a right circular cylinder is inscribed in a sphere of radius r. find the largest possible surface area of such a cylinder
Answers
For a inscribed right circular cylinder in sphere of radius r, largest possible surface area of such a cylinder is equals to the [tex]= πr²( 1 + \sqrt{5}) [/tex].
The maximum or minimum value of a continuous function can be determined by derivatives of the function. If y= f(x) is a continuous function, Then slope of function at extreme points( maximum or minimum) is zero, [tex]\frac{dy}{dx} =0[/tex]. Let's consider r be the radius of the sphere and is a constant. Let R and h be the radius and height of the cylinder that can be inscribed in a sphere of radius r as shown in the above figure. By Pythogoras Theorem, base radius of cylinder as,[tex]R =\sqrt{ r²-\frac{h²}{4}}[/tex]
Surface area of cylinder, S = 2πRh + 2πR²
=> [tex]S= 2πh\sqrt{ r²- \frac{h²}{4}} + 2π(r²- \frac{h²}{4}) \\ [/tex]. We need to calculate height of cylinder such that surface area is maximum. For surface area to be maximum, [tex] \frac{dS}{dh} =0[/tex]
[tex]2πh \frac{1}{2\sqrt{ r² - \frac{h²}{4}}}( \frac{ - 2h}{4} ) + 2π\sqrt{ r² - \frac{h²}{4}} + 2π \frac{(-2 )h}{4} = 0 \\ [/tex]
=>[tex] \frac{- h²}{4\sqrt{ r² - \frac{h²}{4}} } + \sqrt{ r² - \frac{h²}{4}} - \frac{h}{2} = 0[/tex]
=> [tex]- h² + 4 ( r² - \frac{h²}{4} ) - 2h\sqrt{ r² - \frac{h²}{4}} = 0 \\ [/tex]
[tex]- 2h² + 4r² - 2h\sqrt{ r² - \frac{h²}{4}} = 0[/tex]
=>[tex]- 2h²+ 4r² = 2h\sqrt{r²- \frac{h²}{4}}[/tex]
Squaring on both sides, [tex]4h^{4} + 16r⁴ - 16r²h² = 4h² ( r² - \frac{h²}{4}) = 0 \\ [/tex]
[tex]h^{4} + 4r⁴ - 4r²h² = h² r² - \frac{h⁴}{4} = 0 \\ [/tex]
=> [tex]5h^{4} + 4r⁴ - 5r²h² = 0 [/tex]
which is Quadratic equation with h² variable, so using quadratic formula for determining the values of h², [tex]=4r^{2} ( \frac{5 ± \sqrt{5}}{10})[/tex]. Since we need largest possible surface area, we will consider, [tex]h² = 4r²( \frac{ 5 - \sqrt{5}}{10})[/tex]
=> [tex]r² - \frac{h²}{4} = r²( \frac{ 5 + \sqrt{5}}{10})[/tex]
Substituting the values in the surface area equation, [tex]S= 2πh\sqrt{ r² - \frac{h²}{4}} + 2π(r² - \frac{h²}{4}) \\ [/tex]
[tex]= 4πr²(\sqrt{ \frac{5 - \sqrt{5}}{10}})(\sqrt{ \frac{ 5 + \sqrt{5}}{10}} )+ 2πr²( \frac{ 5 + \sqrt{5}}{10}) \\ [/tex]
[tex]= 2πr²( \frac{ 4\sqrt{5}}{10} + \frac{ 5 + \sqrt{5}}{10} ) \\ [/tex]
=> [tex]= πr²( 1 + \sqrt{5}) [/tex]
Hence, required area is [tex]= πr²( 1 + \sqrt{5}) [/tex].
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Related Questions
PLEASE HELP! SHOW WORK AND EXPLAIN ON HOW YOU GOT THE ANSWER! I WILL MARK YOU BRAINLIEST IF YOU DO!
Answers
The value of the surd form √14 is 3. 74
What is square root?The square root of a number is a value that, when multiplied by itself, equals the original number.
