A cylinder is inscribed in a right circular cone of height 2.5 and radius (at the base) equal to 5.5. what are the dimensions of such a cylinder which has maximum volume?
Answers
Check the picture below.
so hmm if we look at the cone straight from the front and check a cross-section of it, it'd look like the cross-section of the right-hand-side of the picture.
Now, with the cylinder having radius "r" and height "h", those two triangles we see in greenish color are similar, thus we can use proportions to correlate "h" and "r", so let's do that.
[tex]\cfrac{2.5-h}{r}~~ = ~~\cfrac{h}{5.5-r}\implies 13.75-5.5h-2.5r+rh=rh \\\\\\ 13.75-2.5r=5.5h\implies \boxed{\cfrac{13.75-2.5r}{5.5}=h}[/tex]
now, let's plug that in the volume of the cylinder's equation
[tex]V(r)=\pi r^2\cdot \cfrac{13.75-2.5r}{5.5}\implies V(r)=\cfrac{\pi }{5.5}(13.75r^2-2.5r^3) \\\\\\ \cfrac{dV}{dr}=\cfrac{\pi r(27.5-7.5r)}{5.5}\hspace{5em}0~~ < ~~r ~~ < ~~ 2.5[/tex]
now, our "r" variable can't be 0 or 2.5, as the cylinder tiptoes up and down, because if that ever happens, our cylinder's volume goes poof.
[tex]0=\cfrac{\pi r(27.5-7.5r)}{5.5}\implies 0=27.5-7.5r\implies 7.5r=27.5 \\\\\\ r=\cfrac{27.5}{7.5}\implies r=\cfrac{11}{3}[/tex]
the way I do region testing for the 1st derivative, is I usually grab the critical point and subtract hmm say 0.0000001 from it and put it in a variable in my calculator, then I add 0.0000001 to it and put it in another variable, then I use that variable in the 1st derivative calculation. Anyhow, doing so and running a 1st derivative test, we'd see that the region to the left of 11/3 is positive whilst the region to the right is negative, as you can see in the picture at the bottom, yielding a maximum extremum, r = 0 is another critical point but as we noticed the constraints is out of bounds, so not in consideration.
so V(r) is at a maximum when r = 11/3
[tex]\blacksquare~~ r=\cfrac{11}{3} ~~\blacksquare \hspace{5em} \cfrac{13.75-2.5\left( \frac{11}{3} \right)}{5.5}=h~\hfill \implies ~\hfill \blacksquare~~ \cfrac{5}{6}=h ~~\blacksquare[/tex]
Related Questions
HELP ASAP
Which of the following are equations for the line shown below? Check all that
apply
Answers
The equation of the line which is plotted on the graph and passes from the points (2,5) and (-2,-3) is y=2x-6.
What is the equation of line?The equation of the line is the way of representation of a line in the equation form.
The equitation of the lines represents the line by the set of points on which the line lies or passes through in the coordinate system.
The formula to find equation of line is,
(y-y₁)={(y₂-y₁)/(x₂-x₁)}(x-x₁)
Here, x and y are the coordinate and subscript (1,2) used for the first and second point.
The points from which the line passes through are (2,5) and (-2,-3). Put the values,
(y-y₁)={(y₂-y₁)/(x₂-x₁)}(x-x₁)
(y-(-2))={(-3-5)/(-2-2)}(x-2)
y+2={-8/-4}(x-2)
y+2=2(x-2)
y+2=2x-4
y=2x-4-2
y=2x-6
Thus, the equation of the line which is plotted on the graph and passes from the points (2,5) and (-2,-3) is y=2x-6.
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Ravi buys lunch at a restaurant.
An 8% sales tax is added, and he tips the waiter 20% of the cost of the food.
Ravi pays a total of $17.92 for his lunch, including tax and tip.
What is the price of Ravi's lunch, before tax and tip?
Answers
The price of Ravi's lunch, before tax and tip was added to the total cost of the meal is $13.19.
What is the price of the lunch?Both the tax and the tip paid would increase the cost of the meal. Thus, their dollar value has to be subtracted from the total amount he pays.
