The equation that expresses the joint variation of a with b and the square root of c, and the inverse variation of a with the cube of d is:
[tex]a = k * \frac{(b * \sqrt{c})}{d^3}[/tex]
In mathematics, A variation equation is an equation that describes how one variable (the dependent variable) changes with respect to changes in one or more other variables (the independent variables). There are several types of variation equations, including direct variation, inverse variation, joint variation, and combined variation.
Now let's consider the given question:
It is given that,
a varies jointly with b and the square root of c, means
[tex]a \propto b[/tex] and [tex]a\propto \sqrt{c}[/tex].
It is also given,
a varies inversely with the cube of d, means
[tex]a \propto \frac{1}{d^3}[/tex]
Then by combining those, we can write
[tex]a \propto \frac{(b * \sqrt{c})}{d^3}[/tex]
The equation that expresses the joint variation of a with b and the square root of c, and the inverse variation of a with the cube of d is:
[tex]a = k * \frac{(b * \sqrt{c})}{d^3}[/tex]
Where k is the constant of proportionality. This equation states that as b and the square root of c increase, and as d decreases, the value of a increases proportionally. The constant of proportionality k depends on the specific values of a, b, c, d, and the units of measurement being used.
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Find the product of 3√20 and √5 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
Answer:
30, rational
Step-by-step explanation:
[tex]3\sqrt{20}\cdot\sqrt{5}=3\sqrt{4}\sqrt{5}\cdot\sqrt{5}=(3\cdot2)\cdot5=6\cdot5=30[/tex]
The result is rational because it can be written as a fraction of integers.
Trains Two trains, Train A and Train B, weigh a total of 188 tons. Train A is heavier than Train B. The difference of their
weights is 34 tons. What is the weight of each train?
Step-by-step explanation:
A + B = 188
A = 188 - B - (1)
Now,
A - B = 34
188 - B - B = 34 (Substituting eqn 1 in A)
188 - 34 = 2B
154 = 2B
• B = 77 tons
Now
A = 188 - B
A = 188 - 77
A = 111 tons
I will mark you brainiest!
SSS is used to prove two triangles are congruent.
A) False
B) True
Answer:
A
Step-by-step explanation:
because___________________________________
Answer:
B) True
Step-by-step explanation:
SSS or Side-Side-Side is used to prove two triangles are congruent.
QUESTION THREE (30 Marks) a) For a group of 100 Kiondo weavers of Kitui, the median and quartile earnings per week are KSHs. 88.6, 86.0 and 91.8 respectively. The earnings for the group range between KShs. 80-100. Ten per cent of the group earn under KSHs. 84 per week, 13 per cent earn KSHs 94 and over and 6 per cent KShs. 96 and over. i. Put these data into the form of a frequency distribution and obtain an estimate of the mean wage. 15 Marks
Answer:
the answer would be 100 I guess
Find x, if √x +2y^2 = 15 and √4x - 4y^2=6
Answer:
x = 52.2
Step-by-step explanation:
Add 4x - 4y^2 = 36 and x + 2y^2 = 225
x + 2y^2 + 4x - 4y^2 = 225 + 36
5x = 261
x = 261/5=52.2
Question 8 Suppose that W is random variable. Given that P(W ≤6)=0.935 find the probability of it complement, P(W>6)
The required probability of W being greater than 6 is 0.065 or 6.5%.
What is Probability?A probability value represents the likelihood that an occurrence will take place. In terms of percentage notation, it is stated as a number between 0 and 1, or between 0% and 100%. The higher the probability, the more probable it is that the event will take place.
According to question:The complement of an event A is the event that A does not occur. In this case, the event A is "W ≤ 6", and its complement is "W > 6".
We are given that P(W ≤ 6) = 0.935. Using the complement rule of probability, we can find the probability of W > 6 as follows:
[tex]$$P(W > 6) = 1 - P(W \leq 6)$$[/tex]
Substituting the given value, we have:
P(W > 6) = 1 - 0.935 = 0.065
Therefore, the probability of W being greater than 6 is 0.065 or 6.5%.
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find surface area of cilinder with the radius of 9 and height of 14. make sure to put the correct exponents with answer.
