Mathematics, 11.11.2020 18:10 AllisonMcruther

For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 44 in.3 and is constructed using the least amount of metal. (Round your answers to two decimal places.)

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Answer from: hftyd2046

$1.91\ \text{in}$

$3.84\ \text{in}$

Step-by-step explanation:

V = Volume of cylinder = $44\ \text{in}^3$

h = Height of cylinder

r = Radius of cylinder

Volume of cylinder is given by

$V=\pi r^2h\\\Rightarrow h=\dfrac{V}{\pi r^2}\\\Rightarrow h=\dfrac{44}{\pi r^2}$

Total surface area of a cylinder is given by

$S=2\pi r^2+2\pi rh\\\Rightarrow S=2\pi r^2+2\pi r\times\dfrac{44}{\pi r^2}\\\Rightarrow S=2\pi r^2+\dfrac{88}{r}$

Differentiating with respect to radius

$\dfrac{dS}{dr}=4\pi r-\dfrac{88}{r^2}$

Equating with zero

$4\pi r-\dfrac{88}{r^2}=0\\\Rightarrow 4\pi r=\dfrac{88}{r^2}\\\Rightarrow r^3=\dfrac{88}{4\pi}\\\Rightarrow r=(\dfrac{22}{\pi})^{\dfrac{1}{3}}\\\Rightarrow r=1.91\ \text{in}$

Double derivative of S

$\dfrac{d^2S}{dr^2}=4\pi+1760$

So is minimum at $\dfrac{dS}{dr}=0$

$h=\dfrac{44}{\pi r^2}=\dfrac{44}{\pi 1.91^2}\\\Rightarrow h=3.84\ \text{in}$

So the radius and height of the cylinder is $1.91\ \text{in}$ and $3.84\ \text{in}$ respectively such that the least amount of metal is used.

Answer from: Quest

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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rauschg

rauschg · hand

know the answer? add it here!

step-by-step explanation:

Answer from: Quest

the answer is: d) o, f, and kr.

atomic radius of bromine (br) is 114 pm.

atomic radius of oxygen (o) is 73 pm.

atomic radius of fluorine (f) is 72 pm.

atomic radius of krypron (kr) is 112 pm.

the atomic radius of a chemical element is a measure of the size of its atom.

the atomic radius varies with increasing atomic number, but usually increases because of increasing of number of electrons.

the atomic radius decreases across the periods because an increasing number of protons, because greater attraction between the protons and electrons.

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