It is represented by the symbol √ and is a way to find the side length of a square given its area or the length of a diagonal in a right triangle.
From the information given, we have that;
The expression is √14
We can see that the value √14 is a surd which can no longer be simplified into a whole or rational number.
To determine the square root, we use the calculator
√14
3. 7416
In the nearest tenth, we get;
3. 74
Hence, the value is 3. 74
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The length of PR is 18. Find the length of ST.
Round your answer to the nearest tenth.
Answers
The length of ST obtained using Pythagorean Theorem is about 13.2 units
What is the Pythagorean Theorem?Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is the sum of the square of the other two legs of the triangle.
The length of the segment [tex]\overline{PR}[/tex] = 18
[tex]\overline{RS}[/tex] = [tex]\overline{SP}[/tex] = 16 (Radial length of the same circle)
Triangle ΔSRP is an isosceles triangle, therefore, ∠SRP ≅ ∠SPR
[tex]\overline{ST}[/tex] ≅ [tex]\overline{ST}[/tex] (Reflexive property of congruence)
Triangles ΔRST and ΔPST are congruent by the Hypotenuse Leg, HL rule of congruent triangles
Therefore; RT ≅ PT
RT + PT = 18
RT + RT = 18
2 × RT = 18
RT = 18/2 = 9
RT = 9
Therefore, ST, can be obtained using Pythagoras Theorem as follows;
ST = √(16² - 9²) = 5·√7 ≈ 13.2
The length of the side ST is about 13.2 units long
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The term of an AP is 75 and the 25th is also 75. Find the common difference and the sum of the 100terms
Answers
According to arithmetic progression, the sum of the first 100 terms is 7500 and the common difference is 0.
In this case, we know that the first term (a1) is 75, and the 25th term (a25) is also 75. So we can use the formula with n = 25:
common difference = (a25 - a1) / (25 - 1)
common difference = (75 - 75) / 24
common difference = 0
Since the common difference is 0, this means that all the terms in the sequence are the same. In other words, this is not really an arithmetic progression - it's just a sequence of 100 terms, all of which are equal to 75.
To find the sum of the first 100 terms, we can use the formula:
sum = (n/2)(first term + last term)
where n is the number of terms, and the last term is the 100th term.
In this case, we know that the first term (a1) is 75, and all the terms are the same. So the last term (a100) is also 75. Therefore, we can use the formula with n = 100:
sum = (100/2)(75 + 75)
sum = 50 x 150
sum = 7500
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The circle with center F is divided into sectors. In circle F, EB is a diameter. The radius
of circle F is 4 units.
What is the length of the arc corresponding to the subtended central angle ∠EFD?
Answers
The answer of the given question based on the circle is , the answer is option (d) 8/3π.
What is Arc?In geometry, an arc is portion of circumference of a circle. It is curved line that connects two points on circle. The length of arc is proportional to the angle that it subtends at the center of the circle.
Since EFB is a straight line and EB is a diameter of the circle, angle EFB subtends the entire circle at F. Therefore, the angle subtended by arc EFD is the difference between the angles subtended by arcs EF and EB.
Arc EF subtends a full circle at F, so its angle is 2π radians. Arc EB subtends a straight angle at F, so its angle is π radians.
Therefore, the angle subtended by arc EFD is:
2π - π = π
The length of arc is given by formula L = rθ, where r is radius of circle and θ is angle (in radians) subtended by arc at center of circle.
In this case, r = 4 and θ = π, so the length of the arc corresponding to angle EFD is:
L = rθ = 4π
Therefore, the answer is (d) 8/3π.
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Determine if the following system of equations has no solutions,infinitely many solutions or exactly one solution -x+5y=-8 2x-10y=10
Answers
Answer:
Step-by-step explanation:
[Part A] Which is the ratio for sin A, written as a fraction in simplest form?
Answers
The ratio for sin A, written as a fraction in simplest form, is 3 / (2√3).
