Final answer:
The question asks for the original vector function given its derivative and an initial condition. Through integration and applying the initial condition, the vector function is found to be r(t) = t⁵i + t⁶j + (t²/2 + 1/2)k.
Explanation:
The question involves finding the vector function r(t) given its derivative r'(t) = 5t⁴i + 6t⁵j + t k and an initial condition r(1) = i + j. To find r(t), integrate each component of r'(t) with respect to t. The integration results in:
The i component: ∫5t4 dt = t⁵/1 + C₁The j component: ∫6t5 dt = t⁶/1 + C₂The k component: ∫t dt = t²/2 + C₃Therefore, r(t) = (t⁵/1 + C₁)i + (t⁶/1 + C₂)j + (t²/2 + C₃)k. Substituting t = 1 and using the initial condition r(1) = i + j to find the constants C₁, C₂, and C₃, we get:
C₁ = 0C₂ = 0C₃ = 1/2Finally, r(t) = t⁵i + t⁶j + (t²/2 +1/2) k.
The function [tex]\( r(t) \)[/tex] is:
[tex]\[ r(t) = t^5i + t^6j + \frac{1}{2}t^2k - \frac{1}{2} \][/tex]
[tex]So, \( r(t) = t^5i + t^6j + \frac{1}{2}t^2k - \frac{1}{2} \).[/tex]
To find [tex]\( r(t) \) given \( r'(t) \) and \( r(1) \)[/tex], we need to integrate[tex]\( r'(t) \)[/tex]with respect to ( t ) and then use the given initial condition ( r(1) ) to determine the constant of integration.
Given:
[tex]\[ r'(t) = 5t^4i + 6t^5j + tk \][/tex]
[tex]\[ r(1) = i + j \][/tex]
First, let's integrate [tex]\( r'(t) \)[/tex] with respect to [tex]\( t \)[/tex] to find [tex]\( r(t) \)[/tex]:
[tex]\[ r(t) = \int 5t^4i \, dt + \int 6t^5j \, dt + \int tk \, dt \][/tex]
[tex]\[ r(t) = \frac{5}{5}t^5i + \frac{6}{6}t^6j + \frac{1}{2}t^2k + C \][/tex]
[tex]\[ r(t) = t^5i + t^6j + \frac{1}{2}t^2k + C \][/tex]
Now, we apply the initial condition [tex]\( r(1) = i + j \)[/tex]:
[tex]\[ r(1) = (1)^5i + (1)^6j + \frac{1}{2}(1)^2k + C = i + j \][/tex]
[tex]\[ i + j + \frac{1}{2}k + C = i + j \][/tex]
Comparing the coefficients of [tex]\( i \), \( j \), and \( k \)[/tex], we get:
Coefficient of [tex]\( i \): \( 1 = 1 \)[/tex] (matches)
Coefficient of [tex]\( j \): \( 1 = 1 \)[/tex] (matches)
Coefficient of [tex]\( k \): \( \frac{1}{2} = 0 \)[/tex] (doesn't match)
So, to make the coefficients match, [tex]\( C = -\frac{1}{2} \)[/tex].
Therefore, the function [tex]\( r(t) \)[/tex] is:
[tex]\[ r(t) = t^5i + t^6j + \frac{1}{2}t^2k - \frac{1}{2} \][/tex]
[tex]So, \( r(t) = t^5i + t^6j + \frac{1}{2}t^2k - \frac{1}{2} \).[/tex]
The indicated function y1(x) is a solution of the given differential equation. use reduction of order or formula (5) in section 4.2, y2 = y1(x) e−∫p(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). 9y'' − 12y' + 4y = 0; y1 = e2x/3
The indicated function y1(x) is a solution of the given differential equation.The general solution is [tex]y = c_1 e^{2x}- c_2e^{-6x}/8[/tex]
What is a differential equation?An equation containing derivatives of a variable with respect to some other variable quantity is called differential equations.
The derivatives might be of any order, some terms might contain the product of derivatives and the variable itself, or with derivatives themselves. They can also be for multiple variables.
