Find the volume of the cylinder. Round your answer to the nearest tenth.

Archy [21]

(a) If the particle's position (measured with some unit) at time t is given by s(t), where

$s(t) = \dfrac{5t}{t^2+11}\,\mathrm{units}$

then the velocity at time t, v(t), is given by the derivative of s(t),

$v(t) = \dfrac{\mathrm ds}{\mathrm dt} = \dfrac{5(t^2+11)-5t(2t)}{(t^2+11)^2} = \boxed{\dfrac{-5t^2+55}{(t^2+11)^2}\,\dfrac{\rm units}{\rm s}}$

(b) The velocity after 3 seconds is

$v(3) = \dfrac{-5\cdot3^2+55}{(3^2+11)^2} = \dfrac{1}{40}\dfrac{\rm units}{\rm s} = \boxed{0.025\dfrac{\rm units}{\rm s}}$

(c) The particle is at rest when its velocity is zero:

$\dfrac{-5t^2+55}{(t^2+11)^2} = 0 \implies -5t^2+55 = 0 \implies t^2 = 11 \implies t=\pm\sqrt{11}\,\mathrm s \imples t \approx \boxed{3.317\,\mathrm s}$

(d) The particle is moving in the positive direction when its position is increasing, or equivalently when its velocity is positive:

$\dfrac{-5t^2+55}{(t^2+11)^2} > 0 \implies -5t^2+55>0 \implies -5t^2>-55 \implies t^2 < 11 \implies |t|$

In interval notation, this happens for t in the interval (0, √11) or approximately (0, 3.317) s.

(e) The total distance traveled is given by the definite integral,

$\displaystyle \int_0^8 |v(t)|\,\mathrm dt$

By definition of absolute value, we have

$|v(t)| = \begin{cases}v(t) & \text{if }v(t)\ge0 \\ -v(t) & \text{if }v(t)$

In part (d), we've shown that v(t) > 0 when -√11 < t < √11, so we split up the integral at t = √11 as

$\displaystyle \int_0^8 |v(t)|\,\mathrm dt = \int_0^{\sqrt{11}}v(t)\,\mathrm dt - \int_{\sqrt{11}}^8 v(t)\,\mathrm dt$

and by the fundamental theorem of calculus, since we know v(t) is the derivative of s(t), this reduces to

$s(\sqrt{11})-s(0) - s(8) + s(\sqrt{11)) = 2s(\sqrt{11})-s(0)-s(8) = \dfrac5{\sqrt{11}}-0 - \dfrac8{15} \approx 0.974\,\mathrm{units}$

7 0

2 years ago

What’s the value of tan (3arc cos 3/4)

Volgvan

<h2>Answer: -1.469</h2>

This trigonometric function can be written as:

$tan (3 cos^{-1} (\frac{3}{4}))$ (1)

Firstly, we have to solve the inner parenthesis:

$cos^{-1} (\frac{3}{4})= 41.409$ (2)

Substituting (2) in (1):

$tan (3(41.409))=tan(124.228)$ (4)

Finally we obtain the value:

$tan(124.228)=-1.469$

6 0

2 years ago

Read 2 more answers

Which package has a better price a 10 pack of 2.1 ounce bars which costs $15.37 or a 12 pack of 1.4 ounce bars which cause $15.3

jasenka [17]

Find which is a better package between the 2
10 pack of 2.1 ounce bar = costs 15.37 dollars
=> 2.1 x 20 = 21 there are a total of 21 ounce of bar
=> 21 / 15.37
=> 1.366 dollars / ounce

12 pack of 1.4 ounce bars = costs 15.35
=> 1.4 x 12 = 16.8, there are 1a total of 6.8 ounces of bar
=> 16.8 / 15.35
=> 1.094 dollars / ounce

Thus the second choice is the better package between the two. The 12 pack of 1.4 ounce bars.

8 0

3 years ago

When we use the formula for the area of a triangle, why do we use the height and not the hypotenuse?

ira [324]

Answer:

area is a half x l x h

Step-by-step explanation:

we cannot use the hypotenuse because it wouldn't bring the area

5 0

2 years ago

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Convert 5.6 into vulgar fraction

zheka24 [161]

Answer:

The answer is 28/5

Step-by-step explanation:

5.6 can be written as 56/10 into vulgar fraction.

Thus, The answer is 56/10 = 28/5

-TheUnknownScientist 72

8 0

2 years ago