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].
[(1 1/5−2/5)⋅(−3/4)3]÷(−9)
Given that:
[{1(1/5)−(2/5)⋅(−3/4)3}]÷(−9)
= [{(5+1)/5 - (2/5).(-3/4)3}]÷(-9)
= [{(6/5) - (2/5).(-3/4)3}]÷(-9)
= [{(6/5) - (2/5).(-3/4)3}]÷(-9)
= [{(6/5) - (2/5).(-3*3)/4}]÷(-9)
= [{(6/5) - (2/5) . (-9/4)}]÷(-9)
= [{(6/5) - (2/5) × (-9/4)}]÷(-9)
= [{{6/5) - {{2*(-9)}/{5*4}]}]÷(-9)
= [{(6/5) - {-9/(5*2)}}]÷(-9)
= [{(6/5) - (-9/10)}]÷(-9)
Take the LCM of 5 & 10 is 10.
= [{(6*2 + 9*1)/10}]÷(-9)
= [{(12+9)/10}]÷(-9)
= [{-(21/10)}]÷(-9)
= -(21/10) ÷ (-9)/1
= -(21/10) × (1/9)
= {(-21*1)/(10*9)]
= {-21/90)
= {(-21÷3)/(90÷3)}
= -9/30
= {(-9÷3)/(30÷3)}
= (-3/10) = -3/10 Ans.
what is the y intercept of the graph
what is the slope
Answer:
y-intercept = 1
slope = 4
Step-by-step explanation:
The y-intercept is where the line intersects with the y.
And the slope is rise over run, so it rises 4 and walks over 1, therefore the slope is [tex]\frac{4}{1}[/tex] which simplifies to 4.
Hope this helped! :)
Answer:
y intercept (0,1)
slope 4
Step-by-step explanation:
The y intercept is where the graph crosses the y axis
The graph crosses at y =1
(0,1)
We can find the slope by taking rise over run
The graph rises 4 units ( goes up 4 units)
Then goes to the right 1 units
The slope is 4/1 = 4
3
Johnny has been hired to draw a mural on the window of the pet store. The scale drawing he is using is shown below.
4.5 cm
6 cm
If 1 centimeter represents 10 inches on the mural, what is the actual height and width of the hamster that he will be drawing on the
mural?
OA.
66 in by 49.5 in
ОВ.
90 in by 120 in
Od
6 in by 4.5 in
OD. 60 in by 45 in
Answer:
OD 60 in by 45 in
Step-by-step explanation:
Simplify the expression
-4 + 3 + 5 =
Answer:
4
Step-by-step explanation:
-4+3=-1
-1+5=4
⦁ Avery noticed that 9/10 of the apples in a basket are red. There are 45 red apples in the basket. ⦁ Represent the situation with an equation. Use a for the variable. ⦁ Solve the equation in Part (a). Show your work. ⦁ What does the solution of the equation represent in the situation? ⦁
We want to write and solve an equation that represents the given situation. The equation we will find is (9/10)*a = 45, and its solution is a = 50.
So we know that 9/10 of the apples in the basket are red, and we know that there are 45 red apples.
Then, if we define a as the total number of apples, we know that (9/10) times a must be equal to 45, we can write this as:
(9/10)*a = 45
This is the equation that we must solve.
We can solve this if we multiply both sides by (10/9), it gives;
(10/9)*(9/10)*a = (10/9)*45
a = (10/9)*45 = 50
This means that there are 50 apples in the basket.
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find three solutions for 9x - 2y = 18
Answer:
[tex](x,y) = \left(5,\dfrac{27}2 \right), \left(0,-9), \left(9, -\dfrac 92 \right)[/tex]
Step-by-step explanation:
2m>30
what is the answer to this
Answer:
m>15
Step-by-step explanation:
Divide both sides by 2
Hello there! :-)
This is a one-step inequality, which means we can solve it in one step.
The goal is to find m.
To isolate m, we divide both sides by 2:
m>15
Remember to divide both sides by 2.
Therefore, m>15 (this means m is greater than 15)
Hope it helps. Use the comment section to clarify your doubts.
Answered by
~A felicitous teen who helps others :-)
Good luck with your studies. :)
Q.
A circle has a radius of 21 cm. Then determine the area!
EASY
Hello there! :-)
The formula for finding the area of a circle is
[tex]A=\pi r^{2}[/tex]Notice that only the radius is squared.
We are given the radius of the circle: 21 cm.
We plug the value of r into the formula:
A=π(21)²A=π(441)A=1,385 cmHope this helps. Use the comment section to clarify your doubts.
Answered by
~A teen whose hobby is helping people with their homework
#LetItSnow
Good luck.
Answer:
A=1.386 cm²
Step-by-step explanation:
[tex]\sf\bf A = \pi \times r \times r[/tex]
[tex]\sf\bf = \frac{22}{7} \times 21 \times 21[/tex]
[tex]\sf\bf = \frac{22}{\cancel{7}} \times \cancel{21} \times 21[/tex]
[tex]\sf\bf = 22 \times 3 \times 21[/tex]
[tex]\sf\bf = 66 \times 21[/tex]
[tex]\sf\bf = 1.386 \: {cm}^{2}[/tex]
The table below shows the deposits (+) and withdrawals ( - ) each week for a newly opened bank account with a starting balance of $100.
