The temperature dropped rapidly overnight. Starting at 80°F, the temperature dropped 3° per minute
Answers
Answer:
Your full equation is then y = -3*x + 80.
Step-by-step explanation:
This sounds like a linear equation that can be put in "y = m*x + b" form because there is a constant Y-intercept value ("b," the starting point) and a constant slope ("m," the rate of change).
The trick here is to pick out these values from the word problem. The first thing to look for are key words like "starting at," "at time 0," or "when x = 0." These describe when the line crosses the Y axis (which is where x = 0). This number is also the "b" from the equation for a line: y = m*x + b. Your problem contains the phrase "starting at 80 degrees Fahrenheit," so our Y-intercept is b = 80 degrees Fahrenheit.
The next thing to look for is the slope, m. The slope of the line explains how fast it changes. A very big (or very negative) value of m means a steeper line; a value of m closer to zero means a line that looks more horizontal (i.e. it doesn't change very fast). In your problem, the temperature "dropped 3 degrees per minute." Because the temperature is getting lower as time passes, the slope of the line will be negative. So, m = -3 degrees Fahrenheit per minute.
Related Questions
A museum director wanted to hang the paintings in a wall. To the nearest foot, how tall does the walk need to be?
Answers
Find the general solution for the following differential equation using the method of undetermined coefficients d²y/dx - 36 y = cosh3x.
Answers
The general solution for the given differential equation is the sum of the complementary function and the particular solution:
\(y = y_h + y_p\\\\= C_1e^{6x} + C_2e^{-6x} + (-1/70)e^{3x} + (-1/70)e^{-3x}\)
where C₁ and C₂ are arbitrary constants determined by the initial or boundary conditions of the problem.
We are given the differential equation: d²y/dx - 36y = cosh(3x).
In this case, the homogeneous equation is d²y/dx - 36y = 0.
The characteristic equation associated with the homogeneous equation is obtained by replacing the derivatives with their corresponding algebraic expressions. In our case, we have r² - 36 = 0. Solving this quadratic equation, we find the roots to be r = ±6.
Since the roots are distinct and real, the general solution for the homogeneous equation is given by:
\(y_h = C_1e^{6x} + C_2e^{-6x}\)
where C₁ and C₂ are arbitrary constants determined by the initial or boundary conditions of the problem.
The term cosh(3x) can be written as a linear combination of exponential functions using the identities:
\(cosh(ax) = (e^{ax} + e^{-ax})/2, \\\\sinh(ax) = (e^{ax} - e^{-ax})/2.\)
Therefore, \(cosh(3x) = (e^{3x} + e^{-3x})/2.\)
Now, we assume the particular solution has the form:
\(y_p = A_1e^{3x} + A_2e^{-3x}\)
where A₁ and A₂ are undetermined coefficients.
Substituting these derivatives into the original differential equation, we get:
\((9A_1e^{3x} + 9A_2e^{-3x}) - 36(A_1e^{3x} + A_2e^{-3x}) = (e^{3x} + e^{-3x})/2.\)
To satisfy this equation, the coefficients of the exponential terms on both sides must be equal. Therefore, we have the following system of equations:
9A₁ - 36A₁ = 1/2,
9A₂ - 36A₂ = 1/2.
Solving these equations, we find A₁ = -1/70 and A₂ = -1/70.
Thus, the particular solution is:
\(y_p = (-1/70)e^{3x} + (-1/70)e^{-3x}\)
Finally, the general solution for the given differential equation is the sum of the complementary function and the particular solution:
\(y = y_h + y_p\\\\= C_1e^{6x} + C_2e^{-6x} + (-1/70)e^{3x} + (-1/70)e^{-3x}\)
where C₁ and C₂ are arbitrary constants determined by the initial or boundary conditions of the problem.
