This exercise is about optimization and seeks to prove that if the new strategy of packing the discs is adopted, then h/r = [tex]\sqrt[4]{3}[/tex]/n ≈2.21.
What is the proof for the above strategy?We must determine the amount of metal consumed by each end, or the area of each hexagon.
The hexagon is divided into six congruent triangles, each of which has one side (s in the diagram) in common with the hexagon.
Step I
Next, let's derive the length of s = 2r tan π/6 = (2/([tex]\sqrt{3}[/tex])r². From this we can state that the area of each of the triangles are 1/2(sr) = (1/[tex]\sqrt{3}[/tex])r²
while the total area of the hexagon is 6 * (1/[tex]\sqrt{3}[/tex])r² = (2/[tex]\sqrt{3}[/tex])r².
From the above, we can state that the quantity we want to minimize is given as:
A = 2πrh + 2* (2/[tex]\sqrt{3}[/tex])r²
Step 2
Next, we substitute for h and differentiate. This gives us:
da/dr = - (2V/r²) + [tex]\sqrt[8]{3r}[/tex].
Let us equate the above to zero.
[tex]\sqrt[8]{3r} ^{3}[/tex] = 2V = 2πr²h ⇒ h/r =[tex]\sqrt[4]{3}[/tex]/n
The above is approximately 2.21
Because d²A/dr²=[tex]\sqrt[8]{3}[/tex] + 4V/r[tex]^{3}[/tex] > 0 the above minimizes A.
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Maths. Please Help. Will Be Rewarded
Answer:
see explanation
Step-by-step explanation:
under a reflection in the x- axis
a point (x, y ) → (x, - y ) , then
(3, 4 ) → (3, - 4 )
(3, 1 ) → (3, - 1 )
(5, 1 ) → (5, - 1 )
Answer:
Points would be (3,-1) (5,-1) (3,-4)
Step-by-step explanation:
So its asking for a reflection on the x-axis, which means you simply take the shape and move the points to the same spot on the other side of the boldest horrizontal line, or in other words, you can just add a negative to the y variable in the points, because the rule is (x,-y).
On a motor trip, Mr. Williams covered 2/6 of his total trip distance by driving 388 miles. The total distance to be covered in miles
is:
A. 1,438
B. 1,164
C. 1,382
D. 1,218
Answer:
B
Step-by-step explanation:
2/6 x d = 388
d = 388 x 6/2 = 1164 m
Answer:
The total distance covered in miles is B, 1164 miles.
Step-by-step explanation:
Since we know that 2/6 of his total trip distance is 388 miles we can do 388 / 2/6 = 1164. Therefore, his total trip distance is 1164 miles.
Write the following equation symbolically
7 divided by x is less than 8 times y
Answer:
7/x < 8y
Step-by-step explanation:
7/x < 8y
What is the lateral area and the surface area?
Answer:
The lateral surface of an object is all of the sides of the object, excluding its base and top (when they exist). The lateral surface area is the area of the lateral surface. This is to be distinguished from the total surface area, which is the lateral surface area together with the areas of the base and top.Step-by-step explanation:
I hope it's helpful for youSimplify the expression.
Answer:
3
Step-by-step explanation:
((7x-4)^2) - (7x-4)(7x-4) + 3
(7x-4)(7x-4) is 7x-4^2
So its (7x-4)^2 - (7x-4)^2 + 3
(7x-4)^2 cancel out.
Answer is 3
1. A parallelogram has a base of 12 centimeters and a height of 8 centimeters.
What is the area of the parallelogram? *
A. 20 cm2
B. 40 cm2
C. 96 cm2
D. 208 cm2
A hose fills a hot tub at a rate of 4.39 gallons per minute. How many hours will it take to fill a 350-gallon hot tub?
Answer:
1.33 hours(If it doesn't let you type that in, just do 1 hour for rounding purposes)
Step-by-step explanation:
First, we take the 350 gal variable and divide that by 4.39, we now get 79.73
Now, we convert that into hours(60mins per hour), so 79.73 divided by 60.
We get 1.33, which is your answer!
solve the system by elimination 4x-y=2 and x+3y=7
Answer:
x=1 y=2
Step-by-step explanation:
Isolate x for 4x -y =2: x=2+y/4
Substitute x= 2+y/4
(2+13y/4) +3y=7
simplify 2+13y/4) =7
Isolate y for 2+13y/4) =7: y=2
Forx= 2+y/4
Substitute y =2
x=2+2/4
x=1
39-42 Differentiate [tex]f[/tex] and find the domain of [tex]f\text{.}[/tex]
39. [tex]f(x)=\frac{x}{1-\ln (x-1)}[/tex]
[tex]\text{Given that,}\\\\f(x) = \dfrac x{1- \ln (x-1)}\\\\\\f'(x)=\dfrac{(1- \ln(x-1))-x\cdot \left(0-\dfrac 1{x-1} \right)}{(1- \ln(x-1))^2}\\\\\\~~~~~~~~= \dfrac{(x-1)(1- \ln(x-1))+x}{(x-1)(1-\ln(x-1))^2}\\\\\\~~~~~~~~= \dfrac{x-x\ln(x-1)-1+\ln(x-1)+x}{(x-1)(1-\ln(x-1))^2}\\\\\\~~~~~~~~= \dfrac{2x-1-(x-1)\ln(x-1)}{(x-1)(1-\ln(x-1))^2}\\[/tex]
[tex]\text{domain f(x):} ~\{x|x > 1, x\neq e+1 \}[/tex]
What does this mean?