Given differential equation is
y''-4y'+4y=0
and
[tex]y_1(x) = e^{2x}[/tex]
[tex]y_2(x) = y_1(x) \int\limits^a_b {e^{\int pdx} \, / y_1 ^2(x)dx[/tex]
The general form of equation
y''+P(x)y'+Q(x)y=0
Comparing both the equation
So, P(x)= - 4
[tex]y_2(x) = y_1(x) \int\limits^a_b {e^{\int pdx} \, / y_1 ^2(x)dx\\\\\\[/tex]
[tex]y_2(x) = e^{2x}\int e^{-4x} \, / e^{4x}dx[/tex]
[tex]y_2(x) = e^{2x}\int e^{-8x}dx\\\\y_2(x) = -e^{-6x}/8[/tex]
The general solution is
[tex]y = c_1 e^{2x}- c_2e^{-6x}/8[/tex]
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The sum of 7 consecutive odd numbers is 91. What is the sum of the two largest numbers in this set?
The function for the cost of materials to make a shirt is f(x) = five sixths x + 5, where x is the number of shirts. The function for the selling price of those shirts is g(f(x)), where g(x) = 5x + 6. Find the selling price of 18 shirts
The sun’s rays are striking the ground at a 55° angle, and the length of the shadow of a tree is 56 feet. How tall is the tree?
select one:
a. 80.0 feet
b. 45.9 feet ( Incorrect)
c. 34.2 feet (incorrect)
d. 32.1 feet
A map is scaled so that 3 cm on the map is equal to 21 actual miles. if two cities on the map are 5 cm apart, what proportion would you use to solve the problem?
You roll a pair of fair dice until you roll “doubles” (i.e., both dice are the same). what is the expected number, e[n], of rolls?
The expected number of rolls, e[n], to get doubles on a pair of fair dice is 6. This is calculated by recognizing that getting doubles has a probability of 1/6, and the expectation for a geometric distribution is the inverse of the probability (1/p).
Explanation:To solve the problem of finding the expected number of rolls, e[n], to get doubles on a pair of fair dice, we first need to calculate the probability of rolling doubles. Since there are 6 faces on each die, there are a total of 6 x 6 = 36 possible outcomes when rolling two dice.
Out of these 36 possible rolls, there are 6 outcomes that result in doubles: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). This means the probability of getting doubles in one roll is 6/36, which simplifies to 1/6.
Because each roll is independent, we can model the scenario using a geometric distribution, where the expected value, or mean, is given by 1/p, where p is the success probability. Substituting p with 1/6, the expected number of rolls to get doubles would be 1/(1/6) = 6.
Therefore, the expected number of rolls needed to roll doubles is 6.
A spinner is divided into 10 equal sections numbered 1 through 10. If the arrow is spun once, what is the probability it will land on a number less than 3?
The probability with the condition of the spinner landing on a number less than 3 is 0.2
What is a conditional probability?A conditional probability is a probability of an event occuring with a condition that another event had previously occurred. The event in the question is spinning the spinner once while the condition is that the number landed on is less than 3.
The spinner has 10 equal sections numbered 1 through 10.
The conditional probability of landing on a number less than 3 is the same as the probability of landing on either 1 or 2.
There are two sections out of ten that corresponds to numbers less than 3. The probability of landing on a number less than 3 is therefore;
P(Landing on a number less than 3) = P(Landing on 1) + P(Landing on 2)
P((Landing on 1) = 1/10
P(Landing on 2) = 1/10
P(Landing on 1) + P(Landing on 2) = (1/10) + (1/10) = 2/10
(1/10) + (1/10) = 2/10 = 0.2
The probability of landing on a number less than 3 is 0.2
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There is a line through the origin that divides the region bounded by the parabola
y=4x−3x^2 and the x-axis into two regions with equal area. What is the slope of that line?
The average score on a standardized test is 500 points with a standard deviation of 50 points. If 2,000 students take the test at a local school, how many students do you expect to score between 500 and 600 points?