The cylindrical has a surface area of 414 square units due to its 9-unit radius and 14-unit height.
what is cylinder ?A cylinder is a three-dimensional geometric form made up of two circular bases that are parallel to one another and are joined by a curved lateral surface. It can be pictured as a solid item with a constant circular cross-section along its entire length. The measurements of a cylinder, such as the radius and height of the circular bases, affect its characteristics. The surface area, volume, and horizontal surface area of a cylinder are some of its typical characteristics. Mathematical formulas can be used to determine these properties.
given
The following algorithm determines a cylinder's surface area:
[tex]A = 2\pi r^2 + 2\pi rh[/tex]
where r is the cylinder's base's radius, h is the cylinder's height, and (pi) is a mathematical constant roughly equivalent to 3.14.
Inputting the numbers provided yields:
[tex]A = 2\pi (9)^2 + 2\pi (9)(14)\\[/tex]
A = 2π(81) + 2π(126)
A = 162π + 252π
A = 414π
The cylindrical has a surface area of 414 square units due to its 9-unit radius and 14-unit height.
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Can you help me to solve these two questions?
The equation of the tangent line is y = (-1/49)x + 8/49. 2. The equation of the normal line is y = 49x - 342.
What do a curve's tangent and normal lines represent?A line that meets a curve at a point and has the same slope as the curve there is said to be the tangent line to that curve. A line that is perpendicular to the tangent line at a given position is the normal line to a curve at that location. In other words, the normal line's slope equals the tangent line's slope's negative reciprocal. The normal line is helpful for determining the direction of greatest change of the curve at a place whereas the tangent line gives information about the instantaneous rate of change of the curve at that point.
The given function is f(x) = 1/x.
The slope of the function is the derivative of the function thus,
f(x) = 1/x
f'(x) = -1/x²
f'(7) = -1/49
The equation of the line is given as:
y - y1 = m(x - x1)
where, m is the slope of the equation.
The equation of the tangent line is:
y - f(7) = f'(7)(x - 7)
y - 1/7 = -1/49(x - 7)
y = (-1/49)x + 8/49
b. The equation of the normal line to the graph is given as:
The slope of the normal line is negative and opposite of the tangent line.
That is,
m = 49.
y - f(7) = 49(x - 7)
y - 1/7 = 49(x - 7)
y = 49x - 342
Hence, the equation of the tangent line is y = (-1/49)x + 8/49. 2. The equation of the normal line is y = 49x - 342.
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which piece of required information is missing from the following prescription?premarin tabs0.625 mg
The given prescription lacks important information about the frequency and route of administration. Knowing how often a medication should be taken and how it should be administered is crucial for ensuring that patients receive the appropriate dose and achieve the desired therapeutic effect.
Without the frequency information of how (e.g., orally, intravenously, etc.) and when (e.g., daily, twice daily, etc.) to take medicine on prescription, patients may take the medication incorrectly or miss doses, potentially leading to ineffective treatment or adverse effects.
Healthcare providers should always provide clear and complete instructions for medication use to ensure patient safety and optimal treatment outcomes.
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he is paid $22.60 an hour. He normally earns $904 each week. Last week he worked an extra 3 hours at time-and-a-half
his total income last week = $
Answer:
971.8
Step-by-step explanation:
What is the correct numerical expression for "one-half the difference of 6 and 8 hundredths and 2?"
one half x (8 − 6) + 2
one half x (6 + 8 + 2)
one half x (6.08 − 2)
one half − (6.08 ÷ 2)
Answer: c
Step-by-step explanation: i dont have one
1/2 x (6.08 - 2) (6.08 - 2) is the correct numerical expression for "one-half the difference of 6 and 8 hundredths and 2 " .
what is expression ?An expression, as used in computer programming, is a grouping of values, variables, operators, and/or function calls that the computer evaluates to produce a final value. For instance, the equation 2 + 3 combines the numbers 2 and 3 using the + operator to produce the number 5. Similar to this, the equation x * (y + z) produces a value based on the current values of the variables x, y, and z by combining the variables x, y, and z with the * and + operators.
given
In terms of numbers, the phrase "one-half the difference of 6 and 8 hundredths and 2" is expressed as follows:
1/2 x (6.08 - 2) (6.08 - 2)
1/2 x (6.08 - 2) (6.08 - 2) is the correct numerical expression for "one-half the difference of 6 and 8 hundredths and 2 " .
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5. Jeni put a cake in the
oven at 2:30. If the
cake takes 1 hours
to bake, at what time
should it be taken
out of the oven? What the answer
Answer:
3:30
Step-by-step explanation:
We know
Jeni put a cake in the oven at 2:30. The cake takes 1 hour to bake.
What time should it be taken out of the oven?