Value of the lunch after the tip has been removed = (1 - 0.2) x 17.92 = $14.34
Value of the lunch after the tax has been removed = (1 - 0.08) x $14.34 = $13.19
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I NEED HELP! ASAP, THIS IS SO HARD AND MY BRAIN HURTS, IF U HELP ME WITH OTHER QUESTIONS LIKE THIS ILL GIVE U 50+ POINTS
Answers
15/3*8
5*8
40!!
help me pls thank you
Answers
Answer:
i don't think you can create a triangle
Step-by-step explanation:
you need it to be a<b+c
Write an equation for this model and solve for `x`
Answers
x = -0.8
Find the volume of a cone with a diameter of 12 cm and a height of 7 m
Answers
Answer:
263.76 m^3
Step-by-step explanation:
Using 3.14 as pi
pi * radius^2 * height / 3
3.14 * 6^2 * 7 / 3
3.14 * 36 * 7 /3
3.14 * 252 /3
3.14 * 84
263.76
A 0.25 kg baseball is in contact with a bat for 0.0075 s and exerts a force of 18,500 n. what impulse is delivered to the ball? 33 kg • meters per second. 140 kg • meters per second. 4600 kg • meters per second. 74,000 kg • meters per second.
Answers
The impulse delivered to the baseball is 140kgm/s.
Option B) 140 kg • meters per second is the correct answer.
What is Impulse?An impulse is simply a concept that involves the change in the momentum of an object when force is introduced for a period of time.
It is expressed as;
ΔP or Change in Momentum = F × t
Where F is force applied and t is the elapsed time.
Given that;
Mass of baseball m = 0.25kgTime of contact t = 0.0075sForce exerted F = 18500N = 18500kgm/s²Impulse ΔP = ?We substitute our given values into the expression above to determine the impulse.
ΔP = F × t
ΔP = 18500kgm/s² × 0.0075s
ΔP = 18500kgm/s² × 0.0075s
ΔP = 138.75kgm/s = 140kgm/s
Therefore, the impulse delivered to the baseball is 140kgm/s.
Option B) 140 kg • meters per second is the correct answer.
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Answer:
(B)
Step-by-step explanation:
I took the test
find the domain and range pls!
Answers
And range should be, (-♾️, 2)
If you ment something other then this please message me so I can help!
Which solution to the equation startfraction 3 over a 2 endfraction startfraction 2 over a endfraction = startfraction 4 a minus 4 over a squared minus 4 endfraction is extraneous?
Answers
The solution of the rational equation [tex]\frac{3}{a + 2} * \frac{2}{a} = \frac{4 a - 4 }{a^2 - 4}[/tex] is a = 3
How to determine the solution?The equation is given as:
[tex]\frac{3}{a + 2} * \frac{2}{a} = \frac{4 a - 4 }{a^2 - 4}[/tex]
Express a^2 - 4 as a difference of two squares
[tex]\frac{3}{a + 2} * \frac{2}{a} = \frac{4 a - 4 }{(a + 2)(a - 2)}[/tex]
Cancel out the common terms
[tex]\frac{3}{} * \frac{2}{a} = \frac{4 a - 4 }{a - 2}[/tex]
Factor out 2 from 4a - 4
[tex]\frac{3}{} * \frac{2}{a} = \frac{2(a - 2) }{a - 2}[/tex]
Evaluate the quotient
[tex]\frac{3}{} * \frac{2}{a} = 2[/tex]
Divide both sides by 2
[tex]\frac{3}{1} * \frac{1}{a} = 1[/tex]
Cross multiply
a = 3
Hence, the solution of the equation [tex]\frac{3}{a + 2} * \frac{2}{a} = \frac{4 a - 4 }{a^2 - 4}[/tex] is 3
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Answer:
the answer for Which solution to the equation StartFraction 3 Over a + 2 EndFraction + StartFraction 2 Over a EndFraction = StartFraction 4 a minus 4 Over a squared minus 4 EndFraction is EXTRANEOUS? is a: a=-2
Step-by-step explanation:
I need the formula for this graph
Answers
Answer:
y=[tex]\frac{1}{3}[/tex]x+[tex]\frac{1}{3}[/tex] /
Step-by-step explanation:
MATH PROBLEM SOLVE IT TO GET 50 PTS
Answers
Answer:
b = 17
Step-by-step explanation:
The correct equation is 3b - 6 = 45. You get this by saying the books (b) are 3 dollars each (3b) and you are getting a discount of 6 dollars off (- 6). The total after the discount is 45 dollars. You know have to solve this equation for b (the number of books) to get the answer. When you solve this equation, you get b = 17.