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Solve the following inequality algebraically. |x - 1| =>15
Answers
We have the following:
\(|x-1|\ge15\)solving for x:
\(\begin{gathered} x-1\ge15\rightarrow x\ge16 \\ x-1\leq-15\rightarrow x\leq-14 \end{gathered}\)The answer is:
\((-\infty,-14\rbrack\cup\lbrack16,\infty)\)
The line CD is defined by the points C(-2,1) and D(10,7).
Find the equation of the line CD.
Answers
Answer:
The equation of the line is; y = 0.5·x + 2
Step-by-step explanation:
The points that define the line CD = C(-2, 1) and D(10, 7)
The equation of the line can be presented in the form of the general equation of a straight line, y = m·x + c
Where;
m = The slope of the line = \(\dfrac{7 - 1}{10 - (-2)} = \dfrac{1}{2} = 0.5\)
c = The y-intercept
From the obtained slope, m = 0.5, using point D(10, 7), the equation of the line in point and slope form is therefore;
y - 7 = 0.5·(x - 10)
From the above equation of the line in point and slope form, we get the general form of the equation of the line as follows
y - 7 = 0.5·(x - 10) = 0.5·x - 5
y - 7 = 0.5·x - 5
y = 0.5·x - 5 + 7 = 0.5·x + 2
y = 0.5·x + 2
The equation of the straight line in general is y = 0.5·x + 2.
PLEASE HELP BE FAST YOU'LL GET BRAINLIEST IF CORRECT
Answers
Answer:
D
Step-by-step explanation:
Solution unit price of apples for each by 42.67
A. 4/1.5 =2.67
B. 5/2= 2.5
C 7/2.5= 2.8
1). 8/3.5 = 2.12
Because 2.12 < 2.5 < 2.67 < 2.8
so the answer is D
On a certain hot summer's day, 347 people used the public swimming pool. The daily prices are $ 1.75 for children and $ 2.50 for adults. The receipts for admission totaled $ 785.00. How many children and how many adults swam at the public pool that day?
Answers
There were 110 children and 237 adults whο swam at the public pοοl that day.
Let's assume that the number οf children whο used the swimming pοοl is x, and the number οf adults is y.
Accοrding tο the prοblem, the tοtal number οf peοple whο used the swimming pοοl is 347. Therefοre, we have:
x + y = 347 ....(1)
Alsο, the tοtal amοunt cοllected in admissiοn fees is $785.00. We knοw that the admissiοn fee fοr children is $1.75 and fοr adults is $2.50. Therefοre, we can write anοther equatiοn as:
1.75x + 2.5y = 785 ....(2)
We nοw have twο equatiοns with twο unknοwns. We can sοlve fοr x and y by using any suitable methοd. Here, we will use the substitutiοn methοd.
Frοm equatiοn (1), we can express y in terms οf x as:
y = 347 - x
We can substitute this expressiοn fοr y in equatiοn (2), and sοlve fοr x:
1.75x + 2.5(347 - x) = 785
1.75x + 867.5 - 2.5x = 785
-0.75x = -82.5
x = 110
Therefοre, the number οf children whο used the swimming pοοl is 110. We can use equatiοn (1) tο find the number οf adults:
110 + y = 347
y = 237
Therefοre, the number οf adults whο used the swimming pοοl is 237.
Therefοre, there were 110 children and 237 adults whο swam at the public pοοl that day.
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Solve for the input that corresponds to the given output value. (Round answers to three decimal places when appropriate. Enter your answers as a comma-separated list. Note: Even though the question may be completed without the use of technology, the authors intend for you to complete the activity using the technology you will be using in the remainder of the course so that you become familiar with the basic functions of that technology.) t(x)=x²-3x 21 x) 12, 0) 15. t(x)=12 x= x ((x)=15 X= x Solve for the input that corresponds to the given output value. (Round answers to three decimal plac though the question may be completed without the use of technology, the authors intend for you to course so that you become familiar with the basic functions of that technology.) t(x) = 4x2² 3x + 2; t(x) = 12, t(x) = 15 t(x) = 12 X= X t(x) = 15 X= X Need Help? Read It 27
Answers
The input values that correspond to the given output values are x = 7, x = -3, x = 4, x = 5.
