It looks like the equation is
[tex]\dfrac{du}{dt} = 2t + \dfrac12 \sec^2(t) u[/tex]
with initial value [tex]u(0) = \frac\pi2[/tex].
Rearrange the equation to
[tex]\dfrac{du}{dt} - \dfrac12 \sec^2(t) u = 2t[/tex]
Multiply both sides by the integrating factor
[tex]\displaystyle \mu = \exp\left( \int -\frac12 \sec^2(t)\,dt\right) = \exp\left(-\frac12\tan(t)\right)[/tex]
and solve for [tex]u[/tex].
[tex]\implies e^{-\frac12 \tan(t)} \dfrac{du}{dt} - \dfrac12 \sec^2(t) e^{-\frac12 \tan(t)} u = 2t e^{-\frac12 \tan(t)}[/tex]
[tex]\implies \dfrac{d}{dt}\left[e^{-\frac12 \tan(t)} u\right] = 2t e^{-\frac12 \tan(t)}[/tex]
By the fundamental theorem of calculus, integrating both sides yields
[tex]e^{-\frac12\tan(t)} u = e^{-\frac12\tan(t)} u \bigg|_{t=0} + \displaystyle \int_{\xi=0}^{\xi=t} 2\xi e^{-\frac12 \tan(\xi)} \, d\xi[/tex]
[tex]\implies e^{-\frac12\tan(t)} u = 1\times\dfrac\pi2 + \displaystyle \int_{0}^{t} 2\xi e^{-\frac12 \tan(\xi)} \, d\xi[/tex]
[tex]\implies \boxed{\displaystyle u = \frac\pi2 e^{\frac12\tan(t)} + 2e^{\frac12\tan(t)} \int_0^t \xi e^{-\frac12 \tan(\xi)} \, d\xi}[/tex]
X + 2/3 = 8/9 x=? /////////////////////////////////////////
The value of x is 2/9
Given equation x +2/3 = 8/9
We need to find the value of x
Simplifying the equation is
x+2/3 = 8/9
x = 8/9 -2/3
Taking L.C.M of the denominators 3 and 9
The L.C.M is 9
So we will multiply the numerator and denominator with 1 in 8/9
And we will multiply the numerator and denominator with 3 in 2/3
Hence the new equation formed = 8/9-6/9
Solving this equation we get
8-6 / 9
= 2 / 9
Hence the value of x is 2/9
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This Stem-and-Leaf Plot represents the heights of the students on Ralph’s basketball team. One student’s height is missing from the plot. If the mean height of all the students on the team is 61 inches, what is the missing height?
A. 55 in.
B. 59 in.
C. 61 in.
D. 65 in.
The missing height is 65.
What is the missing height?
Mean is the average of a set of numbers. It is determined by adding the numbers together and dividing it by the total number
Mean = sum of the numbers / total number
Missing height = (mean x total number of students) - sum of existing heights
Sum of existing heights = (54 + 55 + 56 + 61 + 63 + 64 + 70 ) = 423
Missing height = (8 x 61) - 423 = 64
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Which expression is not equivalent to 8+(n-4) (8-4+n) 12+n n+4 (8+n)-4
The expression which is not equivalent to (8 + (n - 4)) is (12+n) and this can be determined by using the arithmetic operations.
Given :
Expression - 8 + (n - 4)
To simplify the given expression following steps can be used:
Step 1 - Rewrite the given expression.
= (n - 4) + 8
Step 2 - Now, remove the parenthesis in the given expression.
= n - 4 + 8
Step 3 - Now, write 8 as the addition of 4 and 4.
= n - 4 + 4 + 4
Step 4 - Subtract 4 by 4.
= n + 4
Therefore, the correct answer is (12+n).
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Identify the domain of the function shown in the graph
Using it's concept, the domain of the function shown in the graph is:
B. [tex]-6 \leq x \leq 6[/tex].
What is the domain of a function?The domain of a function is the set that contains all possible input values for the function. In a graph, the domain is the set that contains all values of x.
From this graph, the function is defined for values of x between -6 and 6, hence the domain is given by:
B. [tex]-6 \leq x \leq 6[/tex].
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A freezer is shaped like a rectangular prism. It has a length of 8 feet and a height of 3 feet. The volume is 54 cubic feet. Find the width of the freezer.
