May you plz help me I am struggling with slopes

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

3 years ago

a teacher awarded 50 marks to his least student in a class test. How many marks was awarded to the best student if he was 40% be

LenaWriter [7]

Answer:

90 marks per that student

4 0

3 years ago

Help please will mark brainliest

natima [27]

Answer:

A table that has (0,0), (-1,1), (-4, 2) and undefined for any positive x value

Step-by-step explanation:

Reflecting across the y axis just changes the x values, it makes them negative. so $\sqrt{x}$ has points (0,0), (1,1), (4, 2) and so on. reflecting over the y axis makes them(0,0), (-1,1), (-4, 2) and again so on.

Also good to mention in $\sqrt{x}$ negative x values are undefined, so flipped over the y axis positive x values are undefined.

As for the answer it doesn't look like any of those shown. The first one is close, but the x values would need to swap their signs.

6 0

3 years ago

How do i fond the quotient and the remainder ?

zhenek [66]

The remainder is the integer left over after dividing one integer by another. The quotient is the quantity produced by the division of two numbers.
Hope that helped !

5 0

3 years ago

The sum of a number times 9 and 30 is at least -28

Mars2501 [29]

Answer:

9n + 30 ≥ -28

Step-by-step explanation:

Let the number be n. Then, symbolically, this statement is:

9n + 30 ≥ -28

8 0

3 years ago