The sine function is a mathematical function that takes an angle as its input and returns the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right triangle
In a right triangle, the sine of an angle A is defined as the ratio of the length of the side opposite to angle A to the length of the hypotenuse. Therefore, we have:
sin A = Opposite / Hypotenuse
Substituting the given values, we get
sin A = CB / AB = 8√3 / 16 = √3 / 2
To simplify this fraction further, we can multiply both the numerator and denominator by √3:
sin A = (√3 / 2) × (√3 / √3) = 3 / (2√3)
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The given question is incomplete, the complete question is:
Which is the ratio for sin A, written as a fraction in simplest form?
the graph of p(x)=2(x-4) is translated 2 units right and down 1 unit. What is the equation of the translated function q(x) in slope intercept form
Answers
The equation of the translated function q(x) in slope-intercept form is q(x) = 2x - 13.
What is the equation of the translated function q(x)The function p(x) = 2(x - 4) can be rewritten in slope-intercept form as:
p(x) = 2x - 8
To translate the graph of p(x) two units to the right and one unit down, we need to make the following adjustments to the equation:
To shift the graph two units to the right, we need to replace x with (x - 2).To shift the graph one unit down, we need to subtract 1 from the entire equation.So the equation of the translated function q(x) in slope-intercept form is:
q(x) = 2(x - 2) - 8 - 1
Simplifying, we get:
q(x) = 2x - 13
Therefore, the equation of the translated function q(x) in slope-intercept form is q(x) = 2x - 13.
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ankara güçlendiren 8.sınıf kurumsal deneme 3 cevap anahtarı ank-2303
Answers
Answer:
I found a video that provides the answer key for the Ankara Güçlendiren
Step-by-step explanation:
Solve graphically the system of linear equations:
x+2y=4
−2x+5y=10
Answers
The required, graph of both lines has been shown where the common solution is (0, 2).
I apologize for the error in my previous solution. Here's the corrected solution:
To solve the system of linear equations x + 2y = 4 and -2x + 5y = 10 graphically, we need to plot the graphs of the two equations on the same set of axes and find the point where they intersect.
The graph of both lines has been shown where the common solution is (0, 2).
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Quadrilateral ABCD has diagonals AC and BD. Which information
is not sufficient to prove ABCD is a parallelogram?
(1) AC and BD bisect each other.
(2) AB CD and BC
(3) ABCD and AB
(4) ABCD and BC
Answers
On solving the provided query we have Another characteristic of parallelograms is that one of the diagonals, BC, bisects the other side, AD, as stated in statement (4).
What is equation?A mathematical equation is a formula that connects two claims and uses the equals symbol (=) to denote equivalence. An equation in algebra is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign places a space between the variables 3x + 5 and 14. The relationship between the two sentences that are written on each side of a letter may be understood using a mathematical formula. The symbol and the single variable are frequently the same. as in, 2x - 4 equals 2, for instance.
The facts (3) ABCD and AB are insufficient to demonstrate that ABCD is a parallelogram.
The diagonals are bisected by each other, which is a characteristic of parallelograms, according to statement (1).
According to statement (2), opposing sides of a parallelogram are congruent and parallel, which is another characteristic of parallelograms.
Another characteristic of parallelograms is that one of the diagonals, BC, bisects the other side, AD, as stated in statement (4).
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Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle C?
Answers
The measure of angle C is 80°
What is a cyclic quadrilateral?A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It can also be called inscribed quadrilateral.
A theorem if circle geometry states that the opposite angles of a cyclic quadrilateral are supplementary. This means the sum of. opposite angles of a cyclic quadrilateral is 180°
Therefore 3x-59+2x-1 = 180
5x -60 = 180
5x = 180+60
5x = 240
x = 240/5
x = 48
angle A = 2x+4 = 2×48+4
= 100°
therefore angle C = 180-(angle A)
C = 180-100
= 80°
therefore the measure of angle C is 80°
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Bob and Carl each rented the same kind of moving truck from EZ Move. There was a flat rental fee plus a charge per mile that the truck was driven. Bob’s cost for his truck was $112. 96 for 138 miles. Carl’s cost for his truck was $142. 78 for 209 miles. Which equation can be used to represent the cost of the rental truck?
