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
For more information, refer to the link given below:
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The average of 5 quiz grades is 10. When the lowest grade is dropped and the new average is calculated, it turns out to be 11. What was the score of the dropped grade?
Answer:
The score of the dropped grade is 6
Step-by-step explanation:
To find the average of a set of numbers you need to add all the numbers and divide by how many numbers there are. The problem wants you to solve it in reverse. Let put this into an algebraic equation with a, b, c, d, and e being the variables of the test scores
(a+b+c+d+e)/5=10
Then we can multiply both sides by 5 and get
a+b+c+d+e=50
Lets assume that c is the lowest test score. To calculate the average of that we get
a+b+d+e/4=11
Doing the same thing, we know that a+b+d+e=44
Now compare the two:
a+b+c+d+e=50
a+b+d+e=44
We can now know that c=50-44=6
So, 6 is the score of the dropped quiz. Hope that helped!
If P is the orthocenter of △ABC, AB = 13, BF = 9,and FC = 5.6, find each measure & the perimeter of triangle ABC.
Answer:
[tex]Perimeter = 38.6[/tex]
Step-by-step explanation:
Given
[tex]A\ B = 13[/tex]
[tex]B\ F = 9[/tex]
[tex]F\ C = 5.6[/tex]
Required
Calculate the perimeter of A B C
To calculate the perimeter of A B C, we sum up the sides of the triangle
i.e.
[tex]Perimeter = A B + B\ C + A\ C[/tex]
AB is given in the question already.
So, we need to calculate B F and F C
To solve for BF, we consider triangle B F C
Using Pythagoras Theorem, we have:
[tex]B\ C^2 = B\ F^2 + F\ C^2[/tex]
Substitute values for BF and FC;
[tex]B\ C^2 = 9^2 + 5.6^2[/tex]
[tex]B\ C^2 = 81 + 31.36[/tex]
[tex]B\ C^2 = 112.36[/tex]
[tex]B\ C^2= \sqrt{112.36[/tex]
[tex]B\ C = 10.6[/tex]
Next, we calculate A C.
To calculate A C, we need to calculate A F, considering triangle B F A
Using Pythagoras Theorem, we have:
[tex]A\ B^2 = B\ F^2 + A\ F^2[/tex]
Substitute values for B F and A B;
[tex]13^2 = 9^2 + A\ F^2[/tex]
[tex]169= 81 + A\ F^2[/tex]
[tex]A\ F^2 = 169 - 81[/tex]
[tex]A\ F^2 = 88[/tex]
[tex]A\ F = \sqrt{88[/tex]
[tex]A\ F = 9.4[/tex]
Since F lies on line A C:
[tex]A\ C = A\ F + F\ C[/tex]
[tex]A\ C = 9.4 + 5.6[/tex]
[tex]A\ C = 15.0[/tex]
Recall that:
[tex]Perimeter = A\ B + B\ C + A\ C[/tex]
[tex]Perimeter = 13 + 10.6 + 15.0[/tex]
[tex]Perimeter = 38.6[/tex]
Hence, the perimeter is 38.6
Given that P is the orthocenter of triangle ABC, applying the properties of an orthocenter of a triangle and the Pythagorean Theorem, each measure and the perimeter of △ABC are:
AB = 13BC = 10.6AC = 15Perimeter = 38.6Recall:
An altitude is the line that connects the vertex of a triangle to the opposite side of which it is perpendicular to that side.Orthocenter is the point where all three altitudes intersect each other.Given that P is the orthocenter of triangle ABC, where,
AB = 13BF = 9FC = 5.6Perimeter of △ABC = AB + BC + AC
First, we need to find BC and AC.
△BFC is a right triangle, apply Pythagorean theorem to find BC:
[tex]BC = \sqrt{BF^2 + FC^2}[/tex]
Substitute[tex]BC = \sqrt{9^2 + 5.6^2}\\\\\mathbf{BC = 10.6}[/tex]
△BFA is a right triangle, apply Pythagorean theorem to find FA:
[tex]FA = \sqrt{AB^2 - BF^2}[/tex]
Substitute[tex]FA = \sqrt{13^2 - 9^2}\\\\\mathbf{FA = 9.4}[/tex]
AC = FA + FC
SubstituteAC = 9.4 + 5.6
AC = 15
Perimeter of △ABC = AB + BC + AC
SubstitutePerimeter of △ABC = 13 + 10.6 + 15
Perimeter of △ABC = 38.6 units
Therefore, given that P is the orthocenter of triangle ABC, applying the properties of an orthocenter of a triangle and the Pythagorean Theorem, each measure and the perimeter of △ABC are:
AB = 13BC = 10.6AC = 15Perimeter = 38.6Learn more here:
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58 students on a bus how many students were fit on a six buses
Answer:
348
Step-by-step explanation:
What is the area of a square with sides of length 9?
O A. 54 square units
B. 81 square units
O C. 18 square units
O D. 36 square units
SUE
I WILL GIVE BRAINLIEST! ALSO KEEP AWARE!