To solve this problem, we use the z statistic. The formula for z score is given as:
z = (x – u) / s
Where,
x = sample score
u = the average score = 500
s = standard deviation = 50
First, we calculate for z when x = 500
z = (500 – 500) / 50
z = 0 / 50
z = 0
Using the standard z table, at z = 0, the value of P is: (P = proportion)
P (z = 0)= 0.5
Secondly, we calculate for z when x = 600
z = (600 – 500) / 50
z = 100 / 50
z = 2
Using the standard z table, at z = 2, the value of P is: (P = proportion)
P (z = 2) = 0.9772
Since we want to find the proportion between 500 and 600, therefore we subtract the two:
P (500 ≥ x ≥ 600) = 0.9772 – 0.5
P (500 ≥ x ≥ 600) = 0.4772
Answer:
Around 47.72% of students have score from 500 to 600.
Answer:
To solve this problem, we use the z statistic. The formula for z score is given as:
z = (x – u) / s
Where,
x = sample score
u = the average score = 500
s = standard deviation = 50
First, we calculate for z when x = 500
z = (500 – 500) / 50
z = 0 / 50
z = 0
Using the standard z table, at z = 0, the value of P is: (P = proportion)
P (z = 0)= 0.5
Secondly, we calculate for z when x = 600
z = (600 – 500) / 50
z = 100 / 50
z = 2
Using the standard z table, at z = 2, the value of P is: (P = proportion)
P (z = 2) = 0.9772
Since we want to find the proportion between 500 and 600, therefore we subtract the two:
P (500 ≥ x ≥ 600) = 0.9772 – 0.5
P (500 ≥ x ≥ 600) = 0.4772
Answer:
Around 47.72% of students have score from 500 to 600.
Step-by-step explanation:
What is the midpoint of the line segment (-3,-2) and (1,4)
Answer:
(-1, 1)
Step-by-step explanation:
The midpoint (M) is found by averaging the coordinates of the end points:
M = ((-3, -2) +(1, 4))/2 = ((-3+1)/2, (-2+4)/2) = (-2/2, 2/2)
M = (-1, 1)
The midpoint of the line segment is (-1, 1).
x = 2, y = -1
14
2. x = 0, y = 2.5
1.665
3. x = -1, y = -3
0.44
4. x = 0.5, y =
9.17
5. x = , y =
-1
6. x = √2, y = √2
-11.25
Answer:
b
Step-by-step explanation:
What standard deviation below the mean of normal young adults equals osteoporosis?
Andrei has a job in the circus walking on stilts. Andrei is 11/10 meters tall. The foot supports of his stilts are 23/10 meters high.
How high is the top of Andrei's head when he is walking on his stilts?
A rocket is launched straight up from the ground, with an initial velocity of 224 feet per second. The equation for the height of the rocket at time t is given by:
h=-16t^2+224t
(Use quadratic equation)
A.) Find the time when the rocket reaches 720 feet.
B.) Find the time when the rocket completes its trajectory and hits the ground.
A.) The rocket reaches 720 feet in 5 seconds and 9 seconds.
B.) The rocket completes its trajectory and hits the ground in 14 seconds
Further explanationA quadratic equation has the following general form:
[tex]ax^2 + bx + c = 0[/tex]
The formula to solve this equation is :
[tex]\large {\boxed {x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} } }[/tex]
Let's try to solve the problem now.
Question A:Given :
[tex]h = -16 t^2 + 224t[/tex]
The rocket reaches 720 feet → h = 720 feet
[tex]720 = -16 t^2 + 224t[/tex]
[tex]16 t^2 - 224t + 720 = 0[/tex]
[tex]16 (t^2 - 14t + 45 = 0)[/tex]
[tex]t^2 - 14t + 45 = 0[/tex]
[tex]t^2 - 9t - 5t + 45 = 0[/tex]
[tex]t(t - 9) - 5(t - 9) = 0[/tex]
[tex](t - 5)(t - 9) = 0[/tex]
[tex]t = 5 ~ or ~ t = 9[/tex]
The rocket reaches 720 feet in 5 seconds and 9 seconds.
