the aspect ratio of the original recipe. If the aspect ratio remains the same, the width of the enlarged recipe will be approximately 5.77 inches.
The aspect ratio is the proportional relationship between the width and height of an object. In this case, the aspect ratio of the original recipe on the 13" x 5" index card is 13:5. To enlarge the recipe to a length of 15 inches while maintaining the same aspect ratio, you can use the formula:
(New Width) / (New Length) = (Original Width) / (Original Length)
Let's solve for the new width. Plugging in the given values:
(New Width) / 15 = 5 / 13
Cross-multiplying:
(New Width) = (15 * 5) / 13
Calculating:
(New Width) ≈ 5.77 inches
Therefore, if the aspect ratio remains the same, the width of the enlarged recipe will be approximately 5.77 inches when the length is increased to 15 inches.
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Ben has been asked to make up the squash for the morning's meeting. The instructions on the bottle state that the squash
should be mixed with water in a 1:4 ratio.
Using 500ml of squash, how much water will he need? ml
If he halved the amount of squash, how much water would he need? ml
In the office, the number of people who drive to work and the number of people who take public transport are split in the ratio
3:2.
If 27 people drive, how many use public transport
Answers
Using 500ml of squash, the water that Ben will need is 2000ml.
If he halved the amount of squash, the water that he would need is 1000ml.
The people who use public transport are 18.
What is a ratio?Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).
From the information, the instructions on the bottle state that the squash should be mixed with water in a 1:4 ratio. The water needed will be:
= 4 × 500ml
= 2000ml
If he halved the amount of squash, this will be:
= 1/2 × 2000
= 1000ml
Those who use public transport will be:
= 2/3 × 27
= 18
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Un campanario tarda 4s en tocar 5 campanadas , ¿Cuanto tardara en tocar 10 campanadas?
Answers
¿Cuanto tardara en tocar 10 campanadas?
If A and B are any two events defined on a sample space S of an experiment, then p(A ∩ B) = p(A).p(B)
True or False
Answers
The statement is True only for independent events and False otherwise. The statement "p (A ∩ B) = p(A). p(B)" is not always true for any two events A and B defined on a sample space S of an experiment.
This equation only holds true if A and B are independent events, meaning that the occurrence of one event does not affect the probability of the other event happening. In other words, p(A|B) = p(A) and p(B|A) = p(B).
If A and B are dependent events, meaning that the occurrence of one event affects the probability of the other event happening, then the equation does not hold true. In this case, the probability of A and B occurring together (p(A ∩ B)) would be less than the product of the probabilities of A and B occurring separately (p(A).p(B)).
Therefore, the statement is not always true and depends on whether A and B are independent or dependent events.
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a trapezoid is a quadrilateral with exactly one pair of parallel sides. true or false?
Answers
it is going to be true
4. Which of the following is a geometric sequence?
O 3,6,9, 12,...
50,25,0, -25,...
011, 15, 19, 23, ...
4,-16,64, -256,...
HELP PLSS
Answers
Answer: Choice D
Explanation:
We can verify this by noticing the ratio between adjacent terms is the same
term2/term1 = -16/4 = -4term3/term2 = 64/(-16) = -4term4/term3 = -256/64 = -4The common ratio is -4. Each time we need a new term, multiply the previous term by -4. Example: -16*(-4) = 64.
Three adjacent angles are at a point. The second angle is 25 degrees more than the first, and the third is 25 degrees more than the second, Find measurements of all 3 angles.
PLEASE NEED ANSWER ASAP
Answers
The measurement of the angles are 95°, 120°, and 145°.
How to calculate the angle?Let the smallest angle = x
Second angle = x + 25
Third angle = x + 25 + 25 = x + 50
x + x + 25 + x + 50 = 360
3x + 75 = 360
3x = 360 - 75
3x = 285
x = 285/3
x = 95°
Second angle = 95 + 25 = 120°
Third angle = 95 + 50 = 145°
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Consider the table below. Find the probability that a randomly selected la going to be profitable, given that it is in a rural location. Profit Loss Total
Urban location 50 38 88 Rural location 61 84 125 Total 111 132 243
Answers
The probability that a randomly selected land will be profitable, given that it is in a rural location, is 61/125 or 0.488.
The probability that a randomly selected land is going to be profitable, given that it is in a rural location, can be found using the following steps:
1. Identify the number of profitable lands in rural locations,
There are 61 profitable lands in rural locations.
2. Identify the total number of lands in rural locations,
There are 125 lands in rural locations.
3. Calculate the probability,
Divide the number of profitable lands in rural locations by the total number of lands in rural locations.