DEPOSITS AND
WITHDRAWALS
Week 1+120
Week 2 -100
Week 3 +50
Week 4 P
Week 5 +200
Week 6 -100
Week 7 -25
Week 8 +50
Part A: What is the value of p if the account shows a zero balance after 4 weeks?
From the given table, the positive sign connote deposit while the negative sign connotes withdrawal.
If the account shows a zero balance after 4 weeks, we will take the sum of all deposits and withdrawals for the first four weeks and equate to zero as shown:
[tex]+120+(-100)+50+p=0[/tex]
Open the parenthesis:
[tex]+120-100+50+p=0\\20+50+p=0\\70+p=0[/tex]
Subtract 70 from both sides
[tex]70+p-70=0-70\\p=-70[/tex]
Hence the value of p if the account shows a zero balance after 4 weeks is -70
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mark has a board that is 12 feet long he cuts the board into 8 peices that are the same lenght. how long is each peice
Answer: So, all you have to do is divide in this situation. He starts with a board that is 12 feet long, keep that in mind. If he cuts that into 8 pieces it's like he is dividing. So 12 divided by eight or 12/8 would be 1.5. Your answer is that each piece would be 1.5ft long.
the perimeter of a regular Pentagon is 100m how long is each side
Answer:
20 cm is the answer.
20
Step-by-step explanation:
the parameter of any shape is the sum of all sides and since the pentagon has 5 sides we just have to divide the parameter by 5
100÷5=20
3.-Las 80 habitaciones de un motel se podrían alquilar todas las noches si el director cobrará
40 € o menos por habitación. Si se cobra (40+x) € por habitación, entonces 2x habitaciones
permanecerán vacantes. Si cada habitación alquilada le cuesta al director 10 € al día y cada
habitación no alquilada 2 € al día, ¿qué precio debería cobrar el director por habitación para
maximizar su beneficio diario?
4.- Un fabricante de automóviles vende 2000 coches al mes, con un beneficio medio de 1000 €
por coche. Las prospectivas de mercado indican que por cada 50 € de descuento que el
fabricante ofrezca a los compradores podría vender 200 coches más al mes.
¿Cuánto descuento debería ofrecer para maximizar el beneficio mensual?
3) Para maximizar su beneficio diario el director debería cobrar cada habitación a €56.
4) El fabricante debería ofrecer un descuento de 150€ para maximizar su beneficio mensual.
3- Dado que las 80 habitaciones de un motel se podrían alquilar todas las noches si el director cobrara 40€ o menos por habitación, y si se cobra (40+x) € por habitación, entonces 2x habitaciones permanecerán vacantes, para determinar, si cada habitación alquilada le cuesta al director 10 € al día y cada habitación no alquilada 2 € al día, qué precio debería cobrar el director por habitación para maximizar su beneficio diario, se deben realizar los siguientes cálculos:
80 x 40 - 80 x 2 = 304070 x 50 - 10 x 10 - 70 x 2 = 326060 x 60 - 20 x 10 - 60 x 2 = 328065 x 55 - 15 x 10 - 65 x 2 = 329566 x 54 - 14 x 10 - 66 x 2 = 329264 x 56 - 16 x 10 - 64 x 2 = 329663 x 57 - 17 x 10 - 63 x 2 = 3295Por lo tanto, para maximizar su beneficio diario el director debería cobrar cada habitación a €56.
3) Dado que un fabricante de automóviles vende 2000 coches al mes, con un beneficio medio de 1000 € por coche, y las prospectivas de mercado indican que por cada 50 € de descuento que el fabricante ofrezca a los compradores podría vender 200 coches más al mes, para determinar cuánto descuento debería ofrecer para maximizar el beneficio mensual se debe realizar el siguiente cálculo:
1000 x 2000 = 2.000.000900 x 2400 = 2.160.000800 x 2800 = 2.240.000700 x 3200 = 2.240.000600 x 3600 = 2.160.000850 x 3000 = 2.550.000Por lo tanto, el fabricante debería ofrecer un descuento de 150€ para maximizar su beneficio mensual.
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click on the measure of the triangle.
Answer:
a
Step-by-step explanation:
because a
A right triangle has legs of lengths 7.5 ft and 8.0 ft. Find the length of the hypotenuse.
Answer:
multiply
Step-by-step explanation:
Please help!! I have less than an hour to turn this in!
Answer:
8 × 10¹
Step-by-step explanation:
don't panic. just breathe deeply and then think logically.
0.002 ... what is that ? 2 10ths ? 2 hundredths ? 2 thousandths ? 2 tenthousandths ?
the 3rd position after the decimal point is for 1/1000.
and that means 10^-3. in the same way as 1000 is 10³.
so, it is 2 × 10^-3.
the other rounded number is then 4 × 10⁴.
now, we are multiplying both numbers
2×10^-3 × 4×10⁴
what do we do ?
we combine the same types of factors, like the basic constants : 2×4 = 8
and 10^-3 × 10⁴.
what hairball when we multiply the same base number with exponents ? we add the exponents :
10^-3 × 10⁴ = 10¹ = 10
because -3 + 4 = 1
and so, the result is
8 × 10¹
2x-y=3
X+y=3
Slove equations algebraially
According to his running log, Baldwin averaged 4 miles per week last month and 25% more this month. How much did he average this month?
