What is the circumference of a circle with a radius of 21 cm?

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

In 2013, a study found that U.S. consumers consumed soft drinks with a mean of 42.2 gallons per year and variance of 169. Ninety

melisa1 [442]

Answer:

Ninety-five percent of consumers in the U.S. consumed less than 63.59 gallons.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

The standard deviation is the square root of the variance, so $\sigma = \sqrt{169} = 13$

Also, the mean is 42.2, so $\mu = 42.2$

Ninety-five percent of consumers in the U.S. consumed less than how many gallons?

The 95th percentile, which is the value of X when Z has a pvalue of 0.95. So X when $Z = 1.645$

$Z = \frac{X - \mu}{\sigma}$

$1.645 = \frac{X - 42.2}{13}$

$X - 42.2 = 13*1.645$

$X = 63.59$

Ninety-five percent of consumers in the U.S. consumed less than 63.59 gallons.

4 0

3 years ago

Multiply x^2-8x+15/3x * 8x/x-3

sergejj [24]

You must follow the steps below:

1. First, you should apply the dstributive property:

=(x²-8x+15/3x)(8x/x-3)
=(8x³-64x²+120x)/3x(x-3)

2. As you can see, the common factor in the numerator is: "8x". So, you have:

=8x(x²-8x+15)/3x(x-3)

3. Then, when you simplify, you obtain:

=8(x²-8x+15)/3(x-3)

Therefore, the solution is: 8(x²-8x+15)/3(x-3)

5 0

3 years ago

A man that is 6 feet tall weighs 165 pounds. Use this information to predict the weight of a man that is 5

Zarrin [17]

Answer:

151 pounds

Step-by-step explanation:

7 0

3 years ago

Please help!! Giving BRAINLIEST!

garri49 [273]

Answer: D
Explanation: rad 36 is the same as rad 6^2 and they are perfect roots do then you just add the numbers like usual.

3 0

3 years ago

What is the circumference ofa circle with a radius of 21cm?​

What is the circumference ofa circle with a radius of 21cm?