A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 10 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to two decimal places.)
Answer:
[tex]r = 1.34[/tex]
Step-by-step explanation:
Given
Solid = Cylinder + 2 hemisphere
[tex]Volume = 10cm^3[/tex]
Required
Determine the radius (r) that minimizes the surface area
First, we need to determine the volume of the shape.
Volume of Cylinder (V1) is:
[tex]V_1 = \pi r^2h[/tex]
Volume of 2 hemispheres (V2) is:
[tex]V_2 = \frac{2}{3}\pi r^3 +\frac{2}{3}\pi r^3[/tex]
[tex]V_2 = \frac{4}{3}\pi r^3[/tex]
Volume of the solid is:
[tex]V = V_1 + V_2[/tex]
[tex]V = \pi r^2h + \frac{4}{3}\pi r^3[/tex]
Substitute 10 for V
[tex]10 = \pi r^2h + \frac{4}{3}\pi r^3[/tex]
Next, we make h the subject
[tex]\pi r^2h = 10 - \frac{4}{3}\pi r^3[/tex]
Solve for h
[tex]h = \frac{10}{\pi r^2} - \frac{\frac{4}{3}\pi r^3 }{\pi r^2}[/tex]
[tex]h = \frac{10}{\pi r^2} - \frac{4\pi r^3 }{3\pi r^2}[/tex]
[tex]h = \frac{10}{\pi r^2} - \frac{4r }{3}[/tex]
Next, we determine the surface area
Surface area (A1) of the cylinder:
Note that the cylinder is covered by the 2 hemisphere.
So, we only calculate the surface area of the curved surface.
i.e.
[tex]A_1 = 2\pi rh[/tex]
Surface Area (A2) of 2 hemispheres is:
[tex]A_2 = 2\pi r^2+2\pi r^2[/tex]
[tex]A_2 = 4\pi r^2[/tex]
Surface Area (A) of solid is
[tex]A = A_1 + A_2[/tex]
[tex]A = 2\pi rh + 4\pi r^2[/tex]
Substitute [tex]h = \frac{10}{\pi r^2} - \frac{4r }{3}[/tex]
[tex]A = 2\pi r(\frac{10}{\pi r^2} - \frac{4r }{3}) + 4\pi r^2[/tex]
Open bracket
[tex]A = \frac{2\pi r*10}{\pi r^2} - \frac{2\pi r*4r }{3} + 4\pi r^2[/tex]
[tex]A = \frac{2*10}{r} - \frac{2\pi r*4r }{3} + 4\pi r^2[/tex]
[tex]A = \frac{20}{r} - \frac{8\pi r^2 }{3} + 4\pi r^2[/tex]
[tex]A = \frac{20}{r} + \frac{-8\pi r^2 }{3} + 4\pi r^2[/tex]
Take LCM
[tex]A = \frac{20}{r} + \frac{-8\pi r^2 + 12\pi r^2}{3}[/tex]
[tex]A = \frac{20}{r} + \frac{4\pi r^2}{3}[/tex]
Differentiate w.r.t r
[tex]A' = -\frac{20}{r^2} + \frac{8\pi r}{3}[/tex]
Equate A' to 0
[tex]-\frac{20}{r^2} + \frac{8\pi r}{3} = 0[/tex]
Solve for r
[tex]\frac{8\pi r}{3} = \frac{20}{r^2}[/tex]
Cross Multiply
[tex]8\pi r * r^2 = 20 * 3[/tex]
[tex]8\pi r^3 = 60[/tex]
Divide both sides by [tex]8\pi[/tex]
[tex]r^3 = \frac{60}{8\pi}[/tex]
[tex]r^3 = \frac{15}{2\pi}[/tex]
Take [tex]\pi = 22/7[/tex]
[tex]r^3 = \frac{15}{2 * 22/7}[/tex]
[tex]r^3 = \frac{15}{44/7}[/tex]
[tex]r^3 = \frac{15*7}{44}[/tex]
[tex]r^3 = \frac{105}{44}[/tex]
Take cube roots of both sides
[tex]r = \sqrt[3]{\frac{105}{44}}[/tex]
[tex]r = \sqrt[3]{2.38636363636}[/tex]
[tex]r = 1.33632535155[/tex]
[tex]r = 1.34[/tex] (approximated)
Hence, the radius is 1.34cm
The radius of the cylinder that produces the minimum surface area is 1.34cm and this can be determined by using the formula area and volume of cylinder and hemisphere.