Answer:
2.25 ft
Step-by-step explanation:
Volume = length x height x width
54= 8 x 3 x ?
Combine: 54= 24 x ?
Find ?: 54/24=2.25
Recheck: 24 x 2.25 = 54
Determine the vertical asymptotes for the reciprocal of this function f(x)=x^2 + 3x -28
Answer:
x = -7 and x = 4.
Step-by-step explanation:
f(x) = x^2 + 3x - 28
= (x + 7)(x - 4).
The reciprocal of f(x) can be written as
1 / (x + 7)(x - 4).
When the denominator is zero we have vertical asymptotes
so they are x = -7 and x = 4.
Divide 2/8 by 9/18 . Input your answer as a reduced fraction.\
Answer:
1/2
Step-by-step explanation:
There is a couple ways of thinking about this. Probably the fastest way is the rule, but the rule is not intuitive. The rule is when you divide fraction, you keep the first fraction the same and then multiple and flip the second fraction.
2/8 x 18/9 = 36/72 To reduce the fraction, could keep dividing the top and the bottom by the same number until it is no longer possible, I could use: 2, 4 3, 9, 18, or 36. If I used 36, I would only have to divide once. 36/16 is 1 and 72/36 is 2 so my answer would be 1/2.
I could have divided the top and bottom by 2 and got 18/36 and then divide by 2 again and get 9/18 and then by 9 and get 1/2. As you can, see there are a lot of option.
The second method is in the picture that I sent you, it is a little more work, but it makes sense.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{2}{8}\div\frac{9}{18} }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf Simplify \ \frac{2}{8} \ to \ \frac{1}{4}. \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1}{4}\div\frac{18}{9} }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf Use \ this \ rule: a\div \frac{b}{c}=a\times \frac{c}{b}a \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1}{4}\times\frac{18}{9} }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf Use \ this \ rule: \frac{a}{b} \times \frac{c}d{}=\frac{ac}{bd}. \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1\times18}{4\times9} \ \to \ \ Multiply }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{ \frac{18}{36}= }}\boxed{\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{\frac{1}{2} }} \end{gathered}$}} \end{gathered}$}[/tex]
Bob and Carol are teenagers. Bob is two years older than Carol. If the digits of Carol's age are reversed, the new number would be three times as large as Bob's age. Find Bob's age.
Bob is 2 years older than Carol and the reverse of Carol's age is 3 times Bob's age, which makes Bob's to be 17 years.
How can Bob's age be calculated?Let 1B represent Bob's age and let 1C represent Carol's age, we can write the following equations;
1B = 1C + 2Reversing Carol's age gives;
C1 = 3 × 1BThe multiples of 3 that have the form X1 have 7 as the rightmost number.
Given that 1B is a teenager, we have;
When;
1B = 171C = 17 - 2 = 15The reverse of Carol's age is therefore;
C1 = 51 = 3 × 17Therefore, from the given description, Bob's age 1B = 17 years
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Please help ASAP!! I need to turn this is right now!!!!
Paolo used the Quantitative Reasoning Process to build a budget while he is going to college. He worked hard over the summer and saved $3,314.85 to live on during the next two semesters of school. Paolo will receive a scholarship each semester for $926.65. If he plans to use up all of his savings while at school, how much money can Paolo spend each month, on average, for things like groceries, entertainment, and miscellaneous items?
Assume two semesters equals 8 months total. Also assume Paolo has no other expenses than what he has listed here.
Scholarship for each Semester: $926.65
Tuition & Books Cost per Semester: $1,800
Housing Cost for a two-semester Contract: $1,075
If he has money left, what is the average monthly amount Paolo can afford to spend each month?
If he has overspent, what is the average monthly amount Paolo would be in debt each month?
(Indicate this amount with a negative sign)
The computation shows that the average monthly amount is 61.64.
How to calculate the value?The amount that's left will be:
= 3314.85 + (2 × 926.65) - (1800 × 2) - 1075
= 3314.85 + 1853.3 - 3600 - 1075
= 493.15
The average monthly amount will be:
= 493.15/8
= $61.64
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Rewrite each multiplication or division expression using a base and an exponent
4^5 divided by 4^2=
Using the laws of indices , 4⁵ ÷ 4² can rewritten as 4³.