To solve for the input values corresponding to given output values, we are given multiple equations of the form t(x) = f(x), where f(x) represents a polynomial function. We need to find the values of x that satisfy these equations.
For the first equation,
t(x) = x² - 3x = 21
x² - 3x - 21 = 0
(x - 7)(x + 3) = 0 (using factorization)
x = 7 or x = -3
For the second equation,
t(x) = 12
x² - 3x - 12 = 0
(x - 4)(x + 3) = 0 (using factorization)
x = 4 or x = -3
For the third equation,
t(x) = 15
x² - 3x - 15 = 0
(x - 5)(x + 3) = 0
x = 5 or x = -3
Once we find the solutions to the quadratic equations, we obtain the corresponding input values (x-values) that satisfy the given output values (t(x)-values).
These solutions represent the points at which the polynomial function intersects the specified output values.
Therefore, by setting t(x) equal to the given output values and solving the resulting quadratic equations, we determine the input values that correspond to the desired outputs. However, the output values are x = 7, x = -3, x = 4, x = 5.
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1,030 at 4% compounded semiannually for 2 years
Answers
Answer: $1114.91
Step-by-step explanation:
The formula for compound interest is
\(A= P(1+\frac{r}{n})^{nt} \\\)
Where
A = final amount
P = initial principal balance (1030 for this)
r = interest rate (0.04 for this)
n = number of times interest applied per time period (2 for this)
t = number of time periods elapsed (2 for this)
\(A= 1030(1+\frac{.04}{2})^{(2)(2)} \\\\A= 1030(1+0.02)^{4} \\A=1030(1.02)^4\\A=1114.905125\)
This rounds up to $1114.91
If the company had $4000 worth of office supplies at the beginning of the period. What is the entry required if we find that at the end of the period we have $3900 of supplies remaining.
Answers
The entry required to account for the change in office supplies would depend on the accounting method used. Assuming the company follows the periodic inventory system, where office supplies are expensed as they are used, the entry would be as follows:
At the beginning of the period:
Debit: Office Supplies Expense - $4,000
Credit: Office Supplies - $4,000
At the end of the period:
Debit: Office Supplies - $3,900
Credit: Office Supplies Expense - $3,900
Explanation:
1. At the beginning of the period, the company records the office supplies as an asset (Office Supplies) and recognizes an expense (Office Supplies Expense) for the same amount. This reduces the value of the asset and reflects the cost of supplies used during the period.
2. At the end of the period, when it is determined that $3,900 worth of supplies remains, the company adjusts the office supplies account by reducing it by the remaining amount. This adjustment is necessary to reflect the correct value of supplies on hand at the end of the period.
The entry ensures that the net effect of the transactions is an expense of $100 ($4,000 - $3,900), which represents the cost of supplies consumed during the period.
hi! please help in math!
i need the solution/explanation on how you got the answer
(y + 3) = -8(x - 4)
what is the slope?
Answers
Answer:
slope m = - 8
Step-by-step explanation:
the equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b ) a point on the line
y + 3 = - 8(x - 4) ← is in point- slope form
with slope m = - 8
The slope is :
↬ -8Solution:
Given: \(\bf{y+3=-8(x-4)}\)
To determine the slope, it's important to know the form of the equation first.
There are 3 forms that you should be familiar with.
The three forms of equations of a straight line are:
Slope Intercept (y = mx + b)Point slope (y-y₁) = m(x - x₁)Standard form (ax + by = c)This equation matches point slope perfectly.
The question becomes, how do you work with point slope to find slope?
Point slopeIn point slope, m is the slope and (x₁, y₁) is a point on the line.
Similarly, the slope of \(\bf{y+3=-8(x-4)}\) is -8.