Given :
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 10 cubic centimeters.The volume of a cylinder is given by:
[tex]\rm V = \pi r^2 h[/tex]
The total volume of the two hemispheres is given by:
[tex]\rm V' = 2\times \dfrac{2}{3}\pi r^3[/tex]
[tex]\rm V' = \dfrac{4}{3}\pi r^3[/tex]
Now, the total volume of the solid is given by:
[tex]\rm V_T = \pi r^2 h+\dfrac{4}{3}\pi r^3[/tex]
Now, substitute the value of the total volume in the above expression and then solve for h.
[tex]\rm 10 = \pi r^2 h+\dfrac{4}{3}\pi r^3[/tex]
[tex]\rm h = \dfrac{10}{\pi r^2}-\dfrac{4r}{3}[/tex]
Now, the surface area of the curved surface is given by:
[tex]\rm A = 2\pi r h[/tex]
Now, the surface area of the two hemispheres is given by:
[tex]\rm A'=2\times (2\pi r^2)[/tex]
[tex]\rm A'=4\pi r^2[/tex]
Now, the total area is given by:
[tex]\rm A_T = 2\pi rh+4\pi r^2[/tex]
Now, substitute the value of 'h' in the above expression.
[tex]\rm A_T = 2\pi r\left(\dfrac{10}{\pi r^2}-\dfrac{4r}{3}\right)+4\pi r^2[/tex]
Simplify the above expression.
[tex]\rm A_T = \dfrac{20}{r} + \dfrac{4\pi r^2}{3}[/tex]
Now, differentiate the total area with respect to 'r'.
[tex]\rm \dfrac{dA_T}{dr} = -\dfrac{20}{r^2} + \dfrac{8\pi r}{3}[/tex]
Now, equate the above expression to zero.
[tex]\rm 0= -\dfrac{20}{r^2} + \dfrac{8\pi r}{3}[/tex]
Simplify the above expression in order to determine the value of 'r'.
[tex]8\pi r^3=60[/tex]
r = 1.34 cm
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What is the diference between the
largest and the smallest 4 digit numbers
largest 4 digit number : 9999
smallest 4 digit numbers : 1000
Difference = 9999 - 1000 = 8999
Which of these expressions is the simplified form of the expression (Sin(x)/1-cos^2(x)) tan(x/2)?
Edge 2020
Answer:C
1/1+cos(x)
Step-by-step explanation:
Simplifird form of the given trigonometric expression will be,
[sinx / (1 - cos²x)] × tan(x/2) = 1 /(1 + cosx)
Simplification of a trigonometric expression:
Given expression in the question,
[tex]\frac{\text{sinx}}{1-\text{cos}^2x}\times \text{tan}(\frac{x}{2})[/tex]
= [tex]\frac{\text{sinx}}{\text{sin}^2x}\times \text{tan}(\frac{x}{2} )[/tex]
= [tex]\frac{1}{\text{sin}x}\times \frac{\text{sin}(\frac{x}{2})}{\text{cos}(\frac{x}{2} )}[/tex]
= [tex]\frac{1}{\text{2sin}\frac{x}{2}\text{cos}(\frac{x}{2} ) }\times \frac{\text{sin}(\frac{x}{2})}{\text{cos}(\frac{x}{2} )}[/tex]
= [tex]\frac{1}{\text{2cos}^2{\frac{x}{2} }}[/tex]
Use the identity → ([tex]2\text{cos}^2x=1+\text{cosx}[/tex])
= [tex]\frac{1}{1+\text{cos}x}[/tex]
Hence, [tex]\frac{\text{sinx}}{1-\text{cos}^2x}\times \text{tan}(\frac{x}{2})=\frac{1}{1+\text{cosx}}[/tex] will be the answer.