Question B:The rocket hits the ground → h = 0 feet
[tex]0 = -16 t^2 + 224t[/tex]
[tex]16 (t^2 - 14t ) = 0[/tex]
[tex]t^2 - 14t = 0[/tex]
[tex]t( t - 14 ) = 0[/tex]
[tex]t = 0 ~ or ~ t = 14[/tex]
The rocket completes its trajectory and hits the ground in 14 seconds
Learn moremethod for solving a quadratic equation : https://brainly.com/question/10278062solution(s) to the equation : https://brainly.com/question/4372455best way to solve quadratic equation : https://brainly.com/question/9438071Answer detailsGrade: College
Subject: Mathematics
Chapter: Quadratic Equation
Keywords: Quadratic , Equation , Formula , Rocket , Maximum , Minimum , Time , Trajectory , Ground
The variable z is directly proportional to x and inversely proportional to y. When x is 12 and y is 18 z has the value 2 what is the value of z when x = 19 and y = 22
Explain why rationalizing the denominator does not change the value of the original expression
Answer:
Because basically, you are multiplying by 1
Step-by-step explanation:
Let me explain this with an example. Rationalize the following expression:
[tex]\frac{5}{\sqrt{7} }[/tex]
In order to rationalize the denominator, the numerator and denominator of the fraction must be multiplied by the root of the denominator. So, what happen if you do that? Well first of all you aren't altering the expression because:
[tex]\frac{\sqrt{7} }{\sqrt{7} } =1[/tex]
Right? Because a certain quantity divided by itself is always equal to 1. So basically you are doing this because you want to rewrite the expression without altering its original value, it is the same when you do this:
[tex]9=3^2=3+3+3=\sqrt{81}[/tex]
Therefore, the only thing you do when you rationalize is remove radicals from the denominator of a fraction. Take a look:
[tex]\frac{5}{\sqrt{7} } *\frac{\sqrt{7} }{\sqrt{7} } =\frac{5*\sqrt{7} }{\sqrt{7}*\sqrt{7} } =\frac{5*\sqrt{7}}{(\sqrt{7} )^2} =\frac{5*\sqrt{7} }{7}[/tex]
You can check this new expression is equal to the original using a calculator:
[tex]\frac{5}{\sqrt{7} } \approx1.8898\\\\\frac{5*\sqrt{7} }{7} \approx1.8898[/tex]
please factor this problem x^2+7x-8
When patey pontoons issued 6% bonds on january 1, 2016, with a face amount of $600,000, the market yield for bonds of similar risk and maturity was 7%. the bonds mature december 31, 2019 (4 years). interest is paid semiannually on june 30 and december 31?
Which of the following statements is not true?
An angle bisector can be a median of a triangle.
A perpendicular bisector can be an altitude of a triangle.
A median can be an altitude of a triangle.
All of the statements are true.
how do you write 20484163 in different forms
The brightness of a variable star adds a component to the simple harmonic motion we have studied in this lesson. Since the brightness is variable the vertical axis may no longer be equal to zero. Also included in the variance is a phase shift. In this case, the equation for this function would be: y = a cos w(t – c ) + b.
Suppose we have a variable star whose brightness alternately increases and decreases. For this star, the time between periods of maximum brightness is 6.5 days. The average brightness (or magnitude) of the star is 5.0 and its brightness varies by + 0.25 magnitude.
1. What is the amplitude of the function for this model?
2. What is the period?
3. What is w?
4. What is the vertical shift?
5. Is there a phase shift? If so, what is it?
6. What is the function
What is the general form of the equation for the given circle centered at O(0, 0)? x2 + y2 + 41 = 0 x2 + y2 − 41 = 0 x2 + y2 + x + y − 41 = 0 x2 + y2 + x − y − 41 = 0
The general form of the equation for the given circle centered at O(0, 0) is:
[tex]x^2+y^2-41=0[/tex]
Step-by-step explanation:We know that the standard form of circle is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
where the circle is centered at (h,k) and the radius of circle is: r units
1)
[tex]x^2+y^2+41=0[/tex]
i.e. we have:
[tex]x^2+y^2=-41[/tex]
which is not possible.
( Since, the sum of the square of two numbers has to be greater than or equal to 0)
Hence, option: 1 is incorrect.
2)
[tex]x^2+y^2-41=0[/tex]
It could also be written as:
[tex]x^2+y^2=41[/tex]
which is also represented by:
[tex](x-0)^2+(y-0)^2=(\sqrt{41})^2[/tex]
This means that the circle is centered at (0,0).
