Even though the first student scored 150, the expected average IQ score for the remaining four students is still 100.
Explain average
The average is a statistical measure that represents the central tendency of a set of values. It is calculated by adding up all the values and dividing them by the total number of values. The average provides a single value that represents the typical value in the set and is often used to compare different sets of data or to summarize large amounts of data.
According to the given information
Let X be the random variable representing the IQ score of an individual student. Then, the sample mean of the five students, denoted by Y, can be expressed as:
Y = (X1 + X2 + X3 + X4 + X5) / 5
Given that the first student scored 150, the expected value of the sample mean of the remaining four students, denoted by E(Y|X1=150), can be calculated as follows:
E(Y|X1=150) = E((X2 + X3 + X4 + X5) / 4 | X1 = 150)
Since X1 = 150 is an observed value, we can treat it as a constant and apply the linearity of expectation:
E((X2 + X3 + X4 + X5) / 4 | X1 = 150) = (E(X2 | X1 = 150) + E(X3 | X1 = 150) + E(X4 | X1 = 150) + E(X5 | X1 = 150)) / 4
Now, since the sample is random, the remaining four IQ scores are also independent and identically distributed as X, with the same mean of 100. Therefore, we have:
E(X2 | X1 = 150) = E(X3 | X1 = 150) = E(X4 | X1 = 150) = E(X5 | X1 = 150) = E(X)
Thus, we can simplify the expression as:
E(Y|X1=150) = (4E(X) / 4) = E(X) = 100
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a man can hit his target 25% of the time if he fires 4 shots in succession what is the probability that he will hit his target
The probability that the man will hit his target at least once in 4 shots is calculated using the complementary probability of him missing all 4 shots. Since he hits his target 25% of the time, he misses it 75% of the time.
The probability of missing all 4 shots is (0.75)^4 = 0.3164.
Now, to find the probability of hitting the target at least once, subtract the probability of missing all shots from 1:
1 - 0.3164 = 0.6836 or 68.36%.
So, the probability that the man will hit his target at least once in 4 shots is approximately 68.36%.
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Ms. Yamato's gross pay is $2644. Her deductions total $548.30.
What percent of her gross pay is take-home pay?
A. 84%
B. 79%
C. 21%
D. 18%
Thus, the correct Percent response is B) 79%.
what is a percent?Percentage refers to a portion of every hundred. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, it is frequently shown using the percent sign, "%".
If you have 100 apples and distribute 10 of them, for instance, you have distributed 10% of your total apple supply.
We deduct Ms. Yamato's deductions from her gross income to determine her take-home pay:
$2644 - $548.30 = $2095.702.
By dividing her take-home pay by her gross pay and multiplying the result by 100%, we can determine what proportion of her gross pay is taken home:
($2095.70 / $2644) * 100% = 79%
Thus, the correct response is B) 79%.
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Given a circle has a center at (-6, 14) and a diameter of 30, write the equation of the circle
Answer:
The center of the circle is at (-6, 14), and the radius is 15, so we have this equation:
[tex] {(x + 6)}^{2} + {(y - 14)}^{2} = 225 [/tex]
A researcher wants to determine whether there is a linear relationship between number of rom-com movies and engagements. The table displays a partial computer printout of the regression analysis and is based on a sample of eight subjects.
Predictor Coeff St. Dev t Ratio p-Value
Constant 4.35744984 2.385747738 5.7382918 0.00375268
Rom-com movies 1.34457321 0.526645738
What is the value of the t-test statistic for H0: β = 0
The value of test statistic is 2.553089.
What is a test statistic?A test statistic is a numerical summary of a sample that is used to make inferences about the population from which the sample was drawn. It is often used in hypothesis testing to determine whether a null hypothesis should be rejected or not.
The test statistic is calculated from the sample data and is compared to a distribution of the test statistic under the null hypothesis. If the test statistic falls in a region of the distribution that is unlikely to occur by chance, then the null hypothesis is rejected in favor of an alternative hypothesis.
Value of test Statistic = 1.34457321/0.526645738
= 2.553089
So, value of test statistic is 2.553089
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Please use Triangle Inequality to solve, I'm having a bit of trouble. I'd also appreciate if you just help me.
The value of x for the given triangle through which the perimeter of the given relation is satisfied is 10.
What about perimeter of triangle?
