Suppose a cylinder that holds 3 L of liquid must be created. Determine the radius and height of the cylinder that will minimize the amount of material used in its construction.(Note: 1 L=1,000 cm^3)
Answers
The radius and the height of the dimension that will minimize the amount of material used in the construction are [tex]\mathbf{r = \sqrt[3]{\dfrac{1500}{\pi}} }[/tex] and [tex]\mathbf{h = \dfrac{3000}{\pi ({\dfrac{1500}{\pi}} )^{2/3}}}[/tex] respectively.
How to find the dimension that minimizes a cylinder?The dimension that minimizes the surface area of a cylinder can be determined by:
Drawing the picture of the problem, Write down & identify optimization as well as the constraint equations;Use the derivative of the optimization equation to find the dimensions.Given that:
1 L = 1000 cm³3 L = 3000 cm³The area of the cylinder = (2 πr)h + 2(πr²)
A = 2πrh + 2πr²The volume of the cylinder
V = πr²hLet's identify the constraint equation and Optimization equation:
To minimize the surface area of the can, we have:
Area equation = Optimization equationThe constraint equation is the equation that limits us:
Volume equation = constraint equationSo, Let's solve for h in our volume equation, we have:
3000 = πr² h
h = 3000/πr²
Now, from the Area equation
[tex]\mathbf{A = 2 \pi r(\dfrac{3000}{\pi r^2}) + 2\pi r^2}[/tex]
Taking the derivate and setting it to zero, we have;
Derivative:
[tex]\mathbf{A = \dfrac{6000}{ r}+ 2\pi r^2}[/tex]
[tex]\mathbf{A = 6000 r^{-1} + 2\pi r^2}[/tex]
[tex]\mathbf{A' = -6000 r^{-2} + 4\pi r}[/tex]
[tex]\mathbf{A' = 4\pi r-\dfrac{6000}{ r^{2} }}[/tex]
Setting it to zero, we have:
[tex]\mathbf{0=\dfrac{ 4 \pi r^3 - 6000}{r^2}}[/tex]
Factor out 4
0 = 4(πr³ - 1500)
1500 = πr³
r³ = 1500/π
[tex]\mathbf{r = \sqrt[3]{\dfrac{1500}{\pi}} }[/tex]
The above is the radius that minimizes the surface area of the cylinder;
From [tex]\mathbf{h = \dfrac{3000}{\pi r^2}}[/tex]
[tex]\mathbf{h = \dfrac{3000}{\pi ( \sqrt[3]{\dfrac{1500}{\pi}} )^2}}[/tex]
[tex]\mathbf{h = \dfrac{3000}{\pi ({\dfrac{1500}{\pi}} )^{2/3}}}[/tex]
Thus, the radius and height that minimize the amount of material to be used in its construction are [tex]\mathbf{r = \sqrt[3]{\dfrac{1500}{\pi}} }[/tex] and [tex]\mathbf{h = \dfrac{3000}{\pi ({\dfrac{1500}{\pi}} )^{2/3}}}[/tex] respectively.
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Related Questions
(please help will give brainlyest)
enter the lowest whole number in the numerator to make the comparison ?/7 > 1/2 true.
Answers
14/7 is greater than 1/2
Sure hope this helps you
When you use the slope formula, does it matter what point you choose as (x1 , y1) and what point you choose as (x2, y2)? Explain why or why not.
Answers
Answer:
It does not matter.
Step-by-step explanation:
How to find a slope with just two points,
Step One: Identify two points on the line.
Step Two: Select one to be (x1, y1) and the other to be (x2, y2).
Step Three: Use the slope equation to calculate the slope.
Example
Let's say that points (15, 8) and (10, 7) are on a straight line. What is the slope of this line?
Step One: Identify two points on the line.
In this example, we are given two points, (15, 8) and (10, 7), on a straight line.
Step Two: Select one to be (x1, y1) and the other to be (x2, y2).
It doesn't matter which we choose, so let's take (15, 8) to be (x2, y2). Let's take the point (10, 7) to be the point (x1, y1).
Step Three: Use the equation to calculate the slope.