Round to the nearest hundredth if necessary. Y = 71 x minus 29. 82
y = 25 x minus 66
y = 0. 42 x + 71
y = 0. 42 x + 55
Answers
The equation that represents the cost of the rental truck is y = 0.42x + 71. The Option C is correct.
Which equation represents cost of rental truck?Let x be the number of miles driven
Let y be the cost of the rental truck.
We can set up a system of equations to represent the given information:
For Bob:
y = mx + b
112.96 = 138m + b
For Carl:
y = mx + b
142.78 = 209m + b
Solving for b in both equations, we get:
b = 112.96 - 138m
b = 142.78 - 209m
Setting the two expressions for b equal to each other, we get:
112.96 - 138m = 142.78 - 209m
209m - 138m = 142.78 - 112.96
71 m = 29.82
m = 0.42
Substituting m into equation for b:
b = 71.
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PLEASE HELPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
Answers
Answer:
it's the first one. I
n my thinking
7. Julie had a balance of $1,189.17 in her checking account. She wrote checks for $62.41, $224.14, $12.92, and $357.16. Her deposits were $197.34 and $879.13. What was Julie's new bank balance?
Answers
According to the given information, Julie's new bank balance was $1,609.01
To calculate Julie's new bank balance, we need to subtract the total amount of checks she wrote from her initial balance, and then add the total amount of deposits she made.
Total amount of checks = $62.41 + $224.14 + $12.92 + $357.16 = $656.63
Total amount of deposits = $197.34 + $879.13 = $1,076.47
New bank balance = Initial balance - Total amount of checks + Total amount of deposits
New bank balance = $1,189.17 - $656.63 + $1,076.47
New bank balance = $1,609.01
Therefore, Julie's new bank balance was $1,609.01.
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The cost of 1 hat and 1 bag is £27.
The cost of 2 hats and 1 bag is £42.
(a) How much does 1 hat cost?
(b) How much does 1 bag cost?
Answers
Answer:
(a) 1 hat is $15
(b) 1 bag is $12
Step-by-step explanation:
1 hat at $15 + 1 bag at $12 is equal to $27
2 hats at $30 together + 1 bag at $12 is equal to $42
A certain number is increased by 6 and then multiplied by 4. This equals 94.
Answers
Answer:
22
i took this one
Answer:
17.5
Step-by-step explanation:
94 divided by 4 is 23.5 and I subtracted 6 from 23.5
how many different license plates consist of five symbols either digits or letters
Answers
There are 60,466,176 different license plates that consist of five symbols, either digits or letters.
To calculate the number of different license plates consisting of five symbols, we need to consider the number of choices for each symbol, which can be either digits (0-9) or letters (A-Z).
Determine the number of choices for each symbol.
There are 10 digits (0-9) and 26 letters (A-Z), so there are a total of 10 + 26 = 36 possible choices for each symbol.
Use the counting principle.
Since there are 5 symbols on the license plate and 36 choices for each symbol, we can use the counting principle to determine the number of different license plates.
The counting principle states that if there are n ways to do one thing and m ways to do another, then there are n x m ways to do both.
Calculate the number of different license plates.
The number of different license plates can be calculated as 36 × 36 × 36 × 36 × 36 = [tex]36^5[/tex] = 60,466,176.
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There are a total of 60,466,176 different license plates consisting of five symbols, either digits or letters. This is because there are 26 letters in the English alphabet and 10 digits, so the total number of symbols is 36.
To calculate the number of different license plates consisting of five symbols, we need to consider that each symbol can be either a digit (0-9) or a letter (A-Z). There are 10 digits and 26 letters, so there are 36 possible choices for each symbol.