The perimeter of a triangle is the total length of its boundary, which is the sum of the lengths of its three sides. The perimeter can be thought of as the distance around the triangle, and it is measured in units of length such as centimeters, meters, or feet. The perimeter of a triangle is an important geometric property that is used in many practical applications, such as calculating the amount of fencing needed to enclose a triangular-shaped garden or determining the length of wire required to form a triangular circuit.
According to the given information:
The perimeter of triangle is sum of all sides of the triangle
In which,
2x + 4 + 3x - 8 + x - 2 = 54
6x - 6 = 54
6x = 60
x = 10
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4. A fish tank is in the shape of a cuboid. It
has a base of width 80cm and length of 1.2 m.
Water is put into the tank to a depth of 70cm.
There are 700 000 small fish in the tank. Each
one of them needs a litre of water.
Is there enough water for the fish in the tank?
The volume of water in the tank won't be enough for the fish.
What is volume of a cuboid?A cuboid is a solid shape or a three-dimensional shape. The volume of a cuboid is expressed as;
V = l×w× h
where l is the length, w is the width and h is the height.
The volume of water needed by the fish is 700,000× 1litre = 700,000 litres.
The volume of water the tank can occupy is
V = l× b× h
V = 80× 1200× 70
V = 6,720,000cm³
in litres ;
V = 6,720,000/1000
V = 6720litres
therefore the volume water in the tank is not enough for the fish
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Find the number of permutations with of the letters in each word
The trigonometry functions when calculated are sin(A) = 18/34 and cos(A) = 30/34
Calculating the trigonometry functionsFrom the question, we have the following parameters that can be used in our computation:
The right triangle
Where we have
Opposite = 18Adjacent = 30Hypotenuse = 34Using the above as a guide, we have the following:
sin(A) = 18/34
cos(A) = 30/34
Hence, the values are sin(A) = 18/34 and cos(A) = 30/34
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James fills the rectangular prism 80 percent with sand. How many cubic inches of sand are in the rectangular prism.
The volume of sand in the rectangular prism is 80 percent of the total volume, which can be written as:
0.8 x V = 0.8(L x W x H) cubic inches
What is rectangular prism?A three-dimensional object with six faces, including two at the top and bottom and four lateral faces, called a rectangular prism. The prism's faces are all rectangular in shape. There are three sets of identical faces as a result.
We need to know the dimensions of the rectangular prism to solve the problem. Let's assume the dimensions are length = L inches, width = W inches, and height = H inches.
The volume of the rectangular prism is given by V = L x W x H cubic inches.
If the rectangular prism is filled 80 percent with sand, then 20 percent of the volume is empty space. So, the volume of sand in the rectangular prism is 80 percent of the total volume, which can be written as:
0.8 x V = 0.8(L x W x H) cubic inches
Therefore, we need to know the dimensions of the rectangular prism to find the volume of sand.
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The complete question is:
James fills the rectangular prism 80 percent with sand. How many cubic inches of sand are in the rectangular prism?
Timmy is building a model of a building in his town that is 180 meters. He is using a scale of 1.5 cm. = 3.5 meters. How tall will his model be?
Answer: 77
Step-by-step explanation: You could set up a proportion for this question. 1.5 centimeters = 3.5 meters x centimeters = 180 meters 1.5 x 180 / 3.5 Rounded to the nearest whole number, the answer is 77.
What is the equation of the line 2x+3y+1=0 after it has been reflected across y=-1 and rotated about o(2,3) by 180
The equation of the line 2x + 3y + 1 = 0 after it has been reflected across y = -1 and rotated about O(2,3) by 180 degrees is -2x + y = 15.
To reflect the line 2x + 3y + 1 = 0 across the line y = -1, we need to subtract twice the y-coordinate of each point on the line from the equation of the line.
This is because the reflection of a point across a line is obtained by reflecting it across the line's perpendicular bisector, which is also the line with the same x-coordinates and twice the distance from the point to the line.
Subtracting twice the y-coordinate, we get:
2x + 3(-2y - 1) + 1 = 0
Simplifying the equation, we get:
2x - 6y - 5 = 0
Now, to rotate the line about the point O(2,3) by 180 degrees, we can use the rotation formula:
x' = 2a - x
y' = 2b - y
where (x,y) are the coordinates of a point on the original line, and (x',y') are the coordinates of the same point after rotation about the point (a,b).