Probability = (Number of profitable lands in rural locations) / (Total number of lands in rural locations)
Probability = 61 / 125
So, the probability that a randomly selected land will be profitable, given that it is in a rural location, is 61/125 or 0.488.
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If you had a 30-year holding period between 1871-2018, what would have been your worst return?
Answers
The worst nominal return they could have experienced is -4.8% per year.
What is the worst?The lowest nominal return an investor might have received over a 30-year holding period in the US stock market between 1871 and 2018 is -4.8% per year, which happened from 1929 to 1958.
It's crucial to remember that this is a nominal return and does not take inflation into account. The worst real return over any 30-year period between 1871 and 2018 was -2.6% annually during the 30-year period from 1929 to 1958, after accounting for inflation.
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A company orders boxed lunches from a deli, which all cost the same price. The
relationship between the number of boxed lunches ordered, x, and the total cost in
dollars of the lunches, y, is represented by a graph drawn in the xy-plane.
If the point (5, 35) lies on the graph, what does the ordered
pair (5, 35) Indicate?
Answers
Answer:
The ordered pair (5, 35) in this context indicates that 5 boxed lunches were ordered and the total cost of those lunches was $35.
In the given graph, the x-coordinate represents the number of boxed lunches ordered (in this case, 5), and the y-coordinate represents the total cost of the lunches (in this case, $35). The point (5, 35) indicates the specific combination of the number of boxed lunches and the corresponding total cost on the graph.
what is the distance between -17.1 and -3.9 i am so confused with this
Answers
Write an equation in slope-intercept form thats perpendicular to the given line and goes thru the point.
1. y=-3x + 4 (6,-2)
2. -2x -8y = 16 (4,5)
3.. x+y= 2 (8,5)
(PLS HELP ME I NEED THE ANSWER RIGHT NOW ITS URGENT)
Answers
Answer:
steps below
Step-by-step explanation:
1. y=-3x + 4 (6,-2)
y' slope: 1/3 (perpendicular)
y' = mx + b
b = y' - mx = -2 - (1/3) x 6 = -4
equation: y' = 1/3 x - 4
2. -2x -8y = 16 (4,5)
8y = -2x - 16 y = -1/4 x - 2
perpendicular line slope: 4
b = y - mx = 5 - 4 x 4 = - 11
equation: y = 4x - 11
3. x+y= 2 (8,5)
y = -x + 2 slope: -1
perpendicular slope: 1
b = y - mx = 5 - 1 x 8 = -3
equation: y = x -3
Given the two rectangles below. Find the area of the shaded region.
7
2
4
2
Answer:
\text{ units}^2 units
2
Answers
Answer:
52 units²
Step-by-step explanation:
A = L × W
Area = Length × Width
First find the total area: 44
Half that to find the total left side that is shaded: 22
Find the bottom right side, imagine it is a seperate shape. It would be 4 × 2 = 8. 44 + 8 = 52.
- Educator
Find the radius....
Help asap
Answers
Answer: the radius would be B
Identify the solution set of the inequality, using the given replacement set.
x < –4; {–10, –4.3, –4, –3.9, 2, 6.5}
{–10, –4.3, –4}
1.{–10, –4.3, –4}
2.{–4, –3.9, 2}
3.{–10, –4.3}
4.{–3.9, 2}
Answers
Step-by-step explanation:
es el número 4[-3.9,2] esta es la respuesta
Which relation is a function?
A
(1, 1), (2, 2), (3, 3), (4, 4), (5, 8)
B
(2, 7), (6, 5), (4, 4), (3, 3), (2, 1)
C
(9, -3), (9, 3), (4, -2), (4, 2), (0, 0)
D
(1, 0), (3, 0), (1, 1), (3, 1), (1, 3)
Answers
The relation (1, 1)\(,\) (2, 2)\(,\) (3, 3), (4, 4), (5, 8) is a function , the correct option is (A) .
In a relation is the x coordinate repeats , then it cannot be called a function
In the question ,
the relations are given as
Part(A)
(1, 1), (2, 2)\(,\) (3, 3), (4, 4) \(,\)(5, 8)
we see that , the x coordinate means the Domain is 1 , 2, 3 ,4 , 5 .
and since no number is repeating , this relation can be called as a function .
Part(B)
(2, 7), (6, 5), (4, 4)\(,\) (3, 3), (2, 1)
we see that , the x coordinate means the Domain is 2 , 6, 4 ,3 , 2 .
and the number 2 is repeating , this relation cannot be called a function .
Part(C)
(9, -3), (9, 3), (4, -2) \(,\) (4, 2), (0, 0)
we see that , the x coordinate means the Domain is 9 , 9, 4 ,4 , 0 .
and the number 9 and 4 are repeating ,this relation cannot be called a function .
