Mathematics, 04.04.2020 02:30 herchellann302

A cylindrical can, open at the top, is to hold cm3 of liquid. Find the height, , and the radius, , that minimize the amount of material needed to manufacture the can. Enter the exact answers.

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Answer from: somedudeontheIn

$r=4\ cm,\ h=4\ cm$

Step-by-step explanation:

Minimization

Optimization is the procedure leading to find the values of some parameters that maximize or minimize a given objective function. The parameters could have equality and inequality restrictions. If only equality restrictions hold, then we can use the derivatives to find the possible maximum or minimum values of the objective function.

The problem states we need to minimize the amount of material needed to manufacture the cylindrical can. The material is the surface area of the can. If the can has height h and radius r on the base, then the surface area is

$A=2\pi rh+\pi r^2$

Note there is only one lid at the bottom (open at the top), that is why we added only the surface area of one circle.

That is our objective function, but it's expressed in two variables. We must find a relation between them to express the area in one variable. That is why we'll use the given volume (We'll assume the volume to be $64\pi cm^3$ because the question skipped that information).

The volume of a cylinder is

$V=\pi r^2h$

We can solve it for h and replace the formula into the formula for the area:

$\displaystyle h=\frac{V}{\pi r^2}$

Substituting into the area

$\displaystyle A=2\pi r\cdot \frac{V}{\pi r^2}+\pi r^2$

Simplifying

$\displaystyle A=\frac{2V}{ r}+\pi r^2$

Now we take the derivative

$\displaystyle A'=-\frac{2V}{ r^2}+2\pi r$

Equating to 0

$\displaystyle \frac{-2V+2\pi r^3}{ r^2}=0$

Since r cannot be 0:

$-2V+2\pi r^3=0$

$\displaystyle r=\sqrt[3]{\frac{V}{\pi}}$

Since $V=64\pi$

$\displaystyle r=\sqrt[3]{\frac{64\pi}{\pi}}=4$

$r=4\ cm$

And

$\displaystyle h=\frac{64\pi}{\pi 4^2}=4$

$h=4\ cm$

Summarizing:

$\boxed{r=4\ cm,\ h=4\ cm}$

Answer from: Quest

Answer: 10pi

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

Find the length of ac express your answer in terms of pi

Answer from: Quest

l.ln.lln.k.nkl

step-by-step explanation:

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A cylindrical can, open at the top, is to hold cm3 of liquid. Find the height, , and the radius, , t...

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