Once we've completed step 2, we are ready to calculate the slope using the equation for a slope:
[tex]slope=\frac{y2-y1}{x2-x1} =\frac{8-7}{15-10} =\frac{1}{5}[/tex]
We said that it really doesn't matter which point we choose as (x1, y1) and the which to be (x2, y2). Let's show that this is true. Take the same two points (15, 8) and (10, 7), but this time we will calculate the slope using (15, 8) as (x1, y1) and (10, 7) as the point (x2, y2). Then substitute these into the equation for slope:
[tex]slope=\frac{y2-y1}{x2-x1} =\frac{7-8}{10-15} =\frac{-1}{-5}=\frac{1}{5}[/tex]
We get the same answer as before!
Often you will not be given the two points but will need to identify two points from a graph. In this case, the process is the same, the first step being to identify the points from the graph. Below is an example that begins with a graph.
So therefore it does not matter.
No, it does not matter which point you choose as (x₁, y₁) and which point you choose as (x₂, y₂) when using slope formula. Slope between two points on line remains same regardless of order in which you choose points.
The slope formula is given by slope = (y₂ - y₁) / (x₂ - x₁)
This is because the slope represents the rate at which the y-coordinate changes with respect to the x-coordinate.
If you swap the points (x₁, y₁) and (x₂, y₂), the change in y and the change in x will simply swap as well, but their ratio, which is the slope, will remain unchanged. For example, if the slope between (x₁, y₁) and (x₂, y₂) is m, then the slope between (x₂, y₂) and (x₁, y₁) will also be m.
This property of the slope formula makes it convenient and efficient to calculate the slope between two points, regardless of their order, as it will yield the same result.
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PLEASE HELP ILL MARK BRAINEST!!!!
Answers
Answer:
(-3,8) would NOT fit the table.
Answer: (-3,8)
Step-by-step explanation: click on the file
Luke bought 256 ounces of strawberries. he divided them evenly between 8 different pies.
Answers
Answer:
2 pounds.
Step-by-step explanation:
We have been given that Luke bought 256 ounces of strawberries. He divided them evenly between 8 different pies. We are asked to find the pounds of strawberries that Luke put on each pie.
First of all, we will convert 256 ounces into pounds.
We know that 1 pound equals 16 ounces. To convert 256 ounces into pounds, we need to divide 256 by 16.
[tex]256[/tex] [tex]ounces=\frac{256}{16}[/tex] [tex]pounds=16[/tex] [tex]pounds[/tex]
Now we will divide 16 pounds by 8 to find number of strawberries put on each pie.
[tex]Pounds[/tex] [tex]of[/tex] [tex]strawberries[/tex] [tex]on[/tex] [tex]each[/tex] [tex]pie=\frac{16}{8}[/tex]
[tex]Pounds[/tex] [tex]of[/tex] [tex]strawberries[/tex] [tex]on[/tex] [tex]each[/tex] [tex]pie=2[/tex]
Therefore, Luke put 2 pounds strawberries on each pie.
Arturo is
putting 18 erasers into equal rows. He
says there will be more erasers in 2 equal
rows than in 3 equal rows Is Arturo
correct?
Answers
Answer:
Yes
Step-by-step explanation:
When there are 2 rows
Number of erasers per row = No. of erasers / No. of rowsNumber of erasers per row = 18/2Number of erasers per row = 9When there are 3 rows
Number of erasers per row = No. of erasers / No. of rowsNumber of erasers per row = 18/3Number of erasers per row = 6Therefore, there will be more erasers per row in 2 equal rows than in 3 equal rows.
rewrite the formula p=2L+2w
Answers
P=2(l+w)
PLSSS HELP IF YOU TURLY KNOW THISS
Answers
Answer:
B
Step-by-step explanation:
4/1 * 1/34 * 1 = 41 * 3 = 34/3; 1 1/3Answer:
B) 1 1/3
Step-by-step explanation:
4 x 1/3 = 4/3
4/3 = 1 1/3
you sign up for a new email account. You must choose a secure password that contains 4 letters followed by one digit. How many combinations are there
Answers
Answer:
113
Step-by-step explanation:
26x4=104
1x9=9
104+9=113
Helppppppppppppppppppppp
Answers
Answer:
I'd say think
Step-by-step explanation:
Do 3i =(7²)
3i=49
Answer:
see explanation
Step-by-step explanation:
given a polynomial has roots x = a and x = b , then the corresponding factors are
(x - a) and (x - b)
the polynomial is then the product of the factors , that is
f(x) = (x - a)(x - b)
note that radical and complex roots occur in conjugate pairs.
