Final answer:
To find the position vector given the acceleration, initial velocity, and initial position, we integrate the acceleration vector to find the velocity vector, and then integrate the velocity vector to find the position vector. The resulting position vector is r(t) = (⅔t³ + t)i - sin t j - ¼ cos 2t k.
Explanation:
Finding the Position Vector
The question asks us to find the position vector of a particle given its acceleration vector a(t) = 8t i + sin t j + cos 2t k, initial velocity v(0) = i, and initial position r(0) = j. To find the position vector, we first need to integrate the acceleration vector to find the velocity vector, and then integrate the velocity vector to find the position vector.
Step 1: Find the Velocity Vector
Integrate the acceleration vector for time to get the velocity vector. The indefinite integral of the acceleration vector gives:
Vx = ½8t² + C1Vy = -cos t + C2Vz = ½sin 2t + C3Using the initial velocity v(0) = i, we find C1 = 1, C2 = 0, and C3 = 0. Therefore, the velocity vector is v(t) = (4t² + 1)i - cos t j + ½sin 2t k.
Step 2: Find the Position Vector
Integrate the velocity vector concerning time to get the position vector. The indefinite integral of the velocity vector gives:
Rx = ⅔t³ + t + C4Ry = -sin t + C5Rz = -¼ cos 2t + C6Using the initial position r(0) = j, we find C4 = 0, C5 = 1, and C6 = 0. Thus, the position vector is r(t) = (⅔t³ + t)i - sin t j - ¼ cos 2t k.
Find the ratio of the increase in price to the original price.
$6.60 to $11.00 per case of 12 quarts
Find the sum of a finite geometric sequence from n = 1 to n = 8, using the expression −2(3)^n − 1.
In how many different ways can you make exactly 0.75 using only nickles dimes and quarters if you have at least one of each coin?
If 3t − 7 = 5t , then 6t =
The lengths of a particular type of metal rod are modeled by the equation r = |–24x + 120|, where r represents the lengths of the rod in inches, and x represents the speed at which the rods are produced, in rods per minute. If the manufacturing process is too fast or too slow, the rods will not be long enough. At what speeds will the lengths of the rods be greater than 48 inches?
A. The length of the rods will be greater than 48 inches when x > 3 or x < 7 rods per minute.
B. The length of the rods will be greater than 48 inches when x < 3 or x > 7 rods per minute.
C. The length of the rods will be greater than 48 inches when x > –3 or x < –7 rods per minute.
D. The length of the rods will be greater than 48 inches when 3 < x < 7 rods per minute
Christine is putting money into a savings account. She starts with $550 in the savings account, and each week she adds $60 . Let S represent the total amount of money in the savings account (in dollars), and let W represent the number of weeks Christine has been adding money. Write an equation relating S to W . Then use this equation to find the total amount of money in the savings account after 19 weeks.
Write an equation of the line perpendicular to the line 8x+15y=12 and containing the point (11,17) write the answer in standard form
The slope of a line perpendicular to another line has a slope which is the negative reciprocal of that line or:
m1 = -1/m2
First, we convert the given equation into slope-intercept form of a line: y = mx + b
8x + 15y = 12
15y = -8x + 12
y = (-8/15) x + 0.8
m1 = -8 / 15
Therefore the slope of the perpendicular line is:
m2 = 15 / 8
Since the perpendicular line crosses the point (11, 17), therefore using the slope formula:
m = (y2 – y1) / (x2 – x1)
15 / 8 = (y2 – 17) / (x2 – 11)
1.875 (x2 – 11) = y2 – 17
1.875 x2 – 20.625 = y2 – 17
y2 = 1.875 x2 – 3.625
y = (15/8) x – 3.625
multiplying both sides by 8:
8y = 15x – 29
rewriting in standard form:
15x – 8y = 29 (ANSWER)
For football tryouts at a local school, 12 coaches and 42 players will split into groups. Each group have the same number of coaches and players. What is the greatest number of groups that can be formed? How many coaches and players will be in each of these groups?
The map of a biking trail is drawn on a coordinate grid. The trail starts at P(−3, 2) and goes to Q(1, 2). It continues from Q to R(1, −1) and then to S(8, −1). What is the total length (in units) of the biking trail?
Answer:
Total length of the biking trail is 14 units
Step-by-step explanation:
The trail starts from point P(-3, 2) and touches points Q(1, 2), R(1, -1) and S(8, -1).
We have to calculate the total length of the biking trail.