Hence, the slope is -8.What is 5.37divied by 3 put 5.37 at the top and 3 at the bottom PLEASE I ONLY HAVE 19 MINS TO SUMMIT THIS!!
Answers
hellopppppppppppppppp
Answers
Answer:
Step-by-step explanation:
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Determine the sample space, give a sample point, and find the number of possible outcomes of the following experiments:
1.Drawing a vowel in an English Alphabet.
2. Learning the Primary colors.
3. Choosing an odd numbers from 1 to 11.
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Answers
The sample space and number of possible outcomes are:
1) Sample space = {a, e, i, o, u}
Number of possible outcomes of drawing a vowel = 5 outcomes
2) Sample space = {Red, Yellow, Blue}
Number of possible outcomes = 3
3) Sample space = {1, 3, 5, 7, 9, 11}
Number of possible outcomes = 6
How to find the sample space?A sample space is a collection or set of possible outcomes from a random experiment. The sample chamber is denoted by the symbol 'S'. A subset of the possible outcomes of an experiment are called events. A sample room can contain a set of results according to an experiment.
1) There are a total of 5 vowels in the English alphabets out of a total of 26 alphabets and as such:
Sample space = {a, e, i, o, u}
Number of possible outcomes of drawing a vowel = 5 outcomes
2) The primary colors are namely: Red, Yellow, Blue
Thus:
Sample space = {Red, Yellow, Blue}
Number of possible outcomes = 3
3) There 11 digits from 1 to 11 and a total of 6 odd numbers.
Sample space = {1, 3, 5, 7, 9, 11}
Number of possible outcomes = 6
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What are the x-intercepts for x2 - 5x + 6 = 0? (hint: factor
first)
Answers
Answer:x-intercepts (0,2) ; (0,3)
Step-by-step explanation:
x^2 - 5x + 6 =0
(x-2)(x-3) = 0
[-2 and -3 was the factors of 6, which make them equal to -5 when them add].
x-2=0 x-3=0
x=2 x=3
so x-intercepts (0,2) ; (0,3)
Which expression is equal to 567.083?
A
(5×100)+(6×10)+(7×1)+(8×100)+(3×1000)
B
(5×100)+(6×10)+(7×1)+(8×110)+(3×1100)
C
(5×100)+(6×10)+(7×1)+(8×1100)+(3×11000)
D
(5×100,000)+(6×10,000)+(7×1000)+(8×10)+(3×1)
Answers
Answer:
the answer is D
Step-by-step explanation:
u just have to add them all up
Prove the following:
1 + cos^2 A = 2 ( cos^4A + sin^4A)
Answers
The 2 sides of the equation are not equal; hence, I cannot prove them to be true.
Find the equation of the curve that passes through the points 2,16 and 5,250. Write a system
Answers
If the curve passes through the points (2,16) and (5,250), then its equation is y = 78x - 140.
In order to find the equation of the curve which passes through the points (2,16) and (5,250), we use the "point-slope" form of a linear equation, which is :
⇒ "Point-slope" form is : "y - y₁ = m×(x - x₁)",
where (x₁, y₁) is point on curve, m = slope of curve, and (x, y) = coordinates of any point on curve,
First, we find slope (m) using the two points, (x₁, y₁) = (2, 16), (x₂, y₂) = (5, 250),
Substituting the values,
We get,
⇒ Slope = (y₂ - y₁)/(x₂ - x₁),
⇒ m = (250 - 16)/(5 - 2),
⇒ m = 234/3,
⇒ m = 78,
Now, we use slope and the points to write equation of curve,
We use the point (2,16),
we get,
⇒ x₁ = 2, y₁ = 16, m = 78;
Substituting the values, in point-slope form equation,
We get,
⇒ y - 16 = 78(x - 2),
⇒ y - 16 = 78x - 156,
⇒ y = 78x - 156 + 16,
⇒ y = 78x - 140,
Therefore, the required curve-equation is "y = 78x - 140".