Learn more to simplify the trigonometric expressions here,
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A bell tolls every 30 minutes on the
hour and at half past the hour. How
many times does the bell toll
between the times of 11.45a.m. and
3.10p.m.?
PLSSSSSSSSSSSSSS HELPPPP MEEEEEEE! ANSWER MY OTHER QUESTION!!!!
Answer:
Step-by-step explanation:
huh??
The graph of an absolute value function opens up and has a vertex of (0, -3).
The domain of the function is
The range of the function is
Answer:
domain (-∞, ∞)
range [-3,∞)
Step-by-step explanation:
domain is the x value and range is the y value. Since its an absolute value the y value rises from both sides of the x value in a 45° angle.
Which number lines have points that represent additive inverses? Check all that apply.
===================================================
Explanation:
Choice B is one answer because -2 and 2 are additive inverses that add to -2+2 = 0
Choice D is a similar story. We have -5+5 = 0
In general, if x is some number then -x is its additive inverse. So we can say x+(-x) = 0 or -x+x = 0. In short, additive inverses add to 0.
Answer:
B and D
Step-by-step explanation:
Colin, Sara and Gordon share some sweets in the ratio 2:5:3. Colin gets 26 sweets. How many sweets are there altogether?
Answer:
130
Step-by-step explanation:
Ratio is 2:5:3. Colin get 2/10 of the sweets so 26/x=2/10. Use the butterfly method to get 260=2x. Divide both sides by 2 and get x=130/
PLEASE HELP
Evaluate 6 + 10v – 8w when v=4 & w= -2
Answer:
6 + 10v – 8w = 62
Step-by-step explanation:
Given the expression
6 + 10v – 8w
as
v=4w=-2substituting the values in the expression
6 + 10v – 8w = 6 + 10(4) - 8(-2)
= 6 + 40 + 16
= 62
I’ll give the brainlest answer! Is this relation a function? Please help
Answer:
show me the choices
Step-by-step explanation:
so i qnswer ir
PLEASE HELP
A car dealer offers a 15% discount off the list price x for any car on the lot. At the same time, the manufacturer offers a $1500 rebate for each purchase of a car.
a. Write a function f(x) to represent the price after the discount is applied.
b. Write a function g(x) to represent the price after the rebate is applied.
Suppose the list price of a car is $19,000. Use a composite function to find the price
of the car:
C. if the discount is applied before the rebate;
D. if the rebate is applied before the discount
Answer:
A) f(x) = 0.85x
B) g(x) = x - 1500
C) g(f(x)) = $14650
D) f(g(x)) = $14875
Step-by-step explanation:
A) The list price is x and a 15% discount is applied.
Thus;
f(x) = x - 15%x
f(x) = 0.85x
B) We are told that the manufacturer offers a $1500 rebate for each purchase of a car.
Thus, the function g(x) to represent the price after the rebate is applied is;
g(x) = x - 1500
C)if the discount is applied before the rebate, the function is;
g(f(x))
Now,
f(x) = 0.85(19000)
f(x) = 16150
g(x) = x - 1500
Thus;
g(x) = 16150 - 1500
g(x) = $14650
D) If discount after rebate, then we have; f(g(x))
g(x) = 19000 - 1500
g(x) = 17500
f(g(x)) = 0.85(17500)
f(g(x)) = $14875
The recently released iPhone XS Max comes in three different storage sizes: 64GB, 256GB, and 512GB. The price of the 64GB phone is $1,099. If prices were proportional to the phone's storage size, how much would the 256GB phone and the 512GB phone cost
A rectangular solar panel has a length of 11 inches shorter than 4 times it’s width if the perimeter of the panel is 218 inches what are the dimensions of the panel
37 % of $2927 is what
Answer:
1082.99
Step-by-step explanation:
2927 - 100%;
x - 37%;
=> x = (37*2927)/100 = 1082.99
A proportional relationship is shown in the table below:
x:
0
1.3
2.6
3.9
5.2
y:
0
1
2
3
4.