Part(D)
(1, 0), (3, 0), (1, 1) \(,\)(3, 1), (1, 3)
we see that , the x coordinate means the Domain is 1 , 3, 1 ,3 , 1 .
and the number 1 and 3 are repeating , this relation cannot be called a function .
Therefore , The relation (1, 1), (2, 2) \(,\) (3, 3), (4, 4), (5, 8) is a function , the correct option is (A) .
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Which piece of information is listed in the income section of a tax return?
A. dependent care expenses
B. dividends
C. retirement plan contributions
D. student loan interest
Answers
Answer:
B dividends
Step-by-step explanation:
The rest are monies paid out.
What is the value of x in the equation ?
Answers
Answer:
\(x=2\)
Step-by-step explanation:
This is our equation
\(2.5\left(6x-4\right)=10+4\left(1.5+0.5x\right)\)
First off we can distribute the 2.5 and the 4
Now our equation is \(15x-10=10+6+2x\)
subtract \(2x\) from both sides
\(13x-10=16\)
add \(10\) to both sides
\(13x=26\)
Divide both sides by \(13\)
\(x=2\)
Lucky Duck
What is the probability you will choose each duck described below?
Write the answer as a fraction in lowest terms and place it in the appropriate
box (certain, likely, unlikely, impossible).
The ducks are
numbered from one
through 12!
Answers
Lucky duck probabilities include:
4. Impossible, 0/12.5. Likely, 1/6.6. Certain, 1.7. Likely, 1/12.8. Impossible, 0/129. Likely, 1/1210. Certain, 1/2.11. Likely, 1/2.12. Impossible, 0/12.13. Likely, 2/3.14. Certain, 1.15. Likely, 7/12.16. Likely, 1/3.How to determine probability?A duck with a number:
Every duck has a number, so the probability of choosing a duck with a number is certain (1/1 or 100%).
A duck with a number greater than 10:
is 2/12 or 1/6, since there are 2 ducks with numbers greater than 10 out of a total of 12 ducks.
Duck number 4:
There is only one duck with the number 4, so the probability of choosing this duck is 1/12.
A duck with sunglasses:
There are no ducks with sunglasses so the probability is impossible.
A duck with a hat:
We don't know how many ducks wear hats, so we cannot determine the probability.
An even-numbered duck:
There are 6 even-numbered ducks (2, 4, 6, 8, 10, 12) out of 12 ducks in total, so the probability of choosing an even-numbered duck is 6/12, which simplifies to 1/2.
A duck with a number less than 7:
There are 6 ducks with a number less than 7 (1, 2, 3, 4, 5, 6) out of 12 ducks in total, so the probability of choosing a duck with a number less than 7 is 6/12, which simplifies to 1/2.
A duck with a mustache:
There are no ducks with mustaches, so the probability of getting a duck with a mustache is impossible, 0.
A duck with a bow tie:
Ducks with bowties are 8, so the probability of choosing a duck with bowtie is likely 8/12, 2/3
A duck with a number lower than 25:
All ducks have a number lower than 25, so the probability of choosing a duck with a number lower than 25 is certain (1).
A duck with a number greater than 5:
There are 7 ducks with a number greater than 5 (6, 7, 8, 9, 10, 11, 12) out of 12 ducks in total, so the probability of choosing a duck with a number greater than 5 is 7/12.
A duck with number that's a multiple of 3:
There are 4 ducks with a number that's a multiple of 3 (3, 6, 9, 12) out of 12 ducks in total, so the probability of choosing a duck with a number that's a multiple of 3 is 4/12, which simplifies to 1/3.
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write each number in the form a +bi. simplify your answer completely. a) -42 b)
\( \sqrt{ - 169} \)
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Explanation: none
is the angle corresponding, alternate interior, alternate exterior or consecutive interior
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(d) At a football match, there are 29 800 people, correct to the nearest 100. (i) At the end of the football match, the people leave at a rate of 400 people per minute, correct to the nearest 50 people. Calculate the lower bound for the number of minutes it takes for all the people to leave.
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The lower bound for the number of minutes that it takes for all people to leave is of:
70 minutes.
How to obtain the lower bound for the number of minutes that it takes for each person to leave?At a football match, there are 29 800 people, correct to the nearest 100. The fewest number of people that can be there, hence, is of:
29,750.
(as 29749 would be rounded to 29,700).
At the end of the football match, the people leave at a rate of 400 people per minute, correct to the nearest 50 people, hence the fastest rate is given as follows:
424 people per minute.
(as 425 people would be rounded to 450).
Hence the time it would take is obtained applying the proportion as follows:
29750/424 = 70 minutes.
(lower bound is fewest time, hence lowest number of people leaving at the fastest rate).
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