See attached image
Answer:
It's a fraction
Step-by-step explanation:
You have to figure out the meaning of the letters and find what goes on top
A rectangular picture has a length of 40 inches and a width of 9 inches.
What is the length of the diagonal of the picture in inches?
Enter your answer in the box.
inches
Step-by-step explanation:
the length and the width of a rectangle create a right-angled triangle, with the diagonal being the Hypotenuse of that triangle (the baseline, opposite of the 90° angle).
so, we can use Pythagoras
c² = a² + b²
with c being the Hypotenuse.
so, we have here
diagonal² = 40² + 9² = 1600 + 81 = 1681
diagonal = sqrt(1681) = 41 in
suppose the acceleration function of an object moving along a line is given by a(t) =0.2t. find the position of the object if you know the initial velocity was v(0) =-3 and initial position was s(0) =1
Answer:
See below
Step-by-step explanation:
sf = s0 + v0 t + 1/2 at^2
1 + (-3)t + 1/2 (.2t) t^2 ( weird acceleration value)
s = 1 -3t + .1 t^3 You will need a 't' value to determine the position
the force needed to break a board varies inversely with its length. If tom uses 20lb of pressure to break a 1.5 -ft board, how many pounds of pressure would he need to use to break a 6 -ft board
[tex]\qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\ \textit{\underline{x} varies inversely with }\underline{z^5} ~\hspace{5.5em} \stackrel{\textit{constant of variation}}{x=\cfrac{\stackrel{\downarrow }{k}}{z^5}~\hfill } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{"f" varies inversely with "L"}}{f=\cfrac{k}{L}}\qquad \textit{we also know that} \begin{cases} f=\stackrel{lbs}{20}\\ L=\stackrel{ft}{1.5} \end{cases} \\\\\\ 20=\cfrac{k}{1.5}\implies 30=k~\hfill \boxed{f=\cfrac{30}{L}} \\\\\\ \textit{when L = 6, what is "f"?}\qquad f=\cfrac{30}{6}\implies f=5[/tex]
What is the value of x?
Answer:
[tex]x = 52[/tex]
Step-by-step explanation:
Step 1: Create an expression
[tex]2x + 76 = 180[/tex]
Step 2: Subtract 76 from both sides
[tex]2x + 76 - 76 = 180 - 76[/tex]
[tex]2x = 104[/tex]
Step 3: Divide both sides by 2
[tex]\frac{2x}{2} = \frac{104}{2}[/tex]
[tex]x = 52[/tex]
Answer: [tex]x = 52[/tex]
Answer:
x = 52
Step-by-step explanation:
the 2 inside angles add up to 180 degress since they lie on the same side on the transversal line. so we set the eqaution up:
2x+76=180
2x=104
x=54
in a significance test of one population mean if vairance of population is unknown but the sample size is 30 or more we use z test why?
The student T distribution would be used since we don't know the population variance (by extension the population standard deviation).
However, if n > 30, then the student T distribution is going to look very similar to the standard normal Z distribution. It won't be a perfect match, but it'll be close enough that it's more simple to use the Z distribution. With the Z distribution, you only have one set of values and you don't have to worry about the degrees of freedom.
If you deposit $50 in checks in your account with $100 cash amount, what is your total deposit?
The three angles inside of a triangle add up to HOW MANY degrees?
180
90
360
120
Answer:
180
Step-by-step explanation:
This is 180 because the specific angles in a triangle will unless an irregularly shaped triangle will equal up to 180.
`y=-x+5` `x-y=1` what the anwers
[tex]y=-x+5~~~~~.....(i)\\\\x-y=1~~~~~~.....(ii)\\\\\\(i)-(ii):\\\\~~~~~y-x+y=-x+5-1\\\\\implies 2y=4\\\\\implies y= \dfrac 42\\\\\implies y=2\\\\\text{Substitute y=2 in eq (ii):}\\ \\~~~~ x-2 = 1\\\\\implies x= 1+2\\\\\implies x = 3\\\\\\\text{Hence,}~~ (x,y) = (3,2)[/tex]
What is the volume, in cubic meters, of a cylinder with a height of 3 meters and a base radius of 4 meters, to the nearest tenths place?
