SOMEONE HELP ME PLEASE

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

3 years ago

Draw a model to represent 42/7

ch4aika [34]

Here is a model I have made (I'm not a great artist). There are 42 circles in total, placed in 7 columns. Each column has 6 circles. This model represents 42/7 by splitting 42 into 7 parts, which are the columns, and showing that each column has 6 circles, which is the answer to 42/7.

7 0

2 years ago

Solve algebraically. 3x + y =3 Y = 2x - 7

Annette [7]

3x + y =3
Y = 2x - 7

substitute Y = 2x - 7 into 3x + y =3
3x + y =3
3x +2x - 7 =3
5x = 10
x = 2

Y = 2x - 7 = 2(2) - 7 = -3

answer
(2, -3)

4 0

3 years ago

Read 2 more answers

Dion practiced the long jump . He jumped 3.05 meters,2.74 meters and 3.3 meters.What was the average length of Dion's jumps?

Mademuasel [1]

All you do is subtract 3.05 and 2.74 then do the same thing with the other ones and see what number is between all of them. What grade are you in

5 0

3 years ago

Read 2 more answers

On question six find the answer and also explain. On question eight find the answer and also explain. On question nine I already

serious [3.7K]

6. No it is not equilateral because a. All the angles should be congruent (60) and as we can tell, they are not, and b. you should never assume measurements.

8. GF is the longest because the angle for that side (GFH) is smaller than the others, so it means it has a longer side.

Your answer for 9 is wrong. The answer is yes because the largest angle = the largest side.

10. AC because largest angle = largest side

11. Literally the same thing as 9 and 10. Largest angle = largest side

3 0

3 years ago