To find the total number of different license plates, we use the following steps:
1. Determine the number of possible choices for each symbol (36, as explained above).
2. Since there are five symbols in the license plate, raise the number of choices (36) to the power of 5.
3. Calculate the result.
So, the calculation would be:
36^5 = 60,466,176
There are 60,466,176 different license plates that can be created using five symbols with either digits or letters.
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Carmen deposits $600 into an account that pays simple interest at a rate of 2% per year. How much interest will she be paid in the first 3 years
Answers
Carmen will be paid $36 in interest in the first 3 years on her deposit of $600 at a simple interest rate of 2% per year.
Given
Principal: $600
Annual interest rate: 0.02
Time period: 3
To Find
Interest ?
Solution
To calculate the interest that Carmen will be paid in the first 3 years on her deposit of $600 at a simple interest rate of 2% per year, we can use the formula:
I = P * r * t
where I is the interest, P is the principal (the initial amount deposited), r is the annual interest rate (as a decimal), and t is the time period (in years).
Substituting the given values, we get:
I = 600 * 0.02 * 3
I = $36
Therefore, Carmen will be paid $36 in interest in the first 3 years on her deposit of $600 at a simple interest rate of 2% per year.
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A teacher was interested in the subject that students preferred in a particular school. He gathered data from a random sample of 100 students in the school and wanted to create an appropriate graphical representation for the data.
Which graphical representation would be best for his data?
Stem-and-leaf plot
Histogram
Circle graph
Box plot
Answers
Answer:
A circle graph would be appropriate here.
Convert the following percents to decimals.
150% =
Answers
Answer:
1.5
Step-by-step explanation:
Move the decimal point two places to the left to go from a percentage to a decimal.
the results of a survey show that 40$ of 300 students choose Recycling as the top priority or their generation? how many students choose recycling?
Answers
Find the volume 17.3 10.4
Answers
The volume of the given cone with a height of 17.3 and a diameter of 10.4 is 489.87 cubic cm
The volume of any cone can easily be determined by the formula (pi*r*r)*(h/3) where pi has the value of 3.14, 'r' is the radius of the cone, and 'h' is the given height of the cone.
Here in the given question, we can see that the height(h) given here is 17.3cm and the diameter is 10.4cm with which we can find the radius because we know the radius is the half of diameter so 'r' will be 5.2cm.
Now just putting the values of 'h' and 'r' in the given formula (pi*r*r)*(h/3) we get the volume of the cone which is 489.87 cubic cm.
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Answer:
Step-by-step explanation:
here,
height h= 17.3cm
diameter= 10.4cm
therefore, radius r will be 5.2cm
pi= 3.14
solution:
volume v of cone= 1/3*pi*r2*h
= 1/3*3.14*5.2*5.2*17.3 cm3
= 489.7 cm3
explanation:
here, it is given that the height is 17.3cm and the radius is 5.2cm. we know that the formula for finding the volume of a cone is 1/3*pi*r2*h.
Find the equation of the line that is perpendicular to the line AB and passes through the point. (0,2)
Answers
The equation of a line perpendicular to line AB and passing through point (0,2) is y = -4x + 2.
The given line has a slope of 1/4, so the slope of any line perpendicular to it will be -4 (the negative reciprocal).
Using the point-slope form of a line, we can write the equation of the line passing through the point (0,2) with slope -4 as
y - 2 = -4(x - 0)
Simplifying, we get
y - 2 = -4x
Adding 2 to both sides, we get
y = -4x + 2
Therefore, the equation of the line passing through the point (0,2) which is perpendicular to the line y=1/4x+5 is y = -4x + 2.
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Can you pls help? This is geometry
Answers
Answer:
[tex]x^{2} + y^{2} = 36[/tex]
Step-by-step explanation:
The form of the equation for a circle is [tex](x-h)^{2} + (y-k)^{2} = r^{2}[/tex], where [tex](h, k)[/tex] is the center and [tex]r[/tex] is the radius. Here, (h, k) is (0,0) and r is 6, so the equation would just be [tex]x^{2} + y^{2} = 36[/tex].
