. A right cylindrical drum is to hold 7.35 cubic feet of liquid. Find the dimensions (radius of the base and height) of the drum which would minimize the surface area. What is the minimum surface area

Answer:

[tex]r=1.05\ \text{ft}[/tex]

[tex]h=2.12\ \text{ft}[/tex]

[tex]20.91\ \text{ft}^3[/tex]

Step-by-step explanation:

r = Radius

h = Height

Volume of cylinder = [tex]7.35\ \text{ft}^3[/tex]

[tex]V=\pi r^2h\\\Rightarrow h=\dfrac{V}{\pi r^2}\\\Rightarrow h=\dfrac{7.35}{\pi r^2}[/tex]

Surface area is given by

[tex]A=2\pi r^2+2\pi rh\\\Rightarrow A=2\pi r^2+2\pi r\dfrac{7.35}{\pi r^2}\\\Rightarrow A=2\pi r^2+\dfrac{14.7}{r}[/tex]

Differentiating with respect to radius we get

[tex]\dfrac{dA}{dr}=4\pi r-\dfrac{14.7}{r^2}[/tex]

Equating with zero we get

[tex]0=4\pi r-\dfrac{14.7}{r^2}\\\Rightarrow 4\pi r=\dfrac{14.7}{r^2}\\\Rightarrow r^3=\dfrac{14.7}{4\pi}\\\Rightarrow r=(\dfrac{14.7}{4\pi})^{\dfrac{1}{3}}\\\Rightarrow r=1.05[/tex]

[tex]\dfrac{d^2A}{dr^2}=4\pi-2\times \dfrac{14.7}{r^3}\\\Rightarrow \dfrac{d^2A}{dr^2}=4\pi+2\times \dfrac{14.7}{1.05^3}\\\Rightarrow \dfrac{d^2A}{dr^2}=37.96>0[/tex]

So, the value of the function is minimum at [tex]r=1.05[/tex]

[tex]h=\dfrac{7.35}{\pi r^2}=\dfrac{7.35}{\pi 1.05^2}\\\Rightarrow h=2.12[/tex]

So, the radius and height which would minimize the surface area is 1.05 feet and 2.12 feet respectively.

Surface area

[tex]A=2\pi r^2+2\pi rh\\\Rightarrow A=2\pi \times 1.05^2+2\pi 1.05\times 2.12\\\Rightarrow A=20.91\ \text{ft}^3[/tex]

The minimum surface area is [tex]20.91\ \text{ft}^3[/tex].

WILL GIVE BRAINLIEST!!

Ignore the $

Meyer rolls two fair, ordinary dice with the numbers $1,2,3,4,5,6$ on their sides. What is the probability that at least one of the dice shows a square number?

HURRY PLEASE IM TIMED. Which list of numbers is shown on the number line below? A number line going from negative 3 to positive 2. There are 4 equal spaces between each number. A point is halfway between negative 3 and negative 2. A point is between the second and third mark between negative 2 and negative 1. A point is on the mark 1 space to the right of 0. A point is halfway between 1 and 2. Negative 2.5, 1 and three-fourths, 50 percent, 1.5 Negative 2.5, negative 1 and one-third, 50 percent, 1.5 Negative 2.5, negative 1 and three-fourths, 25 percent, 1.5 Negative 2.5, negative 1 and one-third, 25 percent, 1.5

The triangles are similar.


What is the value of x?


Enter your answer in the box.

x =

what variable is used to represent SLOPE?​

Help me with this bs

IF THE RATIO IN A VALENTINE'S
DAY CANDY BOX OF MILK TO DARK
CHOCOLATES IS 23. HOW MANY
DARK CHOCOLATES ARE THERE IF
THERE ARE 8 MILK CHOCOLATES?

I NEED HELP ASAP PLZZZ
Above, a right triangle is removed from a rectangle. What is the area of the remaining figure?
o 18 in?
31 in
87 in?
96 in

help please! nobody has been answering my questions :( !!!!!!

Kameron who is 4.5 feet tall , is outside during recess

Scientists often write very large or very small numbers using exponents. How might a scientist write the number shown using exponents?

Some one please help

I have attached the question

Write an equation of the line with the given slope and y-intercept.

Examine the graph.


Select each interval where the graph is decreasing.

a.−4b. 3c.2d.−1

What is the area of this rectangle?
2.5 m
2 m

2x {3x [40 - (I0 x 2)]} = ?

Does anybody know what the answer to this question is?
If you know, please tell me. I'm really in a hurry.

Max purchased a solar pool cover for his swimming pool. What is the area of the
solar cover for the swimming pool?
10ft
32 ft
Square Feet

Which points are 19 units apart?
○(-14, 12) and (5, 12)
○(-5, 19) and (-12, 19)
○(-20, 5) and (5, 12)
○(14, 12) and (5, 12)​

please help if u can!!

The entry fee to a fun park is $20. Jackson spent a total of 35 at the park. If z represents the number of rides jackson went on, which equation represents the situation and what is the value of z that makes the equation true?

A= 2.50=35
x=14

B= 2.50z=20
x=8

C= 20z+2.50=35
c=2

D= 20+2.50z=35
x=6

what is (7/4x)+(-3/2y) simplified

Find the midpoint of points A(-3,7) and B(-6,0) graphically. ​

Edwin has 50$ in his bank.He starts spending $3 each week on snacks at lunch. how many weeks will edwin run out of money?

A triangular plot of land has sides of lengths 420 feet, 330 feet, and 180 feet. Approximate the smallest angle between the sides. The choices are rounded to the nearest degree.
A. 30 degree
B. 27 degree
C. 22 degree
D. 24 degree

Stephen's lunch bill is currently at $843. Stephen orders a fruit salad for take out and wants to leave
$2.25 as a tip for his server. He has a $10 bill and a $5 bill. How much change should he receive after
paying for his lunch, the fruit salad, and the tip?

Jacob has a coin and a spinner with 3 equal sections, one colored red, one colored blue, and one colored green. Which tree diagram best represents all possible outcomes of Jacob spins the spinner once and flips the coin once?

please HURRY IM ON A QUIZ

if 2 is substituted for x, which expression will NOT equal 5?

Two spoiled dogs eat a total of 18 treats in a certain amount of time. How many spoiled dogs will eat 72 treats in the same amount of time, assuming each dog eats treats at the same time

what is the awnser to y=3/2x-1

y is directly proportional to the square of x.
When y= 24, x= 2.
Find the value of y when x=4.​

the length of a rectangle is 8 centimeters. what is the width?​