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The given question is incomplete, the complete question is
Find the equation of the curve that passes through the points (2,16) and (5,250).
simply the difference of 8w^2 -8w - 6) -(3w^2 +7w - 4)
Answers
Answer:
can you post the question I don't understand
Can someone help me with this pleaseee…….
Answers
The sides of the quadrilateral arranged from longest to shortest are CD, AB, DA, and BC.
We have,
To arrange the length of the sides of the quadrilateral from longest to shortest, we need to calculate the length of each side of the quadrilateral using the distance formula:
Distance Formula:
If (x1, y1) and (x2, y2) are two points in a plane, then the distance between them is given by:
d = √((x2 - x1)² + (y2 - y1)²)
Using the distance formula, we can calculate the length of each side of the quadrilateral as follows:
AB = √((4 - (-5))² + (5 - 5)²) = 9
BC = √((2 - 4)² + (0 - 5)²) = √(29)
CD = √((-5 - 2)² + (-2 - 0)²) = √(74)
DA = √((-5 - (-5))² + (5 - (-2))²) = 7
Therefore,
The sides of the quadrilateral arranged from longest to shortest are CD, AB, DA, and BC.
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The sales tax is 4.512%. The price for the Atari is $200. What is the total cost of the Atari with sales tax included?
Answers
Answer:
$209 approx
Step-by-step explanation:
Step one:
given data
The sales tax is 4.512%
The price for the Atari is $200
Step two:
let us find the amount of the tax
= 4.512/100*200
=0.04512*200
=$9.024
Then total will be
Total= 200+9.024
Total=$209 approx
Can you solve it?
find h. (system of equations)
Answers
Five employees are available to perform four jobs. The lime it takes each person to perform each job is given in Table 50. Determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs.
TABLE 50
Person
Time (hours)
Job 1
Job 2
Job 3
Job 4
1
22
18
30
18
2
18
—
27
22
3
26
20
28
28
4
16
22
—
14
5
21
—
25
28
Answers
To determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs, we need to consider the time taken by each person to complete each job. Using the given Table 50, we can analyze the data and identify the optimal assignment.
By examining Table 50, we can identify the minimum time taken by each person for each job. Starting with Job 1, we see that Person 4 takes the least time of 16 hours. Moving to Job 2, Person 2 takes the least time of 18 hours. For Job 3, Person 1 takes the least time of 25 hours. Lastly, for Job 4, Person 4 takes the least time of 14 hours.
Therefore, the optimal assignment would be:
- Person 4 for Job 1 (16 hours)
- Person 2 for Job 2 (18 hours)
- Person 1 for Job 3 (25 hours)
- Person 4 for Job 4 (14 hours)
This assignment ensures that the minimum total time is required to perform the four jobs, resulting in a total time of 16 + 18 + 25 + 14 = 73 hours.
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find the area of the region bounded by the given curves. y = 6x2 ln(x), y = 24 ln(x)
Answers
The area of the region bounded by the given curves. y = 6x2 ln(x), y = 24 ln(x) is 2.85 sq.units.
In this question we need to find the area of the region bounded by the given curves. y = 6x^2 ln(x), y = 24 ln(x)
Equating both the equations of the curve,
6x^2 ln(x) = 24 ln(x)
24 ln(x) - 6x^2 ln(x) = 0
x = 1, 2
This means, the curves intersect at x = 1 and x = 2.
So, the required area would be,
A = ∫[1 to 2] [24 ln(x) - 6x^2 ln(x)] dx
First we find the indefinite integral ∫[24 ln(x) - 6x^2 ln(x)] dx
= -6 ∫[-4 ln(x) + x^2 ln(x)] dx
= -6 ∫ln(x) (x^2 - 4) dx
= -6 ln(x) (1/3 x^3 - 4x) + 2/3 x^3 - 24x
So, ∫[1 to 2] [24 ln(x) - 6x^2 ln(x)] dx
= [-6 ln(x) (1/3 x^3 - 4x) + 2/3 x^3 - 24x] _(x = 1 to x = 2)
= 32 ln(2) - 58/3
= 22.18 - 19.33
= 2.85 sq.units.
