Find the area of the triangle below. Be sure to include the correct unit in your answer.

A metal cylinder can with an open top and closed bottom is to have volume 4 cubic feet. Approximate the dimensions that require

Aleksandr-060686 [28]

Answer:

$r\approx 1.084\ feet$

$h\approx 1.084\ feet$

$\displaystyle A=11.07\ ft^2$

Step-by-step explanation:

Optimizing With Derivatives

The procedure to optimize a function (find its maximum or minimum) consists in :

Produce a function which depends on only one variable
Compute the first derivative and set it equal to 0
Find the values for the variable, called critical points
Compute the second derivative
Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum

We know a cylinder has a volume of 4 $ft^3$ . The volume of a cylinder is given by

$\displaystyle V=\pi r^2h$

Equating it to 4

$\displaystyle \pi r^2h=4$

Let's solve for h

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

A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is

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

Replacing the formula of h

$\displaystyle A=\pi r^2+2\pi r \left (\frac{4}{\pi r^2}\right )$

Simplifying

$\displaystyle A=\pi r^2+\frac{8}{r}$

We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero

$\displaystyle A'=2\pi r-\frac{8}{r^2}=0$

Rearranging

$\displaystyle 2\pi r=\frac{8}{r^2}$

Solving for r

$\displaystyle r^3=\frac{4}{\pi }$

$\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet$

Computing h

$\displaystyle h=\frac{4}{\pi \ r^2}\approx 1.084\ feet$

We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative

$\displaystyle A''=2\pi+\frac{16}{r^3}$

We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.

The minimum area is

$\displaystyle A=\pi(1.084)^2+\frac{8}{1.084}$

$\boxed{ A=11.07\ ft^2}$

8 0

2 years ago

Li Wei has several streams of income, as shown in the table. Some streams are active income and some are passive.

WARRIOR [948]

Based on the streams of income that Li Wei has, we can calculate that he has a total passive income of $3,010.

Passive income refers to income that is made without a person actively involving themselves in the activity generating the income.

In this scenario, the passive income sources are:

Stock dividends
Business investment interest
Royalties from a novel

Total passive income therefore is:

= 530 + 1,080 + 1,400

= $3,010

In conclusion, the total passive income is $3,010.

Find out more at brainly.com/question/1246309.

3 0

2 years ago

HELP! what is the unit rate???

UkoKoshka [18]

The unit rate for that equation is .33

8 0

2 years ago

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How do i find the inverse function of h(x) =(5-2x^2)/(3x+10)

sergeinik [125]

Inverse function: you need to exchange the position of x and y, in other words,

x = (5-2y^2)/(3y+10) , then, if you need a representation of y with respect to x, then you need to perform some math.

5 0

3 years ago

What facts are true for the graph of f(x) = 4x5^x

sashaice [31]

I think you need to add a picture?

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3 years ago