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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:

<u>Optimizing With Derivatives </u>

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

You basically look at the bar graph on the right and see which statement is true. Brainliest and thank you!

wel

Answer:

D

Step-by-step explanation:

To check this answer, you could obviously graph the points, but without graphing, you can tell the line would be straight because each x value is multiplied by the same number to get the corresponding y value.

3 x 5 = 15, 4 x 5 = 20, etc. This shows linear growth. If each x value were multiplied by twice itself (3 x 6, 4 x 8, and so on), you wouldn't have a straight line because the number that x is multiplied by changes depending on the value of x.

7 0

2 years ago

Factor: x(3y-5)+3(3y-15)

alina1380 [7]

3xy-5x+9y-45

Step-by-step explanation:

Step by Step Solution

STEP1:STEP2:Pulling out like terms

2.1 Pull out like factors :

3y - 15 = 3 • (y - 5)

Equation at the end of step2: (x • (3y - 5)) + 9 • (y - 5) STEP3:Equation at the end of step 3 x • (3y - 5) + 9 • (y - 5) STEP4:Trying to factor a multi variable polynomial

4.1 Split 3xy-5x+9y-45

into two 2-term polynomials

-5x+3xy and +9y-45

This partition did not result in a factorization. We'll try another one:

3xy-5x and +9y-45

This partition did not result in a factorization. We'll try another one:

3xy+9y and -5x-45

This partition did not result in a factorization. We'll try another one:

3xy-45 and +9y-5x

This partition did not result in a factorization. We'll try another one:

-45+3xy and +9y-5x

This partition did not result in a factorization. We'll try

8 0

8 months ago

The yearbook staff receive 75 submissions for yearbook articles. Paul will accept 2/5

antiseptic1488 [7]

Answer:

2/5 of 75 is 30 thats all i have for because i am haveing trouble with the same question

Step-by-step explanation:

7 0

2 years ago

Help me ASAP!!!!!!!!!!!!!:)

kenny6666 [7]

Answer:

I think it is wrong because it is unlike term

5 0

2 years ago

Read 2 more answers