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1 year ago

the radius r of a sphere is 4cm and is increasing at the rate of 0.5cm . At what rate is the volume?

Mathematics

1 answer:

Aleks04 [339]1 year ago

7 0

Answer:

32π or 100.53 cm^3 to the nearest hundredth.

Step-by-step explanation:

dr/dt = 0.5

V = (4/3)π r^3

So dV/dr = (4/3)* 3 π r^(3-1)

= 4πr^2

Rate of increase in volume:

dV/dt = dV/dr * dr/dt

At r = 4:

Rate of increase in Volume

dV/dt = 4π(4)^2 * 0.5

= 64π * 0.5

= 32π.

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$\left(\begin{array}{c|cc}&$Running&$Down\\---&---&---\\$Running&0.90&0.10\\$Down&0.30&0.70\end{array}\right)$

(a)To determine the probability of the system being down or running after any k hours, we determine the kth state matrix $P^k$ .

(a)

$P^1=\left(\begin{array}{c|cc}&$Running&$Down\\---&---&---\\$Running&0.90&0.10\\$Down&0.30&0.70\end{array}\right)$

$P^2=\begin{pmatrix}0.84&0.16\\ 0.48&0.52\end{pmatrix}$

If the system is initially running, the probability of the system being down in the next hour of operation is the $(a_{12})th$ entry of the P^2$ matrix.$

The probability of the system being down in the next hour of operation = 0.16

(b)After two(periods) hours, the transition matrix is:

$P^3=\begin{pmatrix}0.804&0.196\\ 0.588&0.412\end{pmatrix}$

Therefore, the probability that a system initially in the down-state is running

is 0.588.

(c)The steady-state probability of a Markov Chain is a matrix S such that SP=S.

Since we have two states, $S=[s_1$ s_2]$

$[s_1$ s_2]\left(\begin{array}{ccc}0.90&0.10\\0.30&0.70\end{array}\right)=[s_1$ s_2]$

Using a calculator to raise matrix P to large numbers, we find that the value of $P^k$ approaches [0.75 0.25]:

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$[0.75$ 0.25]\left(\begin{array}{ccc}0.90&0.10\\0.30&0.70\end{array}\right)=[0.75$ 0.25]$

The steady-state probabilities of the system being in the running state and in the down-state is therefore:

$[s_1$ s_2]=[0.75$ 0.25]$

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3 years ago

the radius r of a sphere is 4cm and is increasing at the rate of 0.5cm . At what rate is the volume?​

the radius r of a sphere is 4cm and is increasing at the rate of 0.5cm . At what rate is the volume?