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goblinko [34]
3 years ago
9

The velocity of a particle moving along the x-axis at any time t ≥ 0 is given by

D%20cos%20%28%5Cpi%20t%29-%20t%287-2%5Cpi%20%29" id="TexFormula1" title="v(t) = cos (\pi t)- t(7-2\pi )" alt="v(t) = cos (\pi t)- t(7-2\pi )" align="absmiddle" class="latex-formula">
A. Find the particles maximum acceleration over the interval [0,2]. Use derivatives to support and justify your answer.

B. Find the particles minimum velocity over the interval [0,3] Justify your answer.

I know that acceleration is the derivative of velocity and that:
v'(t)=a(t)=-\pi sin(\pi t)-7+2\pi \\v''(t)=a'(t)=-\pi ^2cos(\pi t)
But I am confused as to actually figure out the minimum and maximum values using cos and sin
Mathematics
1 answer:
8090 [49]3 years ago
5 0

Answer:

A. a(3/2) = 3π − 7

B. v(3) = 6π − 22

Step-by-step explanation:

As you found:

v(t) = cos(πt) − t (7 − 2π)

v'(t) = a(t) = -π sin(πt) − 7 + 2π

v"(t) = a'(t) = -π² cos(πt)

a) When the acceleration is a maximum, a'(t) = 0.

0 = -π² cos(πt)

0 = cos(πt)

πt = π/2 + 2kπ, 3π/2 + 2kπ

πt = π/2 + kπ

t = 1/2 + k

t = 1/2, 3/2

We need to check if these are minimums or maximums.  To do that, we evaluate the sign of a'(t) within the intervals before and after each value.

a'(0) = -π² cos(0) = -π²

a'(1) = -π² cos(π) = π²

a'(2) = -π² cos(2π) = -π²

At t = 1/2, a'(t) changes signs from - to +.  At t = 3/2, a'(t) changes signs from + to -.  Therefore, t = 1/2 is a local minimum and t = 3/2 is a local maximum.  At t = 3/2, the acceleration is:

a(3/2) = -π sin(3π/2) − 7 + 2π

a(3/2) = 3π − 7

Compare to the endpoints:

a(0) = -π sin(0) − 7 + 2π = -7 + 2π

a(2) = -π sin(2π) − 7 + 2π = -7 + 2π

So a(3/2) = 3π − 7 is the global maximum.

b) Use the same steps as before.  When the velocity is a minimum, a(t) = 0.

0 = -π sin(πt) − 7 + 2π

π sin(πt) = -7 + 2π

sin(πt) = -7/π + 2

πt ≈ 3.372 + 2kπ, 6.053 + 2kπ

t ≈ 1.073 + 2k, 1.927 + 2k

t ≈ 1.073, 1.927

Now we evaluate the sign of a(t) in the intervals before and after each value.

a(0) = -π sin(0) − 7 + 2π = -7 + 2π

a(3/2) = -π sin(3π/2) − 7 + 2π = -7 + 3π

a(2) = -π sin(2π) − 7 + 2π = -7 + 2π

At t = 1.073, a(t) changes signs from - to +.  At t = 1.927, a(t) changes signs from + to -.  Therefore. t = 1.073 is a local minimum and t = 1.927 is a local maximum.  At t = 1.073, the velocity is:

v(1.073) = cos(1.073π) − 1.073 (7 − 2π)

v(1.073) = -1.743

Compare to the endpoints:

v(0) = cos(0) − 0 (7 − 2π) = 1

v(3) = cos(3π) − 3 (7 − 2π) = -22 + 6π

Here, v(3) = -22 + 6π is the global minimum.

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Math homework help please thanks!
trapecia [35]

Answer:

x = 5

Step-by-step explanation:

I already answer this on another question you actually made, but here it is again:

To do this, you need to use the Thales theorem.

You can actually apply this theorem and choose two sides of the figure to do it.In this case, I will choose segment LM and MN and look for a ratio with DE and EF like this:

LM/MN = DE/EF

This theorem states that you can stablish a ratio between two segments assuming that you have at least two paralels lines, like DL || EM || FN

Therefore, it's time to replace the values of those segments and solve for X:

LM/MN = DE/EF

14/16 = 6x -9 / 4x + 4

7/8 = 6x - 9 / 4x + 4

7(4x+4) = 8(6x-9)

28x + 28 = 48x - 72

72 + 28 = 48x - 28x

100 = 20x

x = 100/20

x = 5

Read more on Brainly.com - brainly.com/question/13517023#readmore

4 0
3 years ago
Sarah sent half of the Christmas cards to her friends,and Richard sent three eights of them to his friends. If there are 32 card
Paraphin [41]

Answer:

Sarah family will get 2 cards

Step-by-step explanation:

The total number of cards is 32

Sarah sent half of the Christmas cards to her friends,= 1/2 *32= 16 cards for Sarah's friends

Richard sent three eights of them to his friends.

3/8 *32=12 cards for Richards friends

Total = 12+16= 28

Card left= 32-28=4

Since Sarah and Richard want to send an even number of cards to their families. Then Sarah's family and Richard's family will receive 2 apiece.

5 0
2 years ago
Mr. Niles used of a carton of eggs to make breakfast Saturday. He used of the carton on Sunday. How many more eggs did he use on
nikklg [1K]

Answer:

C. 3 eggs

Step-by-step explanation:

Mr. Niles used 5/12 of a carton of eggs to make breakfast Saturday. He used 2/12 of the carton on Sunday. How many more eggs did he use on Saturday?

A. 7 eggs

B. 4 eggs

C. 3 eggs

D. 1 egg

A carton of egg = 12 eggs

Saturday

5/12 of a carton of egg

= 5/12 × 12

= 5 eggs

Mr. Niles used 5 eggs on Saturday

Sunday:

2/12 of a carton of egg

= 2/12 × 12

= 2 eggs

Mr. Niles used 2 eggs on Sunday

How many more eggs did he use on Saturday?

= 5 eggs - 2 eggs

= 3 eggs

C. 3 eggs

7 0
2 years ago
if the smaller side of a rectangle was increased by 7 cm, it would be exactly 55% of the 110 cm longer side. Find the area of th
DanielleElmas [232]

Answer:

5886 cm

Step-by-step explanation:

start by finding 55% of 110 which is 60.5. then subtract by 7 and then you get 53.5

then multiply 53.5 by 110 = 5885 cm

4 0
3 years ago
Find the slope of the line connecting the point (8,12) to the point (16,16)
dangina [55]

Answer:

The slope is <u>2</u>

Step-by-step explanation:

For this we use this slope formula

\frac{y_{2}-y_{1} }{x_{2} -x_{1} }

These are our pairs

(8,   12)             (16,   16)

x1    y1               x2    y2

\frac{16-8}{16-12} = \frac{8}{4}

8/4=2

<u> M=2</u>

6 0
2 years ago
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