Substituting the coordinates of two points on the line, we get:
(0,-1) → x' = 2(2) - 0 = 4, y' = 2(3) - (-1) = 7
(-1/2,0) → x' = 2(2) - (-1/2) = 4 1/2, y' = 2(3) - 0 = 6
Now, we can use the two points (4,7) and (4 1/2,6) to find the equation of the rotated line using point-slope form:
slope = (6-7) / (4 1/2 - 4) = -2
y - 7 = -2(x - 4)
y = -2x + 15
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HELP ME PLEASE!!!!!!!
Answer:
Step-by-step explanation:
Graph # Matching equation
1 |-3x|
2 |-x|
3 -|2x|
You can tell which one matches by finding the slope and whether the V points up or down
A factory has two assembly lines, M and N, that make the same toy. On Monday, only assembly line M was functioning and it made 900 toys.
On Tuesday, both assembly lines were functioning for the same amount of time. Line M made 300 toys per hour and line N made 480 toys per hour. Line N made as many toys on Tuesday as line M did over both days.
Write an equation that can be used to find the number of hours, t, that the assembly lines were functioning on Tuesday.
Answer: First choice 480t = 300t + 900
Step-by-step explanation:
A factory has two assembly lines, M and N, that make the same toy. On Monday, only assembly line M was functioning and it made 900 toys.
On Tuesday, both assembly lines were functioning for the same amount of time. Line M made 300 toys per hour and line N made 480 toys per hour. Line N made as many toys on Tuesday as line M did over both days.
Write an equation that can be used to find the number of hours, t, that the assembly lines were functioning on Tuesday.
ANS
Let say t is time in hrs for Which Both Assembly line worked
M made 300 toys per hr on Tuesday
Toys made on Tuesday at Assembly line M = 300 × t = 300t toys
N made 480 toys per hr on Tuesday
Toys made on Tuesday at Assembly line N = 480 × t = 480t toys
Toys Made by N on Tuesday = Toys made by M on Tuesday + Toys made by M on Monday
480t = 300t + 900
=> 180 t = 900
=> t = 900/180
=> t = 5 hr
N capacity on tuesday Per hour * t = M capacity on tuesday per hour * t + Toys made by M on Monday
N = N capacity on tuesday Per hour
M = M capacity on tuesday Per hour
=> t (N - M ) = 900
=> t = 900/(N-M)
the ricardo household used w cubic feet of water during a summer month express the number of ccf of water they used algebraically
Ricardo used w/100 Ccf of water.
Given that, Ricardo household used w cubic feet of water in a month of summers,
we need to express the number of ccf of water they used algebraically,
Ccf = A CCF is a unit of measurement of volume.
The CCF is used to measure gas consumption, used in the billing of natural gas and water delivered to households.
1 CCF = 100 cubic feet
CCF means Centum Cubic Feet.
1 cubic feet = 0.01 Ccf
Therefore,
w cubic feet = 0.01 × w Ccf
= w / 100
Hence, Ricardo used w/100 Ccf of water.
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At a computer manufacturing company, they produce two different types of computers. They can make 70 laptops per day while making 55 desktops per day. The company has a total of 14 machines to make computers. They can make a total of 905 computers per day. How many machines make laptops and how many make desktops?
Write a system of equations and solve.
Answer:
Yo wassup bro, at this computer factory, they be making two types of computers, laptops and desktops. They churn out 70 laptops a day and 55 desktops a day. They got a total of 14 machines to make these computers. And they make a total of 905 computers a day. We gotta figure out how many machines are making laptops and how many are making desktops, ya know?
Alright, let's set up a system of equations to solve this. Let's call the number of machines making laptops "x" and the number of machines making desktops "y".
So, we got two equations here:
The total number of computers they make in a day is 905, so we can write: x laptops + y desktops = 905.
They got a total of 14 machines, so we can write: x + y = 14.
Now, let's solve this system of equations to find the values of x and y, man. Once we got those, we'll know how many machines are making laptops and how many are making desktops at this computer factory, yo!