(5)
given x = - 3i is a root then x = 3i is also a root
factors then are (x - 3i) , (x + 3i) , (x - 7) , then
f(x) = (x - 7)(x - 3i)(x + 3i) ← expand 2nd pair using foil
= (x - 7)(x² - 9i²) [ i² = - 1 ]
= (x - 7)(x² + 9) ← distribute factors
= x³ + 9x - 7x² - 63
f(x) = x³ - 7x² + 9x - 63
(6)
roots are x = [tex]\sqrt{5}[/tex] , x = - [tex]\sqrt{5}[/tex] , x = i , x = - i , and factors are
(x - [tex]\sqrt{5}[/tex] ) , (x + [tex]\sqrt{5}[/tex] ) , (x - i) , (x + i) and
f(x) = (x - [tex]\sqrt{5}[/tex] )(x + [tex]\sqrt{5}[/tex] )(x - i)(x + i) ← expand pairs of factors using FOIL
= (x² - 5)(x² + 1) ← distribute factors
= [tex]x^{4}[/tex] + x² - 5x² - 5
then
f(x) = [tex]x^{4}[/tex] - 4x² - 5
Sydney walked 3/4 mile on Monday
5/8 mile on Tuesday, and 1/2 mile on
Wednesday. How far did she walk
Monday through Wednesday?
A 1 1/2 mi
B 1 3/4 mi
C 1 7/8
D 2 1/4
Answers
3/4= 6/8
5/8= 5/8
1/2 = 4/8
6+5+4 = 15
15/8= 1 7/8
So the answer is C
1. Kedir bought 18 pieces of glasses out of a box of glasses. If these are 75% of the total, how many glasses did the box contain?
Answers
Answer:
24 pieces of glass
Step-by-step explanation:
Hey guys! Stuck on this problem. Can you please explain? Will give brainliest!
Answers
Answer:
60
Step-by-step explanation:
1. Calculate area of front surface:
Area of rectangle + Area of smaller rectangle
6*3 + 2*1 = 20
2. then multiply the area by the depth (3)
20 * 3 = 60 cm^3
A linear pair must add up to 180 degrees is true or false?
Answers
Answer:
true
Step-by-step explanation:
Suppose a jar contains 11 red marbles and
21 blue marbles. If you reach in the jar and
pull out 2 marbles at random at the same
time, find the probability that both are red.
Answers
Answer:
There are 32 marbles. 11 are red. So odds of a red on the first are 11/32.
Now there are 31 marbles left. 10 are red. So odds of a red on the second is 10/31.
Probability both are red, just multiply …. 11/32x 10/31 = 110/992 or 0.111.
Change the function f(x) = 2x ^ 2 + 8x + 5 into vertex form.
Answers
Answer:
[tex]f(x)=2(x+4)^2-27[/tex]
Step-by-step explanation:
Complete the square
[tex]f(x)=2x^2+8x+5\\f(x)=2(x^2+8x+\frac{5}{2})\\f(x)+2(\frac{27}{2})=2(x^2+8x+\frac{5}{2}+\frac{27}{2})\\f(x)+27=2(x^2+8x+16)\\f(x)+27=2(x+4)^2\\f(x)=2(x+4)^2-27[/tex]
This tells us that the vertex of the parabola is [tex](-4,-27)[/tex].
The number of 22 cent stamps, n, can be found by solving the equation: 22n + 17(20 − n) = 400
How many 22 cent stamps are in the book?