What is the slope of the line that represents this relationship?
Graph the line that represents this relationship.
(13−0)
= 10/13
(10−0)
13,10
Slope of the given line is [tex]\frac{10}{13}[/tex] and the equation of the line is [tex]10x - 13y = 0[/tex].
What is the slope of a line passing through two points?The slope of a line(m) that passes through points (x, y) and (x, y) is
[tex]m = \frac{y_{2} - x_{1}}{x_{2} - x_{1}}[/tex]
The given line passes through points (0, 1.3) and (0, 1).
Therefore, slope of the line is
[tex]m = \frac{y_{2} - x_{1}}{x_{2} - x_{1}} = \frac{1 - 0}{1.3 - 0} = \frac{10}{13}[/tex]
Now, the line passes through the point (2.6, 2).
Putting this equation in slope-intercept form, we get:
[tex]2 = \frac{10}{13}(2.6) + c\\c = 2 - 2\\c = 0[/tex]
The equation of the line will be:
[tex]y = \frac{10}{13}x\\10x - 13y = 0[/tex]
Learn more about slope of a line here: https://brainly.com/question/21504186
#SPJ3
Set up an equation and solve for x
Answer:
x=22
Step-by-step explanation:
3x+3+27+x+3+2x+18=180
6x+48=180
6x=132
x=22
david noticed that 49=7, what are other numbers between 1 and 100 have exactly three factors
A large farm tractor costs $98,424. If a farmer makes 36 equal payments, how much is each payment
Answer:
$2734
Step-by-step explanation:
$98,424 / 36 = $2734
What is the slope of the line? PLEASE HELP!
Let's estimate 534–395.
(a) Round each number to the nearest hundred.
534 rounds to
395 rounds to
(b) Subtract the rounded numbers.
I
Answer:
A)
534 rounds to 500
395 rounds to 400
B) 100
Step-by-step explanation:
A)
534 rounds to 500 because the next digit 3 is less than 5 which means that you round down.
395 rounds to 400 because the next digit 9 is greater than 5 which means that you round up.
B)
500-400=100
Hope this helps! :)
5
y=2x+5y, equals, 2, x, plus, 5
Complete the missing value in the solution to the equation.
(
2
,
♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️
[tex]y = f(x) = 2x + 5[/tex]
[tex]f(2) = 2(2) + 5[/tex]
[tex]f(2) = 4 + 5[/tex]
[tex]f(2) = 9[/tex]
Thus ;
[tex]( \: 2 \: , \: 9 \: )[/tex]
♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️
Answer:
9
Step-by-step explanation:
What is the common difference between successive terms in the sequence? 0.36, 0.26, 0.16, 0.06, –0.04, –0.14
Answer:
-.1
Step-by-step explanation:
Is the sum of any 2 consecutive prime numbers also prime?
Answer:
Yes! (it’s making me write 20 letters so yes is ur answer ok cool)
Help please!!!!
What is the absolute value of the number indicated on the number line below?
A. -2/3
B. -3/5
C. 3/5
D. 2/3
Answer:
b
Step-by-step explanation:
it is 3 out of 5 ticks over to the left
What are the steps in solving the equation 4(2x + 3) = 10x?
Answer:
Step-by-step explanation:
Distributive property, subtraction, division
4(2x + 3) = 10x
8x + 12 = 10x
12 = 2x
6 = x
Anthony receives $12 in allowance every week.He currently has $492 saved.How many weeks has he been saving?
Answer:
He has been saving for 41 weeks.
