The values of [tex]r[/tex] is [tex]\boxed{r = 3 - \sqrt7 }[/tex] and [tex]\boxed{r = 3 + \sqrt 7 }.[/tex]
Further explanation:
Given:
The function is [tex]y = {e^{rx}}.[/tex]
The differential equation is [tex]y'' - 6y' + 2y = 0\,\,\,\,\,or\,\,\,\,\dfrac{{{d^2}y}}{{d{x^2}}} - 6\dfrac{{dy}}{{dx}} + 2y = 0[/tex]
Explanation:
The given function can be expressed as follows,
[tex]y = {e^{rx}}[/tex]
Differentiate the above equation with respect to [tex]x[/tex].
[tex]\begin{aligned}\frac{{dy}}{{dx}} &= \frac{d}{{dx}}\left( {{e^{rx}}} \right)\\&= {e^{rx}} \times \frac{d}{{dx}}\left( {rx} \right)\\&= {e^{rx}} \times r\\&= r{e^{rx}}\\\end{aligned}[/tex]
Again differentiate with respect to [tex]x[/tex].
[tex]\begin{aligned}\frac{{{d^2}y}}{{d{x^2}}} &= \frac{d}{{dx}}\left( {r{e^{rx}}} \right)\\&= r\frac{d}{{dx}}\left( {{e^{rx}}} \right)\\&=r\times {e^{rx}} \times\frac{d}{{dx}}\left( {rx} \right)\\&= {r^2}{e^{rx}}\\\end{aligned}[/tex]
Now solve the differential equation.
[tex]\begin{aligned}y'' - 6y' + 2y &= 0\\{r^2}{e^{rx}} - 6r{e^{rx}} + 2{e^{rx}} &= 0\\{e^{rx}}\left( {{r^2} - 6r + 2} \right) &= 0\\{r^2} - 6r + 2 &= 0\\\end{aligned}[/tex]
Solve the quadratic equation [tex]{r^2} - 6r + 2 = 0.[/tex]
[tex]\begin{aligned}D&= {b^2} - 4ac\\&={\left( { - 6} \right)^2} - 4\left( 1 \right)\left( 2 \right)\\&= 36 - 8\\&= 28\\\end{aligned}[/tex]
The value of [tex]r[/tex] can be obtained as follows,
[tex]\begin{aligned}r&= \frac{{ - b \pm \sqrt D }}{{2a}}\\r&=\frac{{ - \left( { - 6} \right) \pm \sqrt {28} }}{{2 \times 1}} \\ r &= \frac{{6 \pm 2\sqrt7 }}{2}\\r&= 3 \pm \sqrt7\\r&= 3 - \sqrt 7 \,\,\,\,\,\,\,or\,\,\,\,\,\,\,r = 3 + \sqrt7 \\\end{aligned}[/tex]
The values of [tex]r[/tex] is [tex]\boxed{r = 3 - \sqrt7 }[/tex] and [tex]\boxed{r = 3 + \sqrt 7 }.[/tex]
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Derivatives
Keywords: Derivative, value of x, function, differentiate, minimum value, dy, compute, given value of [tex]x, y=6x, x=4, x=1[/tex]
Please help!!
A road trip of 400 miles require a total of 8 hours to drive. For the first portion of the trip, an average speed of 63 miles per hour was maintained; on the last portion of the trip, the average speed was only 443 miles per hour due to highway construction project. How much time, in hours, to the nearest tenth of an hour, was spent driving at each of the two speeds, and what was the total time for the entire trip?
what is 2x+square root x+1=8 solve for x
Shantel is studying for her history test. After 20 minutes she is 1/4 of the way done. How much longer will she study?
Shantel will need to study for 60 minutes more to complete her history review.
Shantel is studying history and after 20 minutes, she is [tex]\frac{1}{4}[/tex] of the way done.
To determine how much longer she will study, we need to calculate how much time is needed to complete the remaining [tex]\frac{3}{4}[/tex] of her study time.
[tex]\frac{1}{4}[/tex] of the study done in 20 minutes means she will study for 4 * 20 = 80 minutes in total.
To find out how much longer she needs to study, subtract the time she has already studied (20 minutes) from the total study time (80 minutes): 80 - 20 = 60 minutes.
Therefore, Shantel will need to study for 60 minutes more to complete her history review.
Find the quotient of 5/68
The total cost of the fatigato family's two cars was 71,482. The cost of one car was 38,295. Write an equation using a variable to represent the cost of the familys other car.
The distance d that a certain particle moves may be calculated from the expression d = at + bt2 where a and b are constants; and t is the elapsed time. what must the dimensions of the quantities a and b be, respectively?
The area of a circle is 99 in.² what’s the radius
what are two practical products you could add to find 990 x 37 ; 297 693 2970 6930 69300 6330 29700
What is the median of the following string of values?
55, 18, 58, 49, 8, 77, 62, 26, 7, 91
A.55
B.52
C.49
D.58
Answer:
Option B is correct
The median of the following string of values is 52
Step-by-step explanation:
Median defined as the median is the middle number.
To find the median of the following string of values:
55, 18, 58, 49, 8, 77, 62, 26, 7, 91
Arrange the string of values from least to greatest.
7, 8, 18 , 26, 49, 55, 58, 62, 77, 91
As you can see that we have two numbers, there is no middle number.