Answers
Answer:
12
Step-by-step explanation:
22n + 17(20 - n) = 400 ← distribute parenthesis and simplify left side
22n + 340 - 17n = 400
5n + 340 = 400 ( subtract 340 from both sides )
5n = 60 ( divide both sides by 5 )
n = 12
there are 12, 22 cent stamps in the book
Given zy = 4 and x² + y² = a, which expression is equal to (x + y)²?
Answers
The value of the given expression will be equal to ( x + y )² = a + 8.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication and division.
Given that:-
xy = 4 and x² + y² = aThe expression will be calculated as:-
( x + y )² = x² + y² + 2 xy
( x + y )² = a + 2 x ( 4 )
( x + y )² = a + 8
Therefore the value of the given expression will be equal to ( x + y )² = a + 8.
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A gym charges membership dues of $25 per month. Part A
Complete the table to show how the total cost in dollars, C, and the
number of months, m, of gym membership are related. What is the cost of the blue section
Answers
you can turn this into an equation: [tex]c=25m[/tex], where c is the cost and m is number of months.
solve for blue.
[tex]c=25m\\\\c=25(14)\\\\c=350[/tex]
The blue section costs $350.
Mr. reeves planted 5 beans and 12 tomato plants in his garden. What is the ratio of bean plants to tomato plants ?
Answers
Answer:
5/12 = 0.416666.... also in percentage 29,41 % of the plant are beans and 70.59 % are tomato plants.
GIVING BRAINLIEST!!!!!
Answers
plane PBC is right answer.
it's a logical answer....
Answer:
Plane PBC
Step-by-step explanation:
Plane PBC contains 4 points : P, B, C, and Q.
The question asks which contains all three of C, Q, and P.
The plane is Plane PBCsimultaneous eqautions
3x+2y=16
2x+y=9
Answers
Answer:
(2, 5 )
Step-by-step explanation:
3x + 2y = 16 → (1)
2x + y = 9 ( subtract 2x from both sides )
y = 9 - 2x → (2)
substitute y = 9 - 2x into (1)
3x + 2(9 - 2x) =16
3x + 18 - 4x = 16
- x + 18 = 16 ( subtract 18 from both sides )
- x = - 2 ( multiply both sides by - 1 )
x = 2
substitute x = 2 into (2)
y = 9 - 2(2) = 9 - 4 = 5
solution is (2, 5 )
6x+4y = 32
6x+3y = 27
we get,
y = 5
putting y = 5 in eqn (2) we get
2x + 5 =9
2x = 9 -5
2x = 4
or x = 2
therefore x = 2 and y =5.
(2,5)
Determine the solution for the following equation: x = 3 x = 5 x = 13 x = 65
Answers
Answer:
x = 3
Step-by-step explanation:
[tex](8x - 8)^{3/2} = 64\\(8x - 8)^{3} = 4096\\x = 3[/tex]
Answer:
A on edge 23
Step-by-step explanation:
because math
An instrument maker creates a triangle in the shape of an isosceles triangle by bending a metal rod. the rod is 13 inches long, and the bottom of the triangle is longer than the two sides. what is the length of the longest side?
the length of the longest slide is inches. the other two sides will each be
inches in length.
Answers
The length of the longest slide is 5 inches. he other two sides will be 4 inches in length.
Length of the longest slideIsosceles triangle
Given:
Total length=13 inches
Hence:
AC=BC<AB
Let AC=BC=x-1 inches
AB=x inches
Thus:
x-1+(x-1)+x=13
3x-2=13
3x=13+2
3x=15
divide both side by 3x
x=15/3
x=5 inches
Hence:
Other two side
x-1=5-1=4 inches
Therefore the length of the longest slide is 5 inches. he other two sides will be 4 inches in length.
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Answer:
4 in
Step-by-step explanation:
I Just Took the Test.
On average how far from the mean age is each player?
Answers
The mean of the age data of the players is 20 and on average the age of each player is 1 far from mean.
What is dot plot?The dot plot is the way of representation of the data. This data is plotted in the form of points or dots over an x-y axis. The dot plot is also known as the strip plot.