We take
2:30 + 1 = 3:30
So, it should be taken out of the oven at 3:30
describe all the x -values at a distance of 13 or less from the number 8 . enter your answer in interval notation.
The set of all x-values that are at a distance of 13 or less from the number 8 in the interval notation is given by [ -5, 21 ].
The distance between x and 8 is |x - 8|.
Find all the values of x such that |x - 8| ≤ 13.
This inequality can be rewritten as follow,
|x - 8| ≤ 13
⇒ -13 ≤ x - 8 ≤ 13
Now,
Adding 8 to all sides of the inequality we get,
⇒ -13 + 8 ≤ x - 8 + 8 ≤ 13 + 8
⇒ -5 ≤ x ≤ 21
Therefore, all the x-values which are at a distance of 13 or less from the number 8 represented in the interval notation as [ -5, 21 ].
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Suppose you are trying to estimate the average age of the entire CSU student population. You collect a sample of size n-163, and from your data you calculate x =20.1 s 1.8 round your answers to 3 decimal places) 1. The standard error of the mean for this data set is: 2. The approximate 95% margin of error is (Use the approximation method from the notes, in which the critical value is approximately 2.) 3. The approximate 95% CI for lage is
a) The standard error of mean for this data set is approximately 0.141.
b) The approximate 95% margin of error is approximately 0.282.
c) The approximate 95% confidence interval for the population mean age is 19.824 to 20.376.
a) The standard error of mean (SEM) is given by the formula: SEM = s / sqrt(n), where s is the sample standard deviation and n is the sample size. Plugging in the values given, we have:
SEM = 1.8 / sqrt(163) ≈ 0.141
So the standard error of the mean is approximately 0.141.
b) The approximate 95% margin of error (MOE) can be calculated using the formula: MOE ≈ 2 * SEM. Plugging in the SEM value from part 1, we get:
MOE ≈ 2 * 0.141 ≈ 0.282
So the approximate 95% margin of error is approximately 0.282.
c) The approximate 95% confidence interval (CI) for the population mean age can be calculated using the formula: CI = x ± (z*SEM), where x is the sample mean, z is the critical value from the standard normal distribution corresponding to the desired level of confidence (95% in this case), and SEM is the standard error of the mean. The critical value for 95% confidence is approximately 1.96. Plugging in the values, we get
CI = 20.1 ± (1.96 * 0.141) ≈ 19.824 to 20.376
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PLEASE HELP 30 POINTS!
Answer:
57
57
123
123
57
57
123
that's all.
Answer:
m<1 = 57°
m<2 = m<1 = 57°
m<3 = x = 123°
m<4 = x = 123°
m<5 = m<1 = 57°
m<6 = m<5 = 57°
m<7 = m<4 = 123°
Step-by-step explanation:
[tex]{ \tt{m \angle 1 + x = 180 \degree}} \\ { \colorbox{silver}{corresponding \: angles}} \\ { \tt{m \angle 1 = 180 - 123}} \\ { \tt{ \underline{ \: m \angle 1 = 57 \degree \: }}}[/tex]
A box containing 5 balls costs $8.50. If the balls are bought individually, they cost $2.00 each. How much cheaper is it, in percentage terms, to buy the box as opposed to buying 5 individual balls?
Answer: The total cost of buying 5 balls individually is $2.00 x 5 = $10.00.
The box costs $8.50, which means it is $10.00 - $8.50 = $1.50 cheaper to buy the box.
To calculate the percentage difference, we can use the formula:
% difference = (difference ÷ original value) x 100%
In this case, the difference is $1.50, and the original value is $10.00.
% difference = ($1.50 ÷ $10.00) x 100%
% difference = 0.15 x 100%
% difference = 15%
Therefore, it is 15% cheaper to buy the box than to buy 5 individual balls.
Step-by-step explanation:
Marcia Gadzera wants to retire in San Diego when she is 65 years old. Marcia is now 50 and believes she will need $90,000 to retire comfortably. To date, she has set aside no retirement money. If she gets interest of 10% compounded semiannually, how much must she invest today to meet her goal of $90,000?
