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Marina CMI [18]
3 years ago
11

A certain magical substance that is used to make solid magical spheres costs $800 per cubic foot. The power of a magical sphere

depends on its surface area, and a magical sphere can be sold for $30 per square foot of surface area. If you are manufacturing such a sphere, what size should you make them to maximize your profit per sphere?
Mathematics
1 answer:
Flauer [41]3 years ago
6 0

Answer:

the sphere should be constructed with a radius R= 0.075 ft in order to maximise the profit

Step-by-step explanation:

since the profit function P is

P= $30/ft²* A -  $800/ft³ * V = a*A + b*V

where A and  V are the area and the volume of the sphere respectively.Then

A= 4*π*R²

and

V= 4/3*π*R³

where R is the radius

replacing in P

P= a*A + b*V =  4*π*a* R² - 4/3*π*b*R³ =

the maximum is found where the derivative of P with respect to R is equal to 0 , therefore:

dP/dR= 8*π*a* R - 4*π*b*R² = 0

then

8*π*a* R - 4*π*b*R² = 0

4*π*R*(2*a - b*R) = 0

since R>0

2*a - b*R=0

R= 2*a/b

replacing values

R= 2*a/b = 2*$30/ft² / $800/ft³ = 0.075 ft

thus the sphere should be constructed with a radius R= 0.075 ft in order to maximise the profit.

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We are given :

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Let's find the percent increase with the help of formula:

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\red{ \rule{35pt}{2pt}} \orange{ \rule{35pt}{2pt}} \color{yellow}{ \rule{35pt} {2pt}} \green{ \rule{35pt} {2pt}} \blue{ \rule{35pt} {2pt}} \purple{ \rule{35pt} {2pt}}

\tt \% \: increase =  100 \times \frac{(143 - 85)}{85}

\tt  \% \: increase = 100 \times  \frac{58}{85}

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2 years ago
Let P = 0.50.30.50.7 be the transition matrix for a Markov chain with two states. Let x0 = 0.50.5 be the initial state vector fo
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Answer:

Probability distribution vector = \left(\begin{array}{c}0.375\\ 0.625 \end{array} \right)

Step-By-Step Explanation

If P=\left(\begin{array}{cc}0.5&0.3\\ 0.5&0.7 \end{array} \right)  is the transition matrix for a Markov chain with two states.  

x_{0}=\left(\begin{array}{c}0.5\\ 0.5 \end{array} \right)  be the initial state vector for the population.

X_{1}=P x_{0}=\left(\begin{array}{cc}0.5&0.3\\ 0.5&0.7 \end{array} \right) \left(\begin{array}{c}0.5\\ 0.5 \end{array} \right) =\left(\begin{array}{c}0.4\\ 0.6 \end{array} \right)  

X_{2}=P^{2} x_{0}=\left(\begin{array}{c}0.38\\ 0.62 \end{array} \right)  

X_{3}=P^{3} x_{0}=\left(\begin{array}{c}0.38\\ 0.62 \end{array} \right)  

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In the long run, the probability distribution vector Xm approaches the probability distribution vector \left(\begin{array}{c}0.375\\ 0.625 \end{array} \right) .

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2 years ago
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Answer:

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Step-by-step explanation:

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The probability of both being grey is  

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The probability two socks match:

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Step-by-step explanation:

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Answer:

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