Answer:
Step-by-step explanation:
An equation y = kx represent a proportional relation and k = [tex]\frac{y}{x}[/tex]
⊕ Yes, the constant of proportionality is 3.5
Evaluate the given integral by making an appropriate change of variables. 3 x − 6y 8x − y dA, R where R is the parallelogram enclosed by the lines x − 6y
Answer:
Step-by-step explanation:
Hope this helps!
Please mark brainliest!
Oh if you need more information here is a more of an explanation
You get x and y in terms of u and v by setting u=x-2y and v=3x-y and then doing some systems of equations. You get the Jacobian by taking the partial derivatives of x and y with respect to u and v (using the equations x=(2v-u)/5 and y=(v-3u)/5) and then taking their determinate.
Cole is saving money at a constant rateSuppose he initially has $190 saved, and after 3 months, he has $265 saved. Which of these expresses the rate at which Cole is saving?
Answer:
His unit rate per month is 25 dollars
Step-by-step explanation:
Cole has saved 25 dollars per month. by subtracting 190 from 265 you are left with 75. when left with 75 you can divide by 3 for the the 3 months hes been saving his money. 75 divided by 3 is 25.
EASY QUESTION WILL GIVE BRAINLIEST!!!
Tyler has a baseball bat that weighs 28 ounces.
Find this weight in kilograms and in grams. (Note: 1 kilogram 35 ounces).
Make a table
How is 2+ (2) similar to adding 2+2 how is different explain your answer
Answer:
2+2 is 4-1=3 quick maths
Step-by-step explanation:
Answer:
2+(2) is the same to adding 2+2 because the "(" " )" just mean the you have to add the 2 + 2 just like----- 2+2
A T-Rex made 16 waffles for his friends for breakfast. 5 were blueberry waffles and the rest were chocolate chip waffles. How many chocolate chip waffles did he make?
Answer: 11 waffles were chocolate chip
Step-by-step explanation:
16-5=11 waffles were chocolate chip
Tom bought a new mirror for his house. One side of the mirror is 38 inches and the other side is 30 inches. What is the length of the diagonal of the mirror? a. 68 inches b. 23 inches c. 34 inches d. 48 inches
Answer:
d. 48 inches
Step-by-step explanation:
We assume the mirror is a rectangle
We solve for the length of a diagonal using Pythagoras Theorem
a² + b² = c²
c = √a² + b²
Where c = diagonal
a = 30 inches
b = 38 inches
c = √30² + 38²
c = √2344
c = 48.414873748 inches
Approximately = 48 inches
HELP ME PLEASE
What is 50% of 36?
A. 9
B. 16
C. 18
D. 34
Answer:
18 C
Step-by-step explanation:
hope this helps!!! :]
Answer:18
Step-by-step explanation:
What is the area of the figure?
A. 36 cm2
B. 26 cm
C. 15 cm
D. 42 cm2
Jay works at a sandwich shop. He needs to make 7 turkey sandwiches. Each sandwich will have 1/3 of a pound of turkey. How many pounds of turkey will Jay need to make the sandwiches? Remember what of means! And remember to press submit after you answer the question
Answer:
Jay will need 2 and 1/3 pounds of turkey.
Step-by-step explanation:
Since Jay uses 1/3 per sandwich and is making 7, multiply 7 by 1/3.
That gets you 7/3s
Convert it into a mixed number.
3 goes into 7 twice, with one third left over.
So, Jay needs 2 1/3.
hope this helped!
Answer:
approximately 2.3 pounds of turkey. Trust the BlueDragon
Step-by-step explanation:
7 times 1/3= 2.33333
A license plate is to consist of digits followed by uppercase letters. Determine the number of different license plates possible if the first and second digits must beâ odd, and repetition is not permitted.A. 71, 610, 739, 200,000 B. 795, 674, 880,000 C. 7, 894, 720 D. 8, 840, 832,000
Answer:
D. 8, 840, 832,000
Step-by-step explanation:
The computation of the number of license plates possible is shown below:
Here the first position could be taken by any of the five odd numbers
The second position could be taken by any of the 4 odd numbers
The third position could be taken by the eight left numbers
The fourth position could be taken by another seven numbers
Als the first digit could be any of the 26 upper case letters and so on
So,
= 5 × 4 × 8 × 7 × 26 × 25 × 24 × 23 × 22
= 8,840,832,000
Therefore option D is correct
Dale has written too many checks and his bank balance is –42.68. The bank will charge him an overdraft fee of $25. How much does he need to deposit so that his bank balance will be $50?
Answer:
$117.68
Step-by-step explanation:
You have to add them all together, because if he has a negative amount, he first needs to get that to 0. He then pays the overdraft fee, keeping him at 0. Lastly, he pays $50 to put it into his account. Good luck!
will mark brainlist
One number is 9 more than twice the other number. Their sum is 54. Find the numbers.
Use the Distributive Property to simplify
1/7 x (56 - 1/4)
Answer:
7.96428571429
Step-by-step explanation:
Round you the nearest number.
Jackson's bakery sold 291 cheesecakes in one day. How much money did Jackson's bakery make in one day from cheesecakes?