Since distance between two points are represented by
d = [tex]\sqrt{(x-x')^{2}+(y-y')^{2}}[/tex]
So length of PQ = [tex]\sqrt{(1+3)^{2}+(2-2)^{2}}[/tex]
PQ = [tex]\sqrt{(4)^{2}}[/tex]
PQ = 4 units
QR = [tex]\sqrt{(1-1)^{2}+(2+1)^{2}}[/tex]
QR = [tex]\sqrt{(3)^{2}}[/tex]
QR = 3 units
RS = [tex]\sqrt{(8-1)^{2}+(-1+1)^{2}}[/tex]
RS = [tex]\sqrt{(7)^{2}}[/tex]
RS = 7 units
Now total biking trail = PQ + QR + RS
= 4 + 3 + 7
= 14 units
Total length of the biking trail is 14 units
Answer:
The answer is 14 units
Step-by-step explanation:
Susan enlarged to scale a rectangle with a height of 4 centimeters and length of 11 centimeters on her computer. The length of the new rectangle is 16.5 centimeters. Find the height of the new rectangle
4 * 1.5 = 6 cm <== Because this is the answer.
A marathon runner will donate $5000 to charity if her time is less than or equal to 3 hours. She will donate $2000 if her time is more than 3 hours but less than or equal to 4 hours. Finally, she will donate $1000 to charity if her time is more than 4 hours.
a.)Write a piecewise function describing this situation.
In your own words, explain the place value relationship when the same two digits are next to each number in a multi-digit number
The relationship of the two of the same digits that are next to each other in a multi-digit number lies in their place value, wherein the digits from the decimal point that is to the left, is 10 times of larger value compared to the digit in their left. And from the decimal point to the right, it is 10 times of small value that the digit in their right.
Let’s see an example:
123.45, wherein 1 is hundreds, two is tens, 3 is ones, and then after the
decimal points, is where 4 is the tenths and 5 is the hundredths.
Take note of each place value that is next to each other since each value
is 10 times in difference.
You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. what is the minimum number of cards you must pick in order to guarantee that you geta)two pair (for example, two kings or two 5s)b)three of a kind (for example, three 7s)
To guarantee two pairs, a player must pick at least 14 cards from a deck of 52 playing cards without replacement. For three of a kind, a minimum of 16 cards is required. This is an example of sampling without replacement, where each selection affects subsequent draws.
Explanation:To determine the minimum number of cards one must pick from a standard deck of 52 playing cards to guarantee getting two pairs, consider the worst-case scenario where you pick one card of each rank before getting any pair. Since there are 13 different ranks, picking 13 single cards wouldn't guarantee a pair, but the 14th card will definitely match one of the previously drawn ranks, thus forming a pair. To ensure two pairs, you could go through another 13 cards without getting a match to your first pair, so the 15th card would be the second pair. Therefore, you must pick at least 14 cards to guarantee two pairs.
For three of a kind, you pick sequentially from the different ranks. After picking one card of each of the 13 ranks, the 14th card will form a pair, and the 15th card could potentially be of a new rank. However, the 16th card drawn must either create a pair with another rank or a 'three of a kind' with the rank that already has two. Thus, the minimum number of cards one must pick to guarantee a three of a kind is 16.
In sampling without replacement, drawn cards are not returned to the deck, making each draw dependent on the previous ones. This contrasts with sampling with replacement, where each draw is independent since cards are returned to the deck and reshuffled after each pick.
Which of these would make the number sentence true below?
3.7427 < 3.74__
A. Three thousandths
B. Two thousandths
C. Three ten-thousandths
D. Seven ten-thousandths
Find the slant height of this square pyramid 6 inches on each side
Leah likes to stretch 5 minutes for every 10 minutes of dancing. How many minutes should she stretch if she is doing a 50 minute dance class?
Leah should stretch for 25 minutes during a 50-minute dance class, as she stretches for 5 minutes for every 10 minutes of dancing.
Leah stretches for 5 minutes for every 10 minutes of dancing. To calculate how much time she should be stretching during a 50-minute dance class, we need to apply a simple ratio. For every 10 minutes of dance, she stretches for 5 minutes, which is half the time spent dancing. We can set up the proportion as follows: 5 minutes of stretching / 10 minutes of dancing = X minutes of stretching / 50 minutes of dancing.
Now, solving for X gives us 5/10 = X/50, which simplifies to X = (5/10) × 50 = 25 minutes. Therefore, Leah should stretch for 25 minutes during her 50-minute dance class.
Find the maximum and minimum values of f(x,y) = 8x+y for the polygonal convex set having vertices at (0, 0), (4, 0), (3, 5), (0, 5).
The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. find the probability that fewer than 7 field mice are found
The probability that fewer than 7 field mice are found in the 5-acre field is 0.0458.
The probability that fewer than 7 field mice are found in the entire 5-acre field is not simply 58.10% multiplied by 5. This is because the mice are not independent on each other. The number of mice on one acre doesn't affect the number found on another.