2㏒_[6](2)+3㏒_[6](3)+㏒_[6](12)
Answer: 4
Step-by-step explanation:
Given:
[tex]2log_62+3log_63+log_612[/tex]
Rewrite using the power property:
[tex]log_62^2+log_63^3+log_612[/tex]
Rewrite by adding the logarithms together (multiplying) since they all have a common base of 6:
[tex]log_6(2^2*3^3*12)[/tex]
Simplify by multiplying:
[tex]log_6(4*27*12)[/tex]
[tex]log_6(1,296)[/tex]
Evaluate the logarithm:
[tex]log_6(1,296)[/tex]
4
Fruit Smoothle
O E. Talia can use 1
cup orange juice
cup yogurt
8 cup strawberries
8 cup blueberries
cup of ice
Blend all ingredients. Serve cold.
Makes 1 serving.
Which statement about the recipe is true? Mark all that apply.
cup strawberries to double the recipe.
O A. Talia can use
OB. Talia can use
OC. Talia can use
12 cup yogurt to make 3 servings.
cup blueberries to make 4 servings.
OD. Talia can use
1
cups orange juice to make 4 servings.
cup of ice to make 5 servings.
A. Talia can use 1 and 3/4 cup strawberries to double the recipe. B. Talia can use 1 and 1/3 cup yogurt to make 3 servings. D. Talia can use 2 cups orange juice to make 4 servings.
What is proportion?In statistics, a proportion is a fraction or a percentage that reflects the proportion of a population's or sample's members who share a particular attribute to the population's or sample's overall size. It is a kind of ratio where the denominator is the entire population or sample size and the numerator is the number of people who possess a specific trait or attribute.
From the given ingredients and the serving size we can see that the correct statement is:
A. Talia can use 1 and 3/4 cup strawberries to double the recipe.
B. Talia can use 1 and 1/3 cup yogurt to make 3 servings.
D. Talia can use 2 cups orange juice to make 4 servings.
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The complete question is:
A 30 pack of copy paper cost $48.30 a 32 pack cost $49.60 which is the better buy (help me explain it)
Answer:
Step-by-step explanation:
the 32 pack because it costs just slightly more
4. The radius of a cylinder is 3x-2 cm. The height of the cylinder is x +3 cm. What is the
surface area of the cylinder? Use the formula A=2x²+2xrh.
02x (3x2+10x-8)
O 27(12x+7x-2)
O 27(12x²-2x+13)
O 27(12x²-5x-2)
Answer:
D: 27(12x² - 5x - 2).
Step-by-step explanation:
The formula for the surface area of a cylinder is: A = 2πr² + 2πrh
Given that the radius of the cylinder is 3x - 2 cm and the height is x + 3 cm, we can substitute these values in the formula and simplify:
A = 2π(3x - 2)² + 2π(3x - 2)(x + 3)
A = 2π(9x² - 12x + 4) + 2π(3x² + 7x - 6)
A = 18πx² - 24πx + 8π + 6πx² + 14πx - 12π
A = 24πx² - 10πx - 4π
A = 2π(12x² - 5x - 2)
Therefore, the answer is option D: 27(12x² - 5x - 2).
Answer:
D
Step-by-step explanation:
i need some help with this problem i don’t really understand it
Answer:
-34x+29
Step-by-step explanation:
[tex]7(5-x)-3(9x+2)=7.5-7x-3.9x-3.2=35-7x-27x-6\\=-34x+29[/tex]
How do you find the period of a cosine function of the form y = cos bx?
The period of a cosine function of the form y = cos bx is equal to T= 2π/b where 'T' is the period of the cosine function and b is the coefficient of x in the function.
This formula tells us that the period of the cosine function is equal to the length of one complete cycle of the function.
it represents the distance along the x-axis for the cosine function to complete one full oscillation.
The period of a cosine function of the form y = cos bx, first identify the coefficient b.
Use the formula T= 2π/b to calculate the period 'T'.
For example,
Consider cosine function y = cos 2x,
The coefficient of x is 2.
Using the formula above, the period is equal to
Period = 2π/2
= π
So the period of the function y = cos 2x is π.
This implies that the cosine function completes one full oscillation every π units along the x-axis.
Therefore, the formula used to calculate the period of the cosine function y = cos bx is given by T= 2π/b.
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I NEED HELP ON THIS ASAP!! PLEASE IT'S DUE TODAY!
The exponential function to represent the number of new participants of the challenge as a function of the day number, is f(x) = 12 × 4ˣ.
How to find the exponential function ?Let f(x) be the number of new participants on day x.
On day 0 (today), Aliyah, Kim, and Reese are the only participants. They each send selfies to 4 friends, so there will be 3 x 4 = 12 new participants on day 1.