The data of the age of the player is,
18,19,20,21,21,21
The number of player are 6. Thus, the mean of the data is,
Mean=(18+19+20+21+21+21)/6
Mean=20
The distance form mean 20 of each player age is, 2,1,0,1,1,1. The mean deviation is,
Mean Deviation= (2+1+0+1+1+1)/6
Mean Deviation=1
Thus, the mean of the age data of the players is 20 and on average the age of each player is 1 far from mean.
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explain how its false
Answers
Answer:
No
Step-by-step explanation:
An exponent signifies the number of times the base is a factor in the product. When products involving the same base are multiplied, the total number of factors is the sum of the number in each of those products.
__
[tex]5^2=5\cdot5\\\\5^9=5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\\\\5^2\cdot5^9=(5\cdot5)\cdot(5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5)=5^{11}[/tex]
The base of the exponent in the product is the value that is a repeated factor. It is 5, not (5·5).
Your friend is not correct.
__
The general rule for exponents is ...
(a^b)(a^c) = a^(b+c)
I need help with this question please answer fast!
Answers
Answer:
Its probably C or B I hope this helps!
offering tons of points and brainliest for the right answer!
A boat can travel 28 miles up a river in the same amount of time it takes to travel 36 miles down the same river. If the current is traveling at 2 miles per hour, what is the speed of the boat in still water?
Answers
Answer:
16 mph
Step-by-step explanation:
You want the speed of a boat in still water if it travels 28 miles upstream in the same time it travels 36 miles downstream in a river that runs 2 mph.
TimeThe relation between time, speed, and distance is ...
t = d/s
If we let b represent the boat speed, the two identical times give the relation ...
28/(b -2) = 36/(b +2)
Solution28(b +2) = 36(b -2) . . . . cross multiply
28b +56 = 36b -72 . . . . eliminate parentheses
128 = 8b . . . . . . . . . . . . add 72 -28b
16 = b . . . . . . . . . . . . divide by 8
The speed of the boat is 16 mph.
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y=8x+7 and -8x+y=7 solve algebretically
Answers
Answer:
infinitely many solution
Step-by-step explanation:
-8x + 8x + 7 = 7
7 = 7
Keyshawn buys 100 baseball cards to start a collection. He purchases 25 more baseball cards
each week. Which equation could be used to represent the number of baseball cards (b) in
Keyshawn's collection after w weeks?
A. w = 25b + 100
B. w = 100b+25
C. b = 25w + 100
D. b = 100w + 25
Answers
Answer:
C. b = 25w + 100
Step-by-step explanation:
Keyshawn starts off with 100 baseball cards. Each week he buys 25 more baseball cards. Given this we want to find an equation that would represent the number of baseball cards (represented by "b") he would have after a number of weeks (represented by "w")
We know that keyshawn buys 25 baseball cards weekly so the number of baseball cards purchased after w weeks can be represented by 25w.
We also know that he started of with 100 baseball cards so the total amount of baseball cards keyshawn has (b) can be represented by the expression b = 25w + 100
Determine the interval (s) on which the given function is increasing
Answers
Answer:
[tex]\{ x: x < 0 \} \cup \{ x: x > 1 \}[/tex]
Step-by-step explanation:
A function is increasing when its gradient is positive (so y-coordinates are increasing).
We have a graph, rather than a function given to us, so we don't need to use any calculus.
Looking at the graph, we can see that before x=0, the graph is going up (gradient is increasing).
Between x=0 and x=1, the function is decreasing.
After x=1, the function is increasing again.
We can use mathematical notation to express this.
[tex]\{ x: x < 0 \} \cup \{ x: x > 1 \}[/tex]
Or
[tex]x < 0\ \text{and}\ x > 1[/tex]
A function assigns the values. The interval on which the given function is increasing is (-∞,0)∩(1,∞).
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
For the given functions graph, the interval on which the given function is increasing is the interval of the function where the slope of the function is positive. Therefore, the interval of the function on which the given function is increasing is (-∞,0) and (1,∞). The interval for the given function can also be written as (-∞,0)∩(1,∞).
Hence, the interval on which the given function is increasing is (-∞,0)∩(1,∞).
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