To find the probability for the entire field, we need to consider all possible combinations of the number of mice on each acre. This can be a complex calculation involving multiple Poisson distributions.
However, if we assume that the distribution of mice across the field is approximately uniform, we can use a simpler approach. We can assume that the probability of finding fewer than 7 mice on each acre is independent of the others and multiply the individual probabilities:
P(total < 7) ≈ (0.5810)^5 ≈ 0.0458
Therefore, under the assumption of approximate uniformity, the probability of finding fewer than 7 field mice in the entire 5-acre field is approximately 0.0458%.
Complete question:
The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. find the probability that fewer than 7 field mice are found on a given acre that is 5.
the table below summarizes the number of children per household for a sample of 29 families
Help me figure this out please.
Addison has 15 fewer pieces of candy than Ronny does. Is this situation modeled by an expression or equation? How do you know?
Equation, because Addison's candy is equal to 15 less than Ronny's candy.
i hope this help you
Answer:
Equation, because Addison's candy is equal to 15 less than Ronny's candy.
Step-by-step explanation:
what are the intercepts of the graph (0,7)(9,0)(-9,0)
I got 43000 and I don't know what I did wrong
estimating wage to nearest dollar would be 22 x 40
would be 22 x 4 = 88 so 880 per week
estimating 50 weeks
88 x 5 = 440 so 44,000 per year
exact would be 21.50 x 40 = 860 per week
860 x 52 = 44,720 per year
Charles can type 675 words in 9 minutes. How many words can Charles types in 13 minutes?
Mr. Brown owned a house a house, which he rented for $60 a month. The house was assessed for $9000. In 1975 the rate of taxation was increased from $25 to $28 per $1000 assessed valuation. By what amount should the monthly rent have been raised to absorb the increase in that year's taxes?
Mr. Brown should increase the monthly rent by $2.25 to cover the additional annual property tax caused by the tax rate increase from $25 to $28 per $1000 assessed valuation on a $9000 house.
The student is asking for a calculation of how much the monthly rent of a house should be increased to cover the additional property tax imposed when the tax rate was increased from $25 to $28 per $1000 assessed valuation. The house is valued at $9000.
First, we calculate the initial annual tax by multiplying the assessed valuation by the initial tax rate:
Initial annual tax = $9000 / 1000 times $25 = $225
Then we calculate the new annual tax with the increased rate:
New annual tax = $9000 / 1000 times $28 = $252
The increase in the annual tax amount is:
Increased tax amount = New annual tax - Initial annual tax = $252 - $225 = $27
To find the monthly increase, we divide by 12:
Monthly rent increase = Increased tax amount / 12 = $27 / 12 = $2.25
Therefore, the monthly rent should be raised by $2.25 to absorb the increase in that year's taxes.
How can you use models find the volume of composite figures
Mabel is installing a square pool in her backyard and wants a circular fence to enclose the pool to create 4 grassy areas around the pool as show in the figure. If the pool is located at the coordinates (1, 5), (5, 8), (4, 1), and (8, 4), what is the amount of fencing Mabel needs to purchase? Round your answer to the nearest tenth. (Like 5.7 or 3.4).
Which of these ordered triples indicates where the plane cuts the x-axis for this equation? 7x +2y +3z =42 A. (14,0,0) B. (7,0,0) C. (21,0,0) or D. (6,0,0)
Answer:
Option D is correct.
Step-by-step explanation:
Given Equation of plane is 7x + 2y + 3z = 42
We need to find ordered triplet where plane cuts the x-axis.
To find point of x-axis when plane cuts it. we put other coordinates equal to 0.
So, put y = 0 and z = 0 in equation plance to get x-coordinate of the required ordered triplet.
7x + 2 × 0 + 3 × 0 = 42
7x + 0 + 0 = 42
7x = 42
[tex]x=\frac{42}{7}[/tex]
x = 6
⇒ ordered triplet = ( 6 , 0 , 0 )
Therefore, Option D is correct.
if the perimeter of a triangle measures 42 feet and the three sides are consecutive numbers , what are the side lengths
42/3 = 14
14-1=13
14+1= 15
13+14+15 = 42
3 numbers = 13, 14 & 15
I REALLY NEED HELP PLEASE!!
Mark is observing the velocity of a runner at different times. After one hour, the velocity of the runner is 6 km/h. After three hours, the velocity of the runner is 2 km/h. Part A: Write an equation in two variables in the standard form that can be used to describe the velocity of the cyclist at different times. Show your work and define the variables used. (5 points) Part B: How can you graph the equations obtained in Part A for the first 5 hours? (5 points)
I really need to finish this today please help!!