Therefore, the area of the region is 2.85 sq.units.
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Two plumbers received a job. At first, one of the plumbers worked alone for 1 hour, and then they worked together for the next 4 hours. After this 40% of the job was complete. How long would it take each plumber to do the whole job by himself if it is known that the first plumber would take 5 more hours to finish the job than the second plumber?
Answers
Answer: https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Rate-of-work-word-problems.faq.hide_answers.1.html
Step-by-step explanation:
here use this link above to answer your question
Refer to the equation 2x - 6y = 12.
(a) Create a table of values for at least 4 points. Show your work.
(b) Use the table of values to graph
the line.
Answers
The points on the line 2x - 6y = 12 is
x y
0 -2
6 0
9 -1
12 2
What is equation of line in slope intercept form?
The most general form of equation of line in slope intercept form is given by y = mx + c
Where m is the slope of the line and c is the y intercept of the line.
Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.
If \(\theta\) is the angle that the line makes with the positive direction of x axis, then slope (m) is given by
m = \(tan \theta\)
The distance from the origin to the point where the line cuts the x axis is the x intercept of the line.
The distance from the origin to the point where the line cuts the y axis is the y intercept of the line.
a)The equation is 2x - 6y = 12
For x = 0,
\(2 \times 0- 6y = 12\\6y = -12\\y = -\frac{12}{6}\\y = -2\)
For x = 6,
\(2 \times 6 - 6y = 12\\6y = 0\\y = 0\)
For x = 9
\(2 \times 9 -6y = 12\\18 - 6y = 12\\6y = 12 - 18\\6y = -6\\y = -\frac{6}{6}\\y = -1\)
For x = 12
\(2 \times 12 - 6y = 12\\-6y = 12 - 24\\-6y = -12\\y = \frac{12}{6}\\y = 2\)
x y
0 -2
6 0
9 -1
12 2
b) The graph has been attached
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Simplify the following expression: 2x − 2y 5z − 2x − y 3z x − 3y 8z 2x − 2y 8z 3y 8z −3y 8z.
Answers
I believe this is the answer: -34yz-51yzx-384y^(2)z^(2)
Find the inverse of f(x)=7.50x
Answers
Identify the area of the figure rounded to the nearest tenth.
Answers
Answer:
91.5\(cm^{2}\)
Step-by-step explanation:
The area of the rectangle:
a = lw
a =(21)(3) = 63
The area of the square:
a = lw
a = (5)(5) = 25
The area of a half circle:
a=\(\frac{\pi r^{2} }{2}\)
a= \(\frac{3.14(1.5^{2}) }{2}\)
a = \(\frac{3.14(2.25)}{2}\)
a = \(\frac{7.065}{2}\)
a = 3.5 rounded to the nearest tenth
63 + 25 + 3.5 =91.5
Unit 4M : Modeling and Analyzing Exponential Functions "Growing Exponentially -Performance Task #2 Be sure to show your work where possible-failure to show work will result in deduction in grade Part I: Meet Jan Jan’s lifelong dream has been to open her own business. After working, sacrificing, and saving, she finally has enough money to open up an ice cream business. The grand opening of her business is scheduled for the Friday of Labor Day weekend. She would like to have a soft opening for her business on the Tuesday before. The soft opening should give her a good idea of any supply or personnel issues and give her time to correct them before the big official opening. A soft opening means that the opening of the business is not officially announced; news of its opening is just spread by word of mouth (see, not all rumors are bad!). Jan needs a good idea of when she should begin the rumor in order for it to spread reasonably well before her soft opening. She has been told that about 10% of the people who know about an event will actually attend it. Based on this assumption, if she wants to have about 70 people visit her store on the Tuesday of the soft opening, she will need 700 people to know about it. 1. Jan plans to tell two people each day and will ask that person to tell two other people each day through the day of the opening, and so on. Assume that each new person who hears about the soft opening is also asked to tell two other people each day through the day of the opening and that each one starts the process of telling their friends on the day after he or she first hears. When should Jan begin telling others about the soft opening in order to have at least 700 people know about it by the day it occurs? 