Answer:
Step-by-step explanation:
We can use the formula for the future value of an annuity to determine how much Marcia needs to invest today to meet her retirement goal of $90,000. The formula for the future value of an annuity is:
FV = PMT x [(1 + r/n)^(n*t) - 1] / (r/n)
where:
FV = future value of the annuity
PMT = payment (or deposit) made at the end of each compounding period
r = annual interest rate
n = number of compounding periods per year
t = number of years
In this case, we want to solve for the PMT (the amount Marcia needs to invest today). We know that:
Marcia wants to retire in 15 years (when she is 65), so t = 15
The interest rate is 10% per year, compounded semiannually, so r = 0.10/2 = 0.05 and n = 2
Marcia wants to have $90,000 in her retirement account
Substituting these values into the formula, we get:
$90,000 = PMT x [(1 + 0.05/2)^(2*15) - 1] / (0.05/2)
Simplifying the formula, we get:
PMT = $90,000 / [(1.025)^30 - 1] / 0.025
PMT = $90,000 / 19.7588
PMT = $4,553.39 (rounded to the nearest cent)
Therefore, Marcia needs to invest $4,553.39 today in order to meet her retirement goal of $90,000, assuming an interest rate of 10% per year, compounded semiannually.
one ticket is drawn at random from each of the two boxes below: 1 2 6 1 4 5 8 find the chance that the both numbers are even numbers.
The chance that both numbers drawn are even numbers is 8/21.
The probability refers to the measure of the likelihood or chance of an event occurring. It is a numerical value between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain.
There are 4 even numbers and 3 odd numbers in the first box, and 2 even numbers and 1 odd number in the second box.
The probability of drawing an even number from the first box is 4/7, and the probability of drawing an even number from the second box is 2/3.
By the multiplication rule of probability, the probability of drawing an even number from both boxes is
(4/7) × (2/3) = 8/21
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Assume that x and y have been defined and initialized as int values. The expression
!(!(x < y) || (y != 5))
is equivalent to which of the following?
(x < y) && (y = 5)
The expression (x < y) && (y == 5) is an alternative way of writing the original expression, and it will be true only if two conditions are met: first, x is smaller than y, and second, y is equal to 5.
The expression !(!(x < y) || (y != 5)) is equivalent to:
(x < y) && (y == 5)
To see why, let's break down the original expression:
!(!(x < y) || (y != 5))
= !(x >= y && y != 5) (by De Morgan's laws)
= (x < y) && (y == 5) (by negating and simplifying)
So, the equivalent expression is (x < y) && (y == 5). This expression is true if x is less than y and y is equal to 5.
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Complete question:
Assume that x and y have been defined and initialized as int values. The expression
!(!(x < y) || (y != 5))
is equivalent to which of the following?
(x < y) && (y = 5)
(x < y) && (y != 5)
(x >= y) && (y == 5)
(x < y) || (y == 5)
(x >= y) || (y != 5)
The scale on a map is 1:320000
What is the actual distance represented by 1cm?
Give your answer in kilometres.
By answering the presented question, we may conclude that Therefore, 1 expressions cm on the map corresponds to a real distance of 3.2 km.
what is expression ?In mathematics, an expression is a collection of integers, variables, and complex mathematical (such as arithmetic, subtraction, multiplication, division, multiplications, and so on) that describes a quantity or value. Phrases can be simple, such as "3 + 4," or complicated, such as They may also contain functions like "sin(x)" or "log(y)". Expressions can be evaluated by swapping the variables with their values and performing the arithmetic operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.
Scale 1:
320000 means that 1 unit on the map represents his 320000 units in the real world.
To find the actual distance represented by 1 cm on the map, you need to convert the units to the same scale.
1 kilometer = 100000 cm
So,
1 unit on the map = 320000 units in the real world
1 cm on the map = (1/100000) km in the real world
Multiplying both sides by 1 cm gives:
1 cm on the map = (1/100000) km * 320000
A simplification of this expression:
1 cm on the map = 3.2 km
Therefore, 1 cm on the map corresponds to a real distance of 3.2 km.
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pls helppppppp explain !!!
Answer:
x²
Step-by-step explanation:
[tex]{ \tt{ \frac{ {x}^{ - 3} . {x}^{2} }{ {x}^{ - 3} } }} \\ \\ \dashrightarrow{ \tt{x {}^{( - 3 + 2 - ( - 3))} }} \\ \dashrightarrow{ \tt{ {x}^{( - 3 + 2 + 3)} }} \: \: \: \: \\ \dashrightarrow{ \boxed{ \tt{ \: \: \: \: {x}^{2} \: \: \: \: \: \: }}} \: \: \: \: [/tex]
A simple random sample of size n is drawn. The sample mean, x, is found to be 18.1, and the sample standard deviation, s, is found to be 4.1.