From day 1 onwards, each new participant will send selfies to 4 friends. Therefore, the number of new participants will quadruple each day. This gives us an exponential growth function:
f(x) = 12 × 4ˣ
where f(x) represents the number of new participants on day x.
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One gallon of paint covers 400 square feet. What is the least amount of paint needed to paint the walls of a room in the shape of a rectangular prism with a length of 17 feet, a width of 15 feet, and a height of 12 feet? Write your answer as a decimal. Gal
We need at least 3.465 gallons of paint to paint the walls of this room.
The area of the two rectangular faces on either end of the prism is:
length x height = 17 x 12 = 204 square feet
The area of the two rectangular faces on the sides of the prism is:
width x height = 15 x 12 = 180 square feet
The area of the top and bottom faces of the prism is:
length x width = 17 x 15 = 255 square feet
The total surface area of the prism is:
2(204) + 2(180) + 2(255) = 1386 square feet
Now, we divide surface area by coverage of one gallon of paint:
1386 / 400 = 3.465
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Complete the table using the information given.
Using the above information, we can calculate the missing figures in the table and this is shown on the table attached.
What is the table about?For the "Shorts" row: It's given that 42 people were wearing shorts.
Since it's mentioned that twice as many people were wearing closed-toed shoes as open-toed shoes, and 30 people were wearing open-toed shoes, we can deduce that 30 people were also wearing closed-toed shoes.So, the total number of people wearing closed-toed shoes is 30, and the total number of people wearing shorts is also 42, since they are the same group of people.Therefore, the "Close-Toed Shoes" column in the "Shorts" row is also 42, and the "Total" column is the sum of open-toed and closed-toed shoes, which is 84.
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See text below
30 people surveyed were wearing open-toed shoes.
1/3 of the people with open-toed shoes were wearing
pants.
• 42 people were wearing shorts.
Twice as many people were wearing closed-toed shoes
than open-toed shoes.
Complete the table using the information given.
Open-Toed Shoes Close-Toed Shoes Total
Shorts ------- ------ --------
Pants ------ ---------- ------
Total ------ ---------- ---------
6²
Which theorem is shown by the diagram above?
a + b = c
C
D
a - b = c
a² + b² = c²
a²-b² = c²
The theorem is shown in a pythagoras theorem is c² = a² + b²
Which theorem is shown in a pythagoras theoremThe theorem shown in the Pythagorean theorem is "a² + b² = c²". This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with discovering it.
The Pythagorean theorem applies to right-angled triangles and states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Mathematically, we can express this as:
c² = a² + b²
Where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (called the legs) of the right-angled triangle.
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Send total home runs during the baseball season for over five other home runs during the first half of the season what is the fraction of home run hit through and a half of the season as a decimal
Terrell hit 0.8 of his home runs during the first half of the baseball season, which is the decimal equivalent of the fraction (4)/(5).
A fraction is a way of representing a part of a whole, where a whole is divided into equal parts and the fraction represents the number of those equal parts that are being considered.
A decimal number is a number expressed in the base-ten positional system, where each digit can take one of ten possible values and the value of each digit is determined by its position relative to the decimal point.
To write a fraction as a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). In this case, the fraction is (4)/(5), which means Terrell hit 4 out of 5 home runs during the first half of the season
(4)/(5) = 0.8
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The given question is incomplete, the complete question is:
Terrell hit a total of 10 home runs during the baseball season. He hit (4)/(5) of the home runs during the first half of the season. Write the fraction of home runs hit during the first half of the season as a decimal.
The length of one leg of a right triangle is 2 times the length of the other, and the
length of the hypotenuse is 12. What is the length of the longest leg?
Answer:
[tex]\frac{24\sqrt{5} }{5}[/tex]
Step-by-step explanation:
Let's start by assigning one of the unknown legs with the variable x.
We know that the other leg is 2 times the length of x, so we can write:
2x
We also know that the length of the hypotenuse is 12.
From here, we can use the Pythagorean Theorem.
Recall that the Pythagorean Theorem is:
[tex]a^2+b^2=c^2[/tex]
where a is the length of one leg, b is the length of the other leg, and c is the length of the hypotenuse.