2. Let x represent the day number and let y be the number of people who know about the soft opening on day x. Consider the day before Jan told anyone to be Day 0, so that Jan is the only person who knows about the opening on Day 0. Day 1 is the first day that Jan told two people about the opening. a. Complete the following table. Day0 1 2 3 4 5 Number of people who know 1 3 13. Graph the points from the table in part 2a. a. Does it make sense to connect the dots on the graph? Why or why not? b. What does point (5, 32) represent in this situation? Describe the point in a form of a function notation. 2c. Describe the domain of the function. What does the domain mean in this situation? What are the restrictions of the domain that arise from the context? d. Describe the range of the function. What does the range mean in context of the situation? What are the restrictions of the range that arise from the context? e. What is the y-intercept in this problem? What does the y-intercept represent in this situation? 4. The scenario above can be modeled by a type of function known as exponential function; in particular, an exponential growth function. An exponential function has the form , , where a is a non–zero real number and b is a positive real number other than 1. An exponential growth function has a value of b that is greater than 1. Values “a” and “b” are the parameters of the function. Write explicit and recursive equations that describe the relationship between x (day) and y(number of people who know) for the situation of spreading the news about the soft opening of Jan’s ice cream store. What type of a sequence does the function represent? What do the parts of the equations represent in terms of the context?
Answers
1)
From the information given,
x represents the day number
y represents the number of people who know about the opening on day x
On the first day, x = 0, y = 1
On the second day, x = 1, y = 3
On the third day, x = 2, y = 9
The rate at which the number of people is increasing is exponential. The general form of an exponential equation is
y = ab^x
We would find a and b by substituting corresponding values of x and y into the equation. We have
For x = 0 and y = 3,
3 = ab^0
a = 3
For x = 1 and y = 3,
3 = ab^1
By substituting a = 3,
3 = 3b
b = 3/3 = 1
The equation is
Determine the following indefinite integral. 2 5+° () 3t? | dt 2 + 3t 2 ) dt =
Answers
The solution is (5 + °) ((2 + 3t²)² / 12) + C for the indefinite integral.
A key idea in calculus is an indefinite integral, commonly referred to as an antiderivative. It symbolises a group of functions that, when distinguished, produce a certain function. The integral symbol () is used to represent the indefinite integral of a function, and it is usually followed by the constant of integration (C). By using integration techniques and principles, it is possible to find an endless integral by turning the differentiation process on its head.
The expression for the indefinite integral with the terms 2 5+°, ( ) 3t?, 2 + 3t 2, and dt is given by;\(∫ 2(5 + °) (3t² + 2) / (2 + 3t²) dt\)
To solve the above indefinite integral, we shall use the substitution method as shown below:
Let y = 2 + \(3t^2\) Then dy/dt = 6t, from this, we can find dt = dy / 6t
Substituting y and dt in the original expression, we have∫ (5 + °) (3t² + 2) / (2 + 3t²) dt= ∫ (5 + °) (1/6) (6t / (2 + 3t²)) (3t² + 2) dt= ∫ (5 + °) (1/6) (y-1) dy
Integrating the expression with respect to y we get,(5 + °) (1/6) * [y² / 2] + C = (5 + °) (y² / 12) + C
Substituting y = 2 +\(3t^2\) back into the expression, we have(5 + °) ((2 + 3t²)² / 12) + C
The solution is (5 + °) ((2 + 3t²)² / 12) + C.
Learn more about indefinite integral here:
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