Step-by-step explanation:
492 is the total, we need to divide 492 by 12 to get how many weeks, 41 is the answer
Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; âw âr , âw âθ when r = 4, θ = Ï2
Answer:
δw/δr = 4π
δw/δθ = -8π
Step-by-step explanation:
Given the following functions
w = xy + yz + zx, x = r cosθ y = r sinθ, z = rθ where r = 4 and θ = π/2
We are to find δw/δr and δw/δθ
δw/δr = δw/δx•δx/δr + δw/δy•δy/δr + δw/δz•δz/δr
δw/δx = y+z
δx/δr = cosθ
δw/δy = x+z
δy/δr = sinθ
δw/δz = y+x
δz/δr = θ
Substitute the given values into the formula
δw/δr = (y+z)cosθ+(x+z)sinθ+(y+x)θ
Substitute the value of x, y and z in terms of theta into the resulting function
δw/δr = (y+z)cosθ+(x+z)sinθ+(y+x)θ
δw/δr = (rsinθ+rθ)cosθ+(rcosθ+rθ)sinθ+(rsinθ+rcosθ)θ
Substitute r = 4 and θ = π/2
δw/δr = (4sinπ/2+4π/2)cosπ/2+(4cosπ/2+4π/2)sinπ/2+(4sinπ/2+4cosπ/2)π/2
Note that cos π/2 = 0 and sinπ/2 = 1
δw/δr = (4+2π)(0)+(0+2π)(1)+(4(1)+4(0))π/2
δw/δr = 0+2π+4π/2
δw/δr = 2π+2π
δw/δr = 4π
For δw/δθ
δw/δθ = δw/δx•δx/δθ + δw/δy•δy/δθ + δw/δz•δz/δθ
δw/δx = y+z
δx/δθ = -rsinθ
δw/δy = x+z
δy/δθ =rcosθ
δw/δz = y+x
δz/δθ = r
Substitute the given values into the formula
δw/δθ = (y+z)-rsinθ+(x+z)rcosθ+(y+x)r
Substitute the value of x, y and z in terms of theta into the resulting function
δw/δθ = (y+z)-rsinθ+(x+z)rcosθ+(y+x)r
δw/δθ = (rsinθ+rθ)-rsinθ+(rcosθ+rθ)rcosθ+(rsinθ+rcosθ)r
Substitute r = 4 and θ = π/2
δw/δθ = (4sinπ/2+4π/2)-4sinπ/2+(4cosπ/2+4π/2)4cosπ/2+(4sinπ/2+4cosπ/2)(4)
Note that cos π/2 = 0 and sinπ/2 = 1
δw/δθ = (4+2π)(-4)+(0+2π)(0)+(4(1)+4(0))(4)
δw/δθ = -16-8π+0+4(4)
δw/δθ = -16+16-8π
δw/δθ = -8π
Marty swans 36 yards in 42 seconds. If he continues to swim at the same rate, how long will it take him to swim 500 yards
AB = BC
5(2x + 2).
3(3x - 1)
Find x:
2
4
-10
16
None of the other answers are correct
Answer:
Step-by-step explanation:
holdup im gonna help
Answer:
x = 2
Step-by-step explanation:
2[3(3x - 1)] = 5(2x + 2)
18x - 6 = 10x + 10
8x = 16
x = 2
Find the equation of the line that is parallel to the given line and passes through the given point.
y = −4x + 7; (9, 5)
The equation is y =__________.
Answer:
y= 4 x
Step-by-step explanation:
y =4x − 7
Choose a point that the parallel line will pass through.
(0 , 0 )
Use the slope-intercept form to find the slope.
Tap for fewer steps...
The slope-intercept form is
y = m x+ b , where m is the slope and b is the y-intercept. y = m x + b
Using the slope-intercept form, the slope is
4 . m = 4
To find an equation that is parallel to y = 4 x − 7 , the slopes must be equal. Using the slope of the equation, find the parallel line using the point-slope formula.
( 0 , 0 ) m = 4
Using the point-slope form
y − y 1 = m ( x −x 1 ) , plug in m = 4 , x 1 = 0 , and y 1 = 0 .
y − ( 0 ) = ( 4 )( x − ( 0 ) )
Solve for y.
Subtract 0 from y .
y = ( 4 ) ( x − ( 0 ) )
Subtract 0 from x .
y= 4 x
HOPE THIS HELPS!
pLEASE MARK bRAINLIEST! :)