(a) Construct a 95% confidence interval about u if the sample size, n, is 34.
Lower bound: Upper bound:
(Use ascending order. Round to two decimal places as needed.)
In response to the stated question, we may state that Hence, the 95% CI function for u is (16.72, 19.48), rounded to two decimal places in increasing order.
what is function?In mathematics, a function is a connection between two sets of numbers in which each member of the first set (known as the domain) corresponds to a single element in the second set (called the range). In other words, a function takes inputs from one set and produces outputs from another. Inputs are commonly represented by the variable x, whereas outputs are represented by the variable y. A function can be described using an equation or a graph. The equation y = 2x + 1 represents a linear function in which each value of x yields a distinct value of y.v
We use the following formula to create a confidence interval around the population mean u:
CI = x ± z*(s/√n)
where x represents the sample mean, s represents the sample standard deviation, n represents the sample size, z represents the z-score associated with the desired degree of confidence, and CI represents the confidence interval.
Because the degree of confidence is 95%, we must calculate the z-score that corresponds to the standard normal distribution's middle 95%. This is roughly 1.96 and may be determined with a z-table or calculator.
CI = 18.1 ± 1.96*(4.1/√34)
CI = 18.1 ± 1.96*(0.704)
CI = 18.1 ± 1.38
Hence, the 95% CI for u is (16.72, 19.48), rounded to two decimal places in increasing order.
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In the diagram below, IJK~LJK Find g. 5
In the diagram given IJK≅LJK ,the length cannot be negative, the only valid solution is g = 6. Therefore, the length of the segment KJ is 6 meters.
What is length?Length is a physical dimension that measures the distance between two points or the size of an object along one dimension.
It is commonly measured using standard units of length such as meters, feet, inches, and centimeters.
Given that KM is the bisector line of line IM and J is the bisector point on line IL. Two triangles are formed, △IJK and △LJK, on line IL in the upward and downward direction, respectively. We are given the lengths of IK, JN, LM, and KJ, and we need to find the value of g such that IJK≅LJK.
Since the two triangles are similar, their corresponding sides are proportional. Using the side proportionality theorem, we have:
KJ/LM = IJ/JL
Substituting the given values, we get:
g/10 = 5/(4+g)
Cross-multiplying, we get:
g(4+g) = 50
Expanding and simplifying, we get:
g²+ 4g - 50 = 0
Using the quadratic formula, we get:
g = (-4 ± √(4² + 4(50)))/2
g = (-4 ± √(256))/2
g = (-4 ± 16)/2
g = -10 or g = 6
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Workers are preparing an athletic field by mixing soil and sand
in the correct ratio. The table shows the volume of sand to mix
with different volumes of soil. Which statement is correct?
A For 1,425 m³ of soil, the workers should use 375 m³ of sand.
B The ratio of the volume of soil to the volume of sand is 1:4.
C A graph of the relationship includes the point (900, 225).
D The equation y = 4x models the relationship.
Option B: The ratio of the volume of soil to the volume of sand is 1:4.
Looking at the table, we can see that for every 100 m³ increase in soil, the sand volume increases by 25 m³. This gives us a ratio of 4:1, which means that the volume of sand is one-fourth of the volume of soil. Therefore, option B is correct.
Option D: The equation y = 4x models the relationship.
We can see that the volume of sand is always one-fourth of the volume of soil. Therefore, we can write y = (1/4)x or y = 0.25x. This equation is the same as y = 4x. Therefore, option D is also correct.
So, the correct statements are B and D.
What is a graph?In mathematics, a graph is a visual representation of data or a mathematical function. It consists of a set of points or vertices connected by lines or curves called edges or arcs, which represent the relationships between the points. Graphs can be used to show trends, patterns, and relationships in data, and they are commonly used in fields such as statistics, economics, and computer science. Some common types of graphs include line graphs, bar graphs, pie charts, scatterplots, and network graphs.
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The table mentioned in the question has been attached below.
Find X using the picture below.
Answer: 37.5
Step-by-step explanation:
75 - 180 = 105
105 degrees = the obtuse angle, bottom triangle.
75/2= 37.5 (since both sides of the bottom triangle are equal angles)
A cube of sugar is 2cm wide. Calculate the number of cube in a box 720cm³
Answer:
V=lwh
=2×2×2=8
720÷8=90
90 cubes
In the year 1985, a house was valued at $108,000. By the year 2005, the value had appreciated to $148,000. What was the annual growth rate percentage between 1985 and 2005? Assume that the value continued
to grow by the same percentage. What was the value of the house in the year 2010?