Simply, take the mean of the two middle numbers.
then;
Median = [tex]\frac{49+55}{2} = \frac{104}{2} = 52[/tex]
Therefore, the median of the following string of values is 52
caryssa makes bracelets to sell at craft fairs. The supplies she needed to start her business cost $45. In addition, the supplies for each bracelet cost her $3. She sells her bracelets for $5. What is the minimum number of braclets she must sell to make a profit?
720,808 in expanded form with exponents
The number 720,808 in expanded form with exponents is written as:
[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]
To express 720,808 in expanded form with exponents, We break it down by place value and then represent each term using powers of 10. The breakdown is as follows:700,000 : [tex]7 \times 10^5 \\[/tex]
20,000 : [tex]2 \times 10^4 \\[/tex]
800 : [tex]8 \times 10^2 \\[/tex]
8 : [tex]8 \times 10^0[/tex]
When we combine these, we get the expanded form:[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]
This representation helps in understanding the value of each digit based on its position in the number.
You and your brother are readibg the same novel you wnat to get ahead of him in the book so you decide to read 30 minutes longer than your brother reads how many minutes will you read if your brother reads for 15 minutes
Answer:
45 minute 30+15= 45 easy
Step-by-step explanation:
If u, v, and w are nonzero vectors in r 2 , is w a linear combination of u and v?
You are at a Carnival .One of the Carnival games ask you to pick a door and then pick a curtain behind the door. There are 3 doors and 4 curtains behind each door.how many choices are possible for the player ?????
There are 12 possible choices for the player.
What is combination?"It is a way of selecting items from a collection "
What is formula of combination?"[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]"
What is n factorial?n! = n × (n - 1) × . . . × 2 × 1
For given question,
There are 3 doors and 4 curtains behind each door.
The number of possible ways of selecting a door from the 3 doors.
Using combination formula,
[tex]^3C_1=\frac{3!}{1!(3-1)!}\\\\ ^3C_1=\frac{3\times 2!}{2!}\\\\ ^3C_1=3[/tex]
The number of possible ways of selecting a curtain from the 4 curtains.
Using combination formula,
[tex]^4C_1=\frac{4!}{1!(4-1)!}\\\\ ^4C_1=\frac{4\times 3!}{3!}\\\\ ^3C_1=4[/tex]
So, the number of choices for the player to pick a door and then pick a curtain behind the door :
[tex]^3C_1\times ^4C_1\\\\=3\times 4\\\\=12[/tex]
Therefore, there are 12 possible choices for the player.
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One book cost £1.99 how much do four books cost
The floor of a one-story building is 14 feet longer than it is wide. The building has 1715 square feet of floor space. Find the length l and width w of the floor.
Final answer:
To find the floor dimensions of a building that is 14 feet longer than it is wide with an area of 1715 sq ft, we solve the quadratic equation w * (w + 14) = 1715, resulting in w=100 feet and length l=114 feet.
Explanation:
The student is asked to find the length (l) and width (w) of a one-story building floor which is 14 feet longer than it is wide, with a total area of 1715 square feet. To solve this, we set up an equation where w represents the width and w + 14 feet represents the length. Since the area of a rectangle is length times width, the equation is w * (w + 14) = 1715.
After solving this quadratic equation, we find that w=100 feet which makes the length l = w + 14 = 114 feet. Therefore, the actual dimensions of the building are 114 ft by 100 feet.
How do I solve this 6x + 3 = 5x + 10
The team manager mixes 0.6 kg of Gatorade powder with water in the 5-gallon water coolers for the football team. Gatorade donates new 7-gallon water jugs. How much powder should the manager use in a new 7-gallon cooler to maintain the same Gatorade mixture?
Why is it impossible to list the set of rational numbers using the roster method?
What is the common ratio for this geometric sequence?
Answer:
Common ratio for the sequence is 3
Step-by-step explanation:
Given sequence is
5, 15, 45, 135,.....
To find out common ratio, divide the second number by the first number
[tex]\frac{15}{5} =3[/tex]
[tex]\frac{45}{15} =3[/tex]
[tex]\frac{135}{45} =3[/tex]
we got 3 that is our common ratio 3
Common ratio for the sequence is 3
It takes 54 minutes for 4 people to paint 6 walls. How many minutes does it take 6 people to paint 7 walls?
Answer:
It will take 42 minutes.
Step-by-step explanation:
In order to solve this we first have to calulate how long will it take one person to paint one wall:
54/6=9minutes
That is between 4 people it will take 9 minutes to paint a wall, one person alone supposing they all work at the same rate will take 36 minutes.
Now we just multiply that by the number of walls to be painted:
36*7=252
And we divide that by the number of people painting:
252/6=42
So it will take 42 minutes to paint 7 walls with 6 people painting.
-0.7 repeating as a decimal in simplest form
Derivative of sin (x^(1/3)) + (sinx)^(1/3)
2. During a 5-day training at your department, the number of visitors tripled each day. If the training started on a Monday with 345 attendees, what was the attendance on Thursday?
The number of attendance on Thursday would be 9,315.
Given that,
During a 5-day training at your department, the number of visitors tripled each day.
Here, the training started on a Monday with 345 attendees.
Hence, the number of visitors on Tuesday is 3 times with 345 attendees.
That is, On Tuesday;
Number of attendees = 345 x 3
= 1,035
On Wednesday;
Number of attendees = 1,035 x 3
= 3,105
On Thursday;
Number of attendees = 3,105 x 3
= 9,315
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10/5 = 2
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water weighs about 8.34 pounds per gallon. About how many ounces per gallon is the weight of the water?