3)
[tex]x^2+y^2+x+y-41=0[/tex]
It could be written in standard form by:
[tex](x+\dfrac{1}{2})^2+(y+\dfrac{1}{2})^2=(\sqrt{\dfrac{83}{2}})^2[/tex]
Hence, the circle is centered at [tex](-\dfrac{1}{2},-\dfrac{1}{2})[/tex]
Hence, option: 3 is incorrect.
4)
[tex]x^2+y^2+x-y=41[/tex]
In standard form it could be written by:
[tex](x+\dfrac{1}{2})^2+(y-\dfrac{1}{2})^2=(\sqrt{\dfrac{83}{2})^2[/tex]
Hence, the circle is centered at:
[tex](\dfrac{-1}{2},\dfrac{1}{2})[/tex]
Solve cos x +sqr root of 2 = -cos x for x over the interval 0,2pi
What is equivalent to the expression "the quotient of five and seven"?
write the smallest numeral possible using the digits 9, 3 and 6
The smallest numeral that can be created from the digits 9, 3, and 6 is 369. This is achieved by arranging the digits in ascending order.
Explanation:The smallest numeral that can be formed using the digits 9, 3, and 6 is 369. In mathematics, when we are to create the smallest possible numeral from a given set of digits, we arrange the digits in increasing order from left to right, that means the smallest digit will be on the left-most side and the largest digit will be on the right-most side.
So, with the digits 9, 3, and 6, we place 3 first as it's the smallest, then 6 as it's the next smallest, and finally 9, resulting in the smallest numeral 369.
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The smallest numeral possible using the digits 9, 3, and 6 is 369, arranged in ascending order.
To write the smallest numeral possible using the digits 9, 3, and 6, we arrange the digits in ascending order. The smallest digit is placed at the beginning, followed by the larger ones. Therefore, the smallest numeral we can create is 369.
Which value is a discontinuity of x^2+7x+1/x^2+2x-15? x=-1 x=-2 x=-5 x=-4
The value of x=-5 is a discontinuity of the function x^2+7x+1 / x^2+2x-15 since it makes the denominator of this function equal to zero.
Explanation:The subject of this question is the discontinuity of a rational function. In Mathematics, a function f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomials, is said to be discontinuous at a particular value of x if and only if q(x) = 0 at that value. From the equation in the question; x^2+7x+1/x^2+2x-15, we can determine its discontinuity by finding the values of x that would make the denominator equal to zero. This is done by solving the polynomial equation x^2+2x-15 = 0 for x. The solutions to this equation represent the values at which the function is discontinuous.
By applying the quadratic formula, (-b ± sqrt(b^2 -4ac))/(2a), where a = 1, b = 2, and c = -15, we get that x = -5, and 3. However, the values given in the question are x=-1, x=-2, x=-5, and x=-4. From these options, only x=-5 makes the denominator zero, thus, x = -5 is a point of discontinuity in the function x^2+7x+1 / x^2+2x-15.
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13-36x^2=-12 which value of x is a solution to he equation
The area of a rectangle wall of a barn is 216 ft.² it's length is 6 feet longer than twice it's width. find the length and width of the wall of the barn
A football is punted from a height of 2.5 feet above the ground with an initial vertical velocity of 45 feet per second.
Write an equation to model the height h in feet of the ball t seconds after it has been punted.
The football is caught at 5.5 feet above the ground. How long was the football in the air?
If the APR of a savings account is 3.6% and interest is compounded monthly, what is the approximate APY of the account?
Answer:
3.66% ( approx )
Step-by-step explanation:
Since, the formula of annual percentage yield is,
[tex]APY = (1+\frac{r}{n})^n-1[/tex]
Where,
r = stated annual interest rate,
n = number of compounding periods,
Here, r = 3.6% = 0.036,
n = 12 ( ∵ 1 year = 12 months )
Hence, the annual percentage yield is,
[tex]APY=(1+\frac{0.036}{12})^{12}-1=1.03659 - 1 = 0.036599\approx 0.0366 = 3.66\%[/tex]