Answer:
x² + y² = 36 (top right answer)
Formula for the circle:
(x - h)² + (y - k)² = r²
h = X center point
k = Y center point
Need help please and thank you
Answers
Answer:
y=2/3x-6 1/3
Step-by-step explanation:
y+1=2/3(x-8)
first do the distributive property.
2/3*x=2/3x
2/3*-8= -5 1/3
2/3x-5 1/3
then we subtract 1 from both sides
y=2/3x-6 1/3
Explanation: I did the problem, i need more points. LIKE THIS
On a map, the distance between two cities is 3.5 inches. The key to the map shows that 1 inch = 7 miles. What is the actual distance between the cities?
Answers
URGENT - Will also give brainliest to simple answer
Find the length of the arc that outlines the sector
Answers
Answer:
arc length ≈ 37.7
Step-by-step explanation:
the arc length is calculated as
arc = circumference of circle × fraction of circle
= 2πr × [tex]\frac{135}{360}[/tex] ( r is the radius )
= 2π × 16 × [tex]\frac{3}{8}[/tex]
= 32π × [tex]\frac{3}{8}[/tex]
= 4π × 3
= 12π
≈ 37.7 ( to the nearest tenth )
a ladder leans against a wall so that its slope is 1.60. the top of the ladder is 8 vertical feet above the ground. what is the approximate horizontal distance from the base of the ladder to the wall? (assume that the positive direction points from the base of the ladder toward the wall.)
Answers
Answer: We can use the trigonometric function tangent to solve this problem. Let x be the horizontal distance from the base of the ladder to the wall. Then we have:
tan(1.60) = x / 8
Multiplying both sides by 8, we get:
x = 8 tan(1.60)
Using a calculator, we find:
x ≈ 8 × 1.518 = 12.14
Therefore, the approximate horizontal distance from the base of the ladder to the wall is 12.14 feet.
Step-by-step explanation:
According to the Democratic Presidential candidates, 51% of young Americans support tuition-free college. Suppose a random sample of 30 UConn students is selected. What is the probability that at least 10 of them support tuition-free college
Answers
The probability that at least 10 of them support tuition-free college is 0.9838.
To answer this question, we will use the binomial distribution formula. Let p = 0.51 be the proportion of young Americans who support tuition-free college, and let n = 30 be the sample size.
The probability of at least 10 UConn students supporting tuition-free college is equal to the sum of the probabilities of 10 or more, 11 or more, 12 or more, ..., up to 30. We can use the normal approximation to the binomial distribution.
Using the normal approximation, we first calculate the mean and standard deviation of the binomial distribution:
mean = np = 30 * 0.51 = 15.3
standard deviation = sqrt(np(1-p)) = sqrt(30 * 0.51 * 0.49) = 2.46
We then standardize the variable X = number of UConn students who support tuition-free college:
Z = (X - mean) / standard deviation
Now we want to find the probability that X is greater than or equal to 10, which is equivalent to finding the probability that Z is greater than or equal to (10 - 15.3) / 2.46 = -2.14.
Using a standard normal table or calculator, we find that the probability of Z being greater than or equal to -2.14 is approximately 0.9838. Therefore, the probability of at least 10 UConn students supporting tuition-free college is approximately 0.9838.
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Evaluate cos^2θ for cosθ = sqrt3/2
Answers
The value of cos(2θ) is 1/2
What is trigonometric identity?Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.
If cos(tetha) = √3/2
this means the adjascent is √ 3 and the hypotenuse is 2
therefore ;
opp = √ 2²- √3)²
= √4-3
= √1
= 1
therefore sin(tetha) = 1/2
cos(2θ) = cos²(tetha) -sin²(tetha)
cos²(tetha) = (√3/2)² = 3/4
sin²(tetha) =( 1/2)² = 1/4
cos(2θ) = 3/4 - 1/4 = 2/4 = 1/2
therefore the value of cos(2θ) is 1/2
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