What is the result of the division in exponent form?
Given the expression; 4⁵ ÷ 4²
To perform the division operation, we use one of the laws of indices;
[tex]n^a/n^b = n^{a-b}[/tex]
Given that;
n = 4a = 5b = 2Now, we apply the laws of indices ;
4⁵ ÷ 4² = 4⁵⁻² = 4³
Therefore, using the laws of indices, 4⁵ ÷ 4² can rewritten as 4³.
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The points (1, 5) and (0, –2) fall on a particular line. What is its equation in slope-intercept form?
Answer:
y = 7x - 2
Step-by-step explanation:
first we need the slope of the line.
the slope is expressed a the ratio (y coordinate change / x coordinate change) when we go from one point on the line to another.
so, let's say we go from (0, -2) to (1, 5), we see
x changes by +1 (from 0 to 1).
y changes by +7 (from -2 to 5)
so the slope is +7/+1 = 7
the slope-intercept form is in general
y = ax + b
a being the slope, b being the y-intercept (the y-value for x = 0).
the point (0, -2) gives us therefore the y-intercept directly : -2.
by using the already established slope and the point (0, -2) for the y-intercept the equation of our line is
y = 7x - 2
Please i really need help
for both qeutions
100 points
Answer:
7. a) slope = 5
7. b) slope = 1/2
8. slope = 1
Step-by-step explanation:
slope = rise/run
rise is the vertical change, run is the horizontal change
7. a)
Pick any two points on your line. (0,0) and (1,5) are easy to work with.
The vertical change between the two points is 5
The horizontal change between those points is 1
so slope = 5/1 = 5
7. b)
Again, you can use (0,0) as one of your points. The point (2,4) is good as the second.
slope = 4/2 = 1/2
8. We don't have a graph, but we are given two points to work with, (1,4) and (5,8). Use the formula given...
m = slope = (y2-y1) / (x2-x1)
= (8 - 4) / (5 - 1)
= 4/4
= 1
I hope that helped!
¿Cuál de los siguientes números NO es un decimal periódico?
A.
11
30
B.
11
400
C.
11
600
D.
111
90
El número decimal que no es un decimal periódico es dado por:
B. [tex]\frac{11}{400}[/tex].
¿Que es un número decimal periódico?Un decimal periódico es un número decimal en el que la misma secuencia de dígitos se repite indefinidamente.
La conversión de las fracciones para decimales són:
11/30 = 0.366666 -> Periódico.11/400 -> 0.0275 -> No es periódico.11/600 -> 0.018333333 -> Periódico, en razón de lá repetición de el 3.11/90 = 0.1222222 -> Periódico.Puede-se aprender más a cerca de números decimales periódicos en https://brainly.com/question/1192529
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Write the following inverse proportion: y is inversely proportional to x, when y = 5 and x = 8.
The inverse proportionality expression is y =40/x
Inverse proportionIf the variable y is inversely proportional to x, this is expressed as;
y = k/x
where
k is the constant of proportionality
If y = 5 and x = 8. then;
5 = k/8
k = 5 * 8
k = 40
Determine the inverse proportionality
Recall that y = k/x
y = 40/x
Hence the inverse proportionality expression is y =40/x
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how do u solve this? what’s the answer?
The functions that have the properties are (d) I and II only
How to determine the functions?The function property is given as:
[tex]a(x) = a^{-1}(x)[/tex]
The above implies that the inverse function and the original function are equal.
From the list of options, we have:
(a) y = x
Make x the subject
x = y --- the same as the original function
(b) y = 1/x
Make x the subject
x = 1/y --- the same as the original function
Hence, the functions that have the properties are (d) I and II only
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The distance that you run around the track in gym class is considered ______ data.
A. discrete
B. neither continuous nor discrete
C. continuous
D. both continuous and discrete
The distance that you run around the track in gym class is considered continuous data.
How to determine the type of data?The variable is given as:
Distance
The distance travelled can take any value between 0 and the length of the track.
Take for instance;
The distance travelled can be 1km and it can also be 1.67 km
This means that the distance is continuous
Hence, the statement that completes the blank is (c) continuous
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