Answer: If I did it correctly it should be 150.8
Step-by-step explanation:
The formula for the volume of a cylinder is height x π x (diameter / 2)2, where (diameter / 2) is the radius of the base (d = 2 x r), so another way to write it is height x π x radius2.
Hope this helps!
what is the primeter of 420 area with 5 m side
Determine whether each relation is a function.
Answer:
yes
Step-by-step explanation:
yes
each input (x) has exactly one output (y)
in other words
it passes the vertical line test
sare sold in packs and boxes.
holds 20 tissues and a box holds 120 tissues.
s P packs of tissues and boxes of tissues.
ys a total of tissues
own a formula for Tin terms of P and B.
Answer:
[tex]derivative n thof \: sin2x [/tex]
can somebody help me please
NO LINKS
Answer:
The function is C(d) = 0.69d+25.95. The equation will be 0.69(57)+2595.
for the last question 0.69d+25.95 = 57, so 0.69d = 31.05, so d is 45
The max distance you can drive with 120 dollars is 45 miles.
which matrix is equal to ( -6 -6.5 1.7 2 -8.5 19.3 )
The matrix that is equal to the considered matrix [-6,-6.5,1.7; 2, -8..5, 19.3 ] is given by: Option D: [-6,-6.5,1.7; 2, -8..5, 19.3 ]
When are two matrices called being equal?Two matrices are equal if and only if their shapes are same, and they've got the same elements for each corresponding row and column positions.
The missing options are specified in the image attached below.
The considered matrix is:
[tex]\left[\begin{array}{ccc}-6&-6.5&1.7\\2&-8.5&19.3\end{array}\right][/tex]
It has 2 rows and 3 columns.
Evaluating each of the options:
Option A and B are wrong as they've got 3 rows and 2 columns, which makes their shape not matching with the shape of the considered matrix.
Option C although has got 2 rows and 3 colums, but its elements are not matching with corresponding elements of the considered matrix.
Option D has 2 rows and 3 colums and its each element matches with the corresponding element of the considered matrix. (ie, for example, element common in first row and first column in option D is -6, and so as for the considered matrix, and similarly, all correspoding elements for each row and column is same in both matrices.)
Thus, the matrix that is equal to the considered matrix [-6,-6.5,1.7; 2, -8..5, 19.3 ] is given by: Option D: [-6,-6.5,1.7; 2, -8..5, 19.3 ]
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Answer:
D hope u get good grade
Step-by-step explanation:
edge 2023
help i need big brain answers!!!! At what position on the number line is the red dot located?
Answer:
it is 57
Step-by-step explanation:
Answer:
A. √57
Step-by-step explanation:
√47 would be off the number line because it's below the number 7
√65 would be past 8 but the red dot comes before the number 8
√72 would be past 8 as well and it would be past √65 but the dot is before number 8
This leaves our answer as √57 beause the square root of 57 is 7.549 (etc.) which is between the numbers 7 and 8, while also being slightly higher than 7.5 :)
Have a good day!
4. Jean borrows $12,000 for 8 years at 6% interest. What is the approximate cost of
the loan? What is Jean's approximate total cost? What is Jean's approximate
monthly payment?
Pls solve!!!
A lot is 60 m by 38 m. A house 32 m by 9 m is built on the lot. How much area is left over?
Answer:
1992
Step-by-step explanation:
60x38=2280
32x9=288
2280-288=1992
Hope it helps!
At least 40% of all arsonists are under 21 years old
Answer:
Step-by-step explanation: yes, we belive that they start young with pyromania. They may see as something powerful
59. If [tex]\lim _{x \rightarrow 1} \frac{f(x)-8}{x-1}=10,[/tex] find [tex]\lim _{x\rightarrow1}f(x)\text{.}[/tex]
Answer:
[tex]\large\lim_{x\rightarrow 1} f\left( x\right)=8[/tex]
Step-by-step explanation:
[tex]\lim_{x\rightarrow 1} \frac{f\left( x\right) -8}{x-1} =10[/tex]
[tex]\lim_{x\rightarrow 1} f\left( x\right) -8=10\times \lim_{x\rightarrow 1}\left( x-1\right)[/tex]
[tex]\lim_{x\rightarrow 1} f\left( x\right) -8=10\times\left( (1)-1\right)[/tex]
[tex]\lim_{x\rightarrow 1} f\left( x\right) -8=10\times\left( 0\right)=0[/tex]
[tex]\lim_{x\rightarrow 1} f\left( x\right) -8=0[/tex]
[tex]\lim_{x\rightarrow 1} f\left( x\right) =8[/tex]
Question
Write the following function in terms of its cofunction.
tan(78)
Answer:
trigonometric function whose value for the complement of an angle is equal to the value of a given trigonometric function of the angle itself the sine is the cofunction of the cosine.