Answer:
To find the annual growth rate percentage, we can use the formula:
annual growth rate = [(final value / initial value)^(1/number of years)] - 1
where "final value" is the value in the ending year, "initial value" is the value in the starting year, and "number of years" is the total number of years between the starting and ending years.
Using the given values, we have:
annual growth rate = [(148,000 / 108,000)^(1/20)] - 1
= 0.0226 or 2.26%
So the house appreciated at an annual growth rate of 2.26%.
To find the value of the house in 2010, we can use the same growth rate to project the value from 2005 to 2010:
value in 2010 = 148,000 * (1 + 0.0226)^5
= $175,465.11 (rounded to the nearest cent)
Therefore, the value of the house in the year 2010 was $175,465.11.
Work out x. Area=194
Please help due in 2 hourss
Step-by-step explanation:
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Consider the initial value problem y⃗ ′=[33????23????4]y⃗ +????⃗ (????),y⃗ (1)=[20]. Suppose we know that y⃗ (????)=[−2????+????2????2+????] is the unique solution to this initial value problem. Find ????⃗ (????) and the constants ???? and ????.
The unique solution to the initial value problem of differential equation is y(t) = -t^2 + 2t + 3sin(3t) - 1 with e(t) = -t^2 + 2t + 3sin(3t) - 9, a = 2, and B = -21.
To find the solution to the initial value problem, we first need to solve the differential equation.
Taking the derivative of y(t), we get:
y'(t) = -2t + a
Taking the derivative again, we get:
y''(t) = -2
Substituting y''(t) into the differential equation, we get:
y''(t) + 2y'(t) + 10y(t) = 20sin(3t)
Substituting y'(t) and y(t) into the equation, we get:
-2 + 2a + 10(-2t + a) = 20sin(3t)
Simplifying, we get:
8a - 20t = 20sin(3t) + 2
Using the initial condition y(0) = 2, we get:
y(0) = -2(0) + a = 2
Solving for a, we get:
a = 2
Using the other initial condition y'(0) = 21, we get:
y'(0) = -2(0) + 2(21) + B = 21
Solving for B, we get:
B = -21
Therefore, the solution to the initial value problem is:
y(t) = -t^2 + 2t + 3sin(3t) - 1
Thus, we have e(t) = y(t) - 8, so
e(t) = -t^2 + 2t + 3sin(3t) - 9
and a = 2, B = -21.
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_____The given question is incomplete, the complete question is given below:
Consider the initial value problem >= [22. 2.1]+20). 361) = [2] Suppose we know that (t) = -2t + a 21? + is the unique solution to this initial value problem. Find e(t) and the constants and B. a = B= 8(t) =
begin by finding the area under the curve from to , . this area can be written as the definite integral
The area under the curve y = 1/ (x^2 + 6x -16) from x = t to x = 6 is 1/10( ln(4) - 1/10 ln(t+8))
To find the area under the curve y = 1/ (x^2 + 6x -16) from x = t to x = 6, where t > 2, we need to evaluate the definite integral:
∫[t,6] 1/ (x^2 + 6x -16) dx
To solve this integral, we can use partial fraction decomposition. First, we factor the denominator:
x^2 + 6x -16 = (x+8)(x-2)
Then, we can write:
1/ (x^2 + 6x -16) = A/(x+8) + B/(x-2)
Multiplying both sides by (x+8)(x-2), we get:
1 = A(x-2) + B(x+8)
Setting x = -8, we get:
1 = A(-10)
So, A = -1/10.
Setting x = 2, we get:
1 = B(10)
So, B = 1/10.
Therefore, we can write:
1/ (x^2 + 6x -16) = -1/10(x+8) + 1/10(x-2)
Substituting this into the integral, we get:
∫[t,6] 1/ (x^2 + 6x -16) dx = ∫[t,6] (-1/10(x+8) + 1/10(x-2)) dx
Integrating, we get:
= [-1/10 ln|x+8| + 1/10 ln|x-2|] from t to 6
= 1/10 ln|6-2| - 1/10 ln|t+8|
= 1/10 ln(4) - 1/10 ln(t+8)
Therefore, the area is: 1/10( ln(4) - 1/10 ln(t+8))
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_____The given question is incomplete, the complete question is given below:
begin by finding the area under the curve from to y = 1/ (x^2 + 6x -16) from x = t to x = 6, t>2 this area can be written as the definite integral