Answer:
4(1.25) = if it’s only one month then it’s 5 miles
Step-by-step explanation:
Please mark brainliest
jorden claims that x^2+3=12 is exponential equation. is jorden correct
Explain how the ratios 5 tacos for every 2 guest and 2 tacos for every 5 guests are different. Include a model in your explanation.
Answer:
5 tacos for 2 guests would mean more for each person, while 2 tacos for 5 guests would mean less for each person
Mrs Ang bought k apples at 50% each from a supermarket. How much
change did she receive if she paid $10 to the cashier? Express your answer
in cents.
Answer:
The apples may have costed 0.50 cents each, and she may have bought 10, which would result in her paying 10 dollars, and receiving 5 dollars in change.
Step-by-step explanation:
Multiply 10 (the amount she gave to the cashier) by 0.50%, it equals 5, so that's how much she paid, and since she gave a 10 dollar bill, she should get 5 dollars in change.
T^T
How do you solve this??
Answer:
look it up in internet and find th right answers
Answer:
g(x)=3|x| is the first box
f(x)=1/3|x| is the second box
Step-by-step explanation:
you can solve these type of questions by looking at the slope which is in front of the x.
since 1/3 is smaller than 3 it's going to have lower slope resulting in 3 to incline faster
Camden has a loyalty card good for a 17% discount at his local hardware store. What number should he multiply the prices on the tags by to find the price he would have to pay, before tax, in one step?
Answer:
0.83Step-by-step explanation:
Let the price is x.
With 17% discount it becomes:
x - 17% of x =x - 0.17x = 0.83xThe number we need is 0.83
Solution,
Assume That The Price is x
When Will Put It in Equation,
[tex] \frak{ \bigstar}{ \rm{ {We \: Will \: Get,}}}[/tex]
[tex]{ \longmapsto{ \sf{x-17 \% \: of x =}}}[/tex][tex]{ \longmapsto { \sf{x-0.17x}}}[/tex]0.83Hence, Required Price is “0.83”
Which statement is generally true about retirement?
A. Your income will increase, while your expense will stay the same.
B. Your income will decrease, while your expenses will increase.
C. Your income will increase, while your expenses will decrease.
D. Your income will decrease, while your expenses will stay the same.
The general statement that is true about retirement is B. your income will decrease, while your expenses will increase.
Retirement is the leaving of one's of occupation. This means you cease to be involved in active service in a job or occupation especially when one is considered old and of retirement age.
During retirement, retirement benefits and pension are usually made available to retirees. However, one's income will come to a decline which makes most people scared of retirement. Keeping up with the standard of living they are used to becomes difficult as income to sustain such may not be available.
Therefore, the general statement that is true about retirement is B. your income will decrease, while your expenses will increase.
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Answer:
It's B
Step-by-step explanation:
Just took it
Sierra left $4.50 as a tip for a waiter. This was 18% of the bill before the tip. How much was her bill before adding the tip?
Answer:
25
Step-by-step explanation:
what is 4.5 18% of
100 / 18 = 5.56
4.5 x 5.55... = 25
Expression 26-(-17)can be rewritten as
Hey there!
The expression 26-(-17) can be rewritten as
26+17
Remember:
a-(-b)=a+bHope it helps. Use the comment section to clarify your doubts.
Answered by
~A felicitous teen who is preparing for the New Year's Day
Good luck. :-)
I want a way to solve such a problem
Answer:
11.2
Step-by-step explanation:
The average price of an item is $42 with a standard deviation of $3
Find a 98% confidence interval about the mean (one decimal place).
Is a value of X=48 unusually high or is it normal? Choose correct explanation!
Step-by-step explanation:
standard deviation of $3
Find a 98% confidence interval about the mean (one decimal place).
Is a value of X=48 unusually high or is it normal? Choose correct explanation
Find the missing number of each unit rate.
20/4=?/1
10/15=?/1
Answer:
Step-by-step explanation:
divide 20 by four
20/4=5/1
now 10 and 15 can both be divided by 5 since 5 is the greatest common factor
10/15=2/3
Answer:
5 for the first 2/3 for the second
Step-by-step explanation:
just divide the numerator by what ever makes the denominator 1
4/4=1 so divide 20 by 4 which is 5
15/15 = 1 so 10/15 is 2/3
Factor. 9(p-1)+(p-1)^2
Answer:
9(−1)+(−1)(−1)Step-by-step explanation:
[tex]9(p-1)+(p-1)^2[/tex]
[tex]p^2+7p-8[/tex]
[tex]=(p-1)(p+8)[/tex]
Answer:[tex](p-1)(p+8)[/tex]-8=-9 + √2x - 1
find a h and k
Answer:
x=2?
Step-by-step explanation: