Answer:
(a) The particle is moving to the right in the interval [tex](0 \ , \ \displaystyle\frac{\pi}{2}) \ \cup \ (\displaystyle\frac{3\pi}{2} \ , \ 2\pi)[/tex] , to the left in the interval [tex](\displaystyle\frac{\pi}{2}\ , \ \displaystyle\frac{3\pi}{2})[/tex], and stops when t = 0, [tex]\displaystyle\frac{\pi}{2}[/tex], [tex]\displaystyle\frac{3\pi}{2}[/tex] and [tex]2\pi[/tex].
(b) The equation of the particle's displacement is [tex]\mathrm{s(t)} \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(t)} \ + \ 3[/tex]; Final position of the particle [tex]\mathrm{s(2\pi)} \ = \ 3[/tex].
(c) The total distance traveled by the particle is 9.67 (2 d.p.)
Step-by-step explanation:
(a) The particle is moving towards the right direction when v(t) > 0 and to the left direction when v(t) < 0. It stops when v(t) = 0 (no velocity).
Situation 1: When the particle stops.
[tex]\-\hspace{1.7cm} v(t) \ = \ 0 \\ \\ 5 \ \mathrm{sin^{2}(t)} \ \mathrm{cos(t)} \ = \ 0 \\ \\ \-\hspace{0.3cm} \mathrm{sin^{2}(t) \ cos(t)} \ = \ 0 \\ \\ \mathrm{sin^{2}(t)} \ = \ 0 \ \ \ \mathrm{or} \ \ \ \mathrm{cos(t)} \ = \ 0 \\ \\ \-\hspace{0.85cm} t \ = \ 0, \ \displaystyle\frac{\pi}{2}, \ \displaystyle\frac{3\pi}{2} \ \ \mathrm{and} \ \ 2\pi[/tex].
Situation 2: When the particle moves to the right.
[tex]\-\hspace{1.67cm} v(t) \ > \ 0 \\ \\ 5 \ \mathrm{sin^2(t) \ cos(t)} \ > \ 0[/tex]
Since the term [tex]5 \ \mathrm{sin^{2}(t)}[/tex] is always positive for all value of t of the interval [tex]0 \ \leq \mathrm{t} \leq \ 2\pi[/tex], hence the determining factor is cos(t). Then, the question becomes of when is cos(t) positive? The term cos(t) is positive in the first and third quadrant or when [tex]\mathrm{t} \ \epsilon \ (0, \ \displaystyle\frac{\pi}{2}) \ \cup \ (\displaystyle\frac{3\pi}{2}, \ 2\pi)[/tex] .
*Note that parentheses are used to demonstrate the interval of t in which cos(t) is strictly positive, implying that the endpoints of the interval are non-inclusive for the set of values for t.
Situation 3: When the particle moves to the left.
[tex]\-\hspace{1.67cm} v(t) \ < \ 0 \\ \\ 5 \ \mathrm{sin^2(t) \ cos(t)} \ < \ 0[/tex]
Similarly, the term [tex]5 \ \mathrm{sin^{2}(t)}[/tex] is always positive for all value of t of the interval [tex]0 \ \leq \mathrm{t} \leq \ 2\pi[/tex], hence the determining factor is cos(t). Then, the question becomes of when is cos(t) positive? The term cos(t) is negative in the second and third quadrant or [tex]\mathrm{t} \ \epsilon \ (\displaystyle\frac{\pi}{2}, \ \displaystyle\frac{3\pi}{2})[/tex].
(b) The equation of the particle's displacement can be evaluated by integrating the equation of the particle's velocity.
[tex]s(t) \ = \ \displaystyle\int\ {5 \ \mathrm{sin^{2}(t) \ cos(t)}} \, dx \ \\ \\ \-\hspace{0.69cm} = \ 5 \ \displaystyle\int\ \mathrm{sin^{2}(t) \ cos(t)} \, dx[/tex]
To integrate the expression [tex]\mathrm{sin^{2}(t) \ cos(t)}[/tex], u-substitution is performed where
[tex]u \ = \ \mathrm{sin(t)} \ , \ \ du \ = \ \mathrm{cos(t)} \, dx[/tex].
[tex]s(t) \ = \ 5 \ \displaystyle\int\ \mathrm{sin^{2}(t) \ cos(t)} \, dx \\ \\ \-\hspace{0.7cm} = \ 5 \ \displaystyle\int\ \ \mathrm{sin^{2}(t)} \, du \\ \\ \-\hspace{0.7cm} = \ 5 \ \displaystyle\int\ \ u^{2} \, du \\ \\ \-\hspace{0.7cm} = \ \displaystyle\frac{5u^{3}}{3} \ + \ C \\ \\ \-\hspace{0.7cm} = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(t)} \ + \ C \\ \\ s(0) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(0)} \ + \ C \\ \\ \-\hspace{0.48cm} 3 \ = \ 0 \ + \ C \\ \\ \-\hspace{0.4cm} C \ = \ 3.[/tex]
Therefore, [tex]s(t) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(t)} \ + \ 3[/tex].
The final position of the particle is [tex]s(2\pi) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(2\pi)} \ + \ 3 \ = \ 3[/tex].
(c)
[tex]s(\displaystyle\frac{\pi}{2}) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(\frac{\pi}{2})} \ + \ 3 \\ \\ \-\hspace{0.85cm} \ = \ \displaystyle\frac{14}{3} \qquad (\mathrm{The \ distance \ traveled \ initially \ when \ moving \ to \ the \ right})[/tex]
[tex]|s(\displaystyle\frac{3\pi}{2}) - s(\displatstyle\frac{\pi}{2})| \ = \ |\displaystyle\frac{5}{3} \ (\mathrm{sin^{3}(\frac{3\pi}{2})} \ - \ \mathrm{sin^{3}(\displaystyle\frac{\pi}{2})})| \ \\ \\ \-\hspace{2.28cm} \ = \ \displaystyle\frac{5}{3} | (-1) \ - \ 1| \\ \\ \-\hspace{2.42cm} = \displaystyle\frac{10}{3} \\ \\ (\mathrm{The \ distance \ traveled \ when \ moving \ to \ the \ left})[/tex]
[tex]|s(2\pi) - s(\displaystyle\frac{3\pi}{2})| \ = \ |\displaystyle\frac{5}{3} \ (\mathrm{sin^{3}(2\pi})} \ - \ \mathrm{sin^{3}(\displaystyle\frac{3\pi}{2})})| \ \\ \\ \-\hspace{2.28cm} \ = \ \displaystyle\frac{5}{3} | 0 \ - \ 1| \\ \\ \-\hspace{2.42cm} = \displaystyle\frac{5}{3} \\ \\ (\mathrm{The \ distance \ traveled \ finally \ when \ moving \ to \ the \ right})[/tex].
The total distance traveled by the particle in the given time interval is[tex]\displaystyle\frac{14}{3} \ + \ \displaystyle\frac{5}{3} \ + \ \displaystyle\frac{10}{3} \ = \ \displaystyle\frac{29}{3}[/tex].
Help help math I’ll give points thanks
Answer:
x=13
Step-by-step explanation:
8x-20=4x+32
start by moving the terms,
8x-4x=32+20
combine
4x=52
divide each side by 4
x=13
If A is the center of the circle which of these is the radii
Answer:
A-b because radius is the line segments from its center to its perimeter
Step-by-step explanation:
Solve using substitution.
X = 1
6x + y = 1
please help
Answer: (1,-5)
In other words, x = 1 and y = -5
=====================================================
Work Shown:
6x + y = 1
6(1) + y = 1
6 + y = 1
y = 1-6
y = -5
The solution as an ordered pair is (1,-5)
Notice I started with the second equation and substituted in the first equation. The idea is to somehow reduce the two variables into one variable, so you can solve for that remaining variable.
Answer:
6x+y=1
or,6×1+y=1
or,6+y=1
or,y=6-1
ans,y=5
For the trig function f(0)=sin(0) whats the input variable
For the trig function g(0)=cos(0) whats the input variable
For the function f(0)=sin(0), 0 is the input variable while f(0) is the output.
For the function g(0)=cos(0), 0 is the input variable while g(0) is the output.
The standard form of representing trigonometry functions is f(x) = sinx or f(x) = cos x
From the functions:
x is the input variable (also known as the domain)y is the output variable also known as the range;Given the trig function f(0)=sin(0), compared with the general equation, we can see that 0 is the input variable while f(0) is the output.
Similarly for the function g(0)=cos(0), 0 is the input variable while g(0) is the output.