2. Determine whether quadrilateral EFGH can
be classified as a parallelogram.
Justify your reasoning.
Now consider quadrilateral EFGH shown
Answer:
3.4
Step-by-step explanation:
Answer:
yes
Step-by-step explanation:
there are two sets of parrallel lines.
A restaurant bill is $75. You want to leave about a 20% tip. Which is a 20% tip?
Lillian is comparing the amount of water in her bathtub and her sink. Her bathtub is 0.65 full of water. Her sink is 0.9 full of water.
Which statement is true about comparing the amount of water Lillian has in her bathtub and sink? (MGSE4.NF.7)
There is more water in the bathtub because 65 > 9
These amounts cannot be compared because decimals can only be compared if they have the same-sized whole.
These amounts cannot be compared because decimals can only be compared if the have the same place value
There is more water in the sink because 65/100 < 90/100
Answer:
Step-by-step explanation:
There is more water in the bathtub because 65 > 9
These amounts cannot be compared because decimals can only be compared if they have the same-sized whole.
These amounts cannot be compared because decimals can only be compared if the have the same place value
There is more water in the sink because 65/100 < 90/100
The use of performance materiality should:_____________.A. reduce the probability that the sum of immaterial and/or undetected misstatements in the financial statements is greater than materiality for the financial statements as a whole.B. increase the probability that the sum of immaterial and/or undetected misstatements in the financial statements is greater than materiality for the financial statements as a whole.C. reduce the probability that the sum of immaterial and/or undetected misstatements in the financial statements is less than materiality for the financial statements as a whole.D. increase the probability that the sum of immaterial and/or undetected misstatements in the financial statements is less than materiality for the financial statements
Answer:
C. reduce the probability that the sum of immaterial and/or undetected misstatements in the financial statements is less than materiality for the financial statements as a whole.
Step-by-step explanation:
This is the likely answer going by the relationship between the undetected misstatement and the materiality for the financial statements.
riddle #2
I am a five digit even number.
all of my digits are factor of 24.
4/5 of my digits are even.
The digit in my thousands place equals the perimeter of a 2×2 array.
The digits in my ten thousand places is twice the value of the digits in the hundreds place, but half the value of the digits in the thousands places.
The digit in my tens places equal the number of lines of symmetry in an isosceles triangle.
Answer:
28416
Step-by-step explanation:
No digits can be repeated
"All my digits are factors of 24" This means that the possible digits are: 1, 2, 3, 4, 6, 8
"The digits in my thousands place equals the perimeter of a 2x2 array" That would be 8
"The digits in my ten thousand places is twice the value of the digits in the hundreds place, but half the value of the digits in the thousands place" That would be 4, which also means that the ten thousands place is 2
"The digits in my tens place equal the number of lines of symmetry in an isoceles triangle" There is only 1 lines of symmetry
This means that we have 2841(?)
Because this number is even and no digits can be repeated, the only possible digits from those available is 6
Therefore the number is 28416
Grace drew a triangle with sides of equal length. Its measured with sides of 3 cm. What is the perimeter of the triangle?
Answer:
9 cm explained to
Step-by-step explanation:
explained that
I need help please, i don’t know how to solve it?
Answer:
b. 22 square root of 58 is 7.6 multiplied to 3 is 22
Answer:
B 22
Step-by-step explanation:
Which of these figures shows parallel lines? Help please
Answer:
B a parrelel line means there not inerscting
Step-by-step explanation:
14. (01.04)
On the graph of the equation 3x + 2y = 18, what is the value of the y-intercept?
-9
-6
6
9
Answer:
6
Step-by-step explanation:
U divide 18 by 3.
Answer:
9
Step-by-step explanation:
In slope-intercept form of linear equation y = mx + b , value of "b" is y-intercept.
Let solve 3x + 2y = 18 for "y"
2y = - 3x + 18
y = [tex]-\frac{3}{2}[/tex] x + 9 ⇒ 9 is y-intercept.
===========
There is another way to solve this kind of problems:
Coordinates of y-intercept are (0, y) .
3(0) + 2y = 18
2y = 18
y = 9
(0, 9)
PLZZ help me with this I will give brilliantis to whoever gets it right
Answer:
C
Step-by-step explanation:
Yan has already finished Three-fifths of the 45 math problems he was assigned today. Each math problem took him 1Four-fifths of a minute to complete. If this pace continues, how much more time, to the nearest minute, will the rest of the problems take him to finish?
Answer:
49 Minutes
Step-by-step explanation:
1. 27 = 1 4/5 Minutes per 1 question.
2. Each question takes 108 seconds.
3. By this Logic, we can do 108 seconds x 27 questions to get-
4. 48.6 Minutes
5. And finally we round to the nearest minute so we get 49 minutes in total.
Answer:
32 minutes
Step-by-step explanation:
Yan has already finished Three-fifths of the 45 math problems he was assigned today. Each math problem took him 1Four-fifths of a minute to complete. If this pace continues, how much more time, to the nearest minute, will the rest of the problems take him to finish?