Let's substitute the values. We have:
[tex]x^2+(2x^2)=12^2=\\x^2+4x^2=144=\\5x^2=144=\\x^2=\frac{144}{5}=\\x=\frac{12}{\sqrt{5} }[/tex]
Let's rationalize the denominator by multiplying the numerator and denominator by [tex]\sqrt{5}[/tex], like so:
[tex]\frac{12}{\sqrt{5} } =\\\frac{12\sqrt{5} }{5}[/tex]
Therefore, [tex]x=\frac{12\sqrt{5} }{5}[/tex]
Let's solve for 2x:
[tex]2x=\\2(\frac{12\sqrt{5} }{5})=\\ \frac{24\sqrt{5} }{5}[/tex]
So, the length of the longest leg is [tex]\frac{24\sqrt{5} }{5}[/tex]
the table shows several packages of assorted spools of thread available at a store. what is the price per spool of each kind of thread?
I'm sorry, but I cannot provide a definitive answer to your question as I do not have access to the table you are referring to.
However, in general, to find the price per spool of each kind of thread, you would need to know the total price of the package and the number of spools in the package.
To calculate the price per spool, you would divide the total price of the package by the number of spools in the package. For example, if a package costs $10 and contains 50 spools of thread, the price per spool would be $0.20 ($10 ÷ 50 = $0.20).
You can use this method to find the price per spool for each type of thread in the table you are looking at.
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Harry is organizing a picnic. He can spend at most $24. 00 on beverages. Iced tea costs $2. 00 per gallon and lemonade costs $2. 50 per gallon. If x represents the number of gallons of iced tea and y represents the number of gallons of lemonade, which inequality shows the number of gallons of each drink that he can buy? Identify the number of gallons of iced tea that Harry can buy if he buys 5 gallons of lemonade.
Answer:
Harry can buy 5 gallons of lemonade and 5 gallons of iced tea and still have $1.50 left over from his original $24 dollar budget
Step-by-step explanation:
Frank and Joey ordered a large pizza. Frank ate 20% of rye pizza and Joey at 4/5 of the pizza. What percentage of the pizza did they eat in all
The percentage of the large pizza they ate is 100%
To calculate the percentage of pizza they both ate we have to add and then covert the fractions to percentage.
Given from the question
Frank ate 20% of the large pizza and Joey ate 4/5 portion of the large pizza
Converting 4/5 into percentage
For converting into percentage we have to multiply 100 to the fraction
= 4/5 x 100
= 80%
Now, let us proceed by adding the collected two percentages to find out the total percentage of pizza consumed by Frank and Joey
20% + 80%
= 100%
The percentage of the large pizza they ate is 100%.
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Solve the IVP given by y''+y=t, y(0)=1, y'(0)=-2
The solution to the IVP given by y''+y=t, y(0)=1, y'(0)=-2 is y(t) = cos(t) - (3/2) sin(t) + (1/2) t.
To solve the Initial Value Problem, we can use the method of undetermined coefficients, which involves assuming a particular form for the solution to the non-homogeneous equation y'' + y = t, and then finding the coefficients of the terms in that form by substituting it back into the equation.
First, we find the general solution to the homogeneous equation y'' + y = 0
The characteristic equation is r² + 1 = 0, which has solutions r = ±i. Therefore, the general solution to the homogeneous equation is
y_h(t) = c₁ cos(t) + c₂ sin(t),
where c₁ and c₂ are constants determined by the initial conditions.
Next, we assume a particular form for the non-homogeneous solution, based on the form of the right-hand side t. Since t is a linear function, we assume that the particular solution has the form
y_p(t) = a t + b.
Substituting this into the differential equation, we get
y''_p + y_p = t
2a + (at+b) = t.
Equating coefficients, we get
a = 1/2, b = 0.
Therefore, the particular solution is
y_p(t) = (1/2) t.
The general solution to the non-homogeneous equation is then the sum of the homogeneous and particular solutions
y(t) = y_h(t) + y_p(t)
= c₁ cos(t) + c₂ sin(t) + (1/2) t.
To determine the constants c₁ and c₂, we use the initial conditions:
y(0) = c₁ cos(0) + c₂ sin(0) + (1/2) (0) = c₁ = 1,
y'(0) = -c₁ sin(0) + c₂ cos(0) + (1/2) (1) = c₂ - (1/2) = -2,
so c₂ = -3/2.
Therefore, the solution to the IVP is
y(t) = cos(t) - (3/2) sin(t) + (1/2) t.
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