Learn more on input variables here: https://brainly.com/question/11451813
slope y-intercept form of an equation for a line that has an x-intercept of "-3" and goes through (-2,2)
Answer:
y=2x+6.
Step-by-step explanation:
1) the condition 'an x-intercept of "-3"' means the point A with the coordinates (-3;0), then it is possible to make up the required equation according to the rule (the coordinates (-2;2) belong to the point B):
[tex]2) \ \frac{x-X_B}{X_A-X_B}=\frac{y-Y_B}{Y_A-Y_B}, \ where \ X_A=-2;Y_A=2;X_B=-3;Y_B=0.[/tex]
after the substitution the coordinates into the rule:
[tex]\frac{x+3}{-2+3} =\frac{y-0}{2-0}; \ => \ \frac{x+3}{1}=\frac{y}{2}; \ => \ y=2x+6.[/tex]
Is (x + 2) a factor of f(x) = x3 - 7x2 + 2X – 8? Use either the remainder theorem or the factor theorem to explain your reasoning.
[tex]\text{Given that,}\\\\f(x) = x^3 -7x^2 +2x -8\\\\\\\text{(x+2) will be a factor of f(x) if } f(-2) = 0 \\\\\\f(-2) = (-2)^3 -7(-2)^2 +2(-2) -8 = -48 \\\\\\\text{Since}~ f(-2) \neq 0, ~ (x +2) \text{is not a factor of f(x).}[/tex]
Carson bought 50 pounds of rice for his restaurant. He cooked the rice and wanted to evenly divide it into 9 containers to be served later. How many pounds of rice should Carson put in each container?
help plss
Answer:
Step-by-step explanation: there you go
Can someone help me plssssssssss
tim borrowed $7,000 at 8% interest rate for 5 years what was the total interest
Write an equation of the line with a slope of -2 and y-intercept of 3
Answer:
y = -2x + 3
Step-by-step explanation:
the original formula is
y = mx + b
m is the slope
b is the inteception on the y axis
What is the relationship between vertices and edges in a pyramid? In a prism?
Answer: depends on the shape of it but mostly equal
Step-by-step explanation: hope it helps
1. Find the volume of the rectangular prism that has a height of 5 meters and the area of the base is 8m2.
Answer:
40m3
Step-by-step explanation:
Multiply the area x the height
Step-by-step explanation:
the volume of a prism or simply "box" is base area × height.
so, we have
8 × 5 = 40 m³
Find the least common denominator for these fractions. Enter your answer in
the space provided.
Answer:
5/12
Step-by-step explanation:
1/4 and 1/6 are fractions.
As we directly can't add fractions whenever denominators are different, so we will have to find out the Least common multiple of 4 and 6.
The LCM Of 4&6 will be 12, so we will write 12 instead of 4&6.
Now, 4*3=12, that is why we will have to multiply the numerator and denominator with 3.
1*3/4*3 = 3/12
Same thing applies to 1/6. In order to get 12 in the denominator we need to multiply 2 with numerator and denominator both.
1*2/6*2= 2/12
However, we have two new fractions with same denominator now, that is 3/12 and 2/12
We can directly add the numerators now, as we have the same denominator. So 3+2/12= 5/12 would be my final answer. (addition of like fractions)
5/12
can y’all please help me
Answer:
15. d(1,3) e(4,4) g(1,-1) f(4,-2)
16. e(2,-1) f(4,-1) g(2,-3) h(4,-5)
Step-by-step explanation:
These are the coordinates to each point (x,y)
I hope this helps!!
When h = π, the orange graph shifts
units to the
.
Answer: it shifts units to the right.when h=-π it shifts units to the left
Step-by-step explanation: positive pi goes to the right, negative to the left
NO LINKS!!!
Purchasing a TV that is 30% off will save $250.
a. What is the part, percent, and whole.
b. What is the original price of the TV?
Answer:
a. part: $250; percent: 30%; whole: $833.33
b. $833.33
Step-by-step explanation:
a. The part is given as $250; the percentage that represents is given as 30%.
The whole is the ratio of the part to the percent:
$250/30% = $250/0.30 = $833.33 . . . whole
__
b. The original price is the "whole" of which 30% is the part.
original price: $833.33
If Sandy memorizes 9 verses every 2 days, how many verses will Sandy memorize in 4,6, and 8 days. Write equivalent ratios to show the answer.
Answer:
4 Days = 18
6 Days = 27
8 Days = 36
4:18
6:27
8:36
Step-by-step explanation:
Answer:
4 days: 18 verses
6 days: 27 verses
8 days: 36 verses
Step-by-step explanation:
If Sandy memorized 9 verses every 2 days then to find how many she can memorize in 4, 6 and 8 days you multiply.
2 days: 9 verses (Multiply 2 times 2 to get how many she can memorize in 4 days. Multiply 2 times 3 to get 6 days. Multiply 2 times 4 to get 8 days. Then multiply the number of verses with the same number you multiplied the number of days.)
4 days: 18 verses
6 days: 27 verses
8 days: 36 verses
Does favorite food depend on your gender? A survey asks a group of college students about their favorite foods. It also groups the students by their gender. The data are recorded in the contingency table below, and a chi-square Test of Independence at the 5% significance level is performed.
Sushi Pizza Hamburgers Row Total
Men 19 16 21 56
Women 19 11 13 43
Column Total 38 27 34 99
The null and alternative hypotheses are:
H0: The two variables are independent, so gender does not affect favorite food.
Ha: The two variables are dependent, so gender does affect favorite food.
Compute the test statistic, rounded to one decimal place.
Answer:
Yes, it does
Step-by-step explanation:
More Men like the Pizza and Hambugers more than the girls. Both like shushi Equaly but 2 beats 1 am a right.
What is the least multiple 6 and 8 have in commen explain
the first several multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 the first several multiples of 6 are 6: 12, 18, 24, 30, 36, 42, 48, 54, 60
Please help me with this.
Answer:
a) 7 b) -5
Step-by-step explanation:
m=Y-y
X-x
a) (70-35)/(10-5)= 7
m= 7
b) (50--20)/(-10-4)= 70/-14 = -5
What is the range of y = sin x?
A. In
72 T
o
B. -1Sys1
c. -1SIS 1
o
D. All real numbers
Answer:
See below
Step-by-step explanation:
The function [tex]f(x)=sin(x)[/tex] oscillates between -1 and 1, so the range is [tex]-1\leq x\leq 1[/tex].
Solve for x please help me
Answer:
30
Step-by-step explanation:
all angles add up to 180*, 180-90=90, 90=3x, 30=x
Answer:
x = 30
Step-by-step explanation:
One side is 90 so the other two sides have to add up to 90 too because a triangles' angles have to add up to 180.
So, we can write that 2x + x = 90
Then we can solve for x:
2x + x = 90
3x = 90
x = 30
Can someone help me with these 2 geometry questions?? Will mark brainiest :)
Answer:
Thats to hard Do it your self
Jack wanted to go sledding. When he looked at the thermometer at 10:00 am the temperature was -10°F. He called his friends and they agreed to meet at their favorite sledding hill at 3 pm. Right before he left to meet his friends, he checked the thermometer again. The temperature was now 15°F. How many degrees did the temperature change from 10 am to 3 pm?
Answer:
25 degrees
Step-by-step explanation:
If its 15 when hes about to leave and -10 when he first checked it shou. be 25
Answer:
answer is 25
Step-by-step explanation:
...........
What is 40÷1/2 please help me
Answer: Pretty sure it's 80
Step-by-step explanation:
How many lb. of burgers did Mrs. Butterfield make?
Answer:
4 burgers
Step-by-step explanation:
four burgers
If you're making the patties yourself (to save money or because you love making burgers), assume four burgers per pound of meat
Shawn wants to save up for a birthday outfit. He figures he can save $25 each week from the money earns cutting hair. If he plans to spend less than $175 for the outfit, how many weeks will it take him save enough money to buy any outfit in his price range?
Answer:
7 weeks
Step-by-step explanation:
175 divided by 25 is 7
There are two types of packages that are available for hotel stay
- you can use the hotel unlimited for $500 a month
- Or you can spend $50 a day per visit.
Enter an equation that can be used to find the number of visits per month needed for the two packages to cost the same amount
Answer:
for one month 500$
for daily visit 50$
500$=50$*x days
x days=500$/50$
x days = 10 days
10 days of daily visits costs 500$ equal to one month
jjjjjhfvkicdcghgfc ghcvh,glv,kghc ,h..
Answer:
0
Step-by-step explanation:
Given that
P
=
x
−
y
.
Find
P
when:
x
=
15
and
y
=
20
Answer:
the answer would be -5
Step-by-step explanation:
p=15-20 p=-5