Answer:
3b - 6 = 45 and b = 17
Step-by-step explanation:
3b - 6 = 45 is your equation you get
3b is how many books you bought and the price multiplied by one another, then subtracting 6 gives you the discounted price which is 45
3b - 6 = 45
+6 +6
3b = 51
/3 /3
b = 17
a book costs 17 dollars.
so b = 17
Pleasei will give you os many point
Answers
Answer:
x = 39°
Step-by-step explanation:
for right angle triangle ABD
BD^2 = 36 + 64 = 100
BD = 10
use sine law
sin (29) / BD = sin (x)/ DC
sin (29)/ 10 = sin x/ 13
sin x = 13 sin (29)/ 10
First solve for BD:
a² + b² = c²6² + 8² = BD²BD² = 36 + 64BD² = 100BD = √100BD = 10Then use sine rule:
sin(A)/a = sin(B)/binsert values
sin(29)/10 = sin(x)/13exchange sides
sin(x) = 13sin(29)/10simplify
sin(x) = 6.302525/10sin(x) = 0.63change side
x = sin⁻¹(0.63)x = 39.06875°rounded to nearest hundredth
x = 39.07°A pattern follows the rule "Start at 3 and add 10."
Which statement is true about the pattern?
A: Each term is 13 more than the previous term because the tens digit goes up by 1 and the one's digit goes up by 3.
B: Each term is 10 times the previous term because the tens digit goes up by 1 and the ones digit stays the same.
C: For each term, the ones digit stays the same and the tens digit goes up by 1 because 10 has one ten and no ones.
D: For each term, the ones digit goes up by 3 and the tens digit goes up by 1 because you start at 3 and go up by 10 each time.
Answers
B: Each term is 10 times the previous term because the tens digit goes up by 1 and the ones digit stays the same.
Juan rolls a 6-sided number cube 4 times. each time, the result is 2. which best describes the events? dependent: each outcome was the same. simple: one single number cube was rolled. independent: each outcome is unaffected by previous outcomes. simple: one single number cube was rolled. dependent. each outcome was the same. compound: multiple events occurred. independent: each outcome was unaffected by previous outcomes. compound: multiple events occurred.
Answers
Answer: 1/6
because there are 6 side to a cube and you can only have a outcome of 1 so therefore, the answer is 1/6
Answer:
D
Step-by-step explanation:
one of the exterior Angles of a triangle is 125°and the interior opposite angles are in the ratio 2:3 please find the Angles of the triangle
Answers
Given , one exterior angle is 125°
Interior angles are in ration 2:3
Let the angles be 2x and 3x
Now , 125° = 2x + 3x
=> 5x = 125°
=> x = 125/5
=> x = 25°
Now the angles are ,
2x = 2(25)=50° &
3x = 3(25) = 75°
Hope this helps you :)
Have a good day :)
Answer:
Angles of the triangles are 55° , 50° , 75°
Step-by-step explanation:
Exterior angle property of triangle:The exterior angle of a triangle equals the sum of opposite interior angles.
Ratio of interior opposite angle = 2 : 3
Interior opposite angles are 2x , 3x
2x + 3x = 125
5x = 125
x = 125/5
[tex]\boxed{x = 25}[/tex]
2x = 2*25 = 50°
3x = 3*25 = 75°
Let the third angle of the triangle = y
Sum of all angles of triangle = 180
50 + 75 + y = 180
125 + y = 180
y = 180 - 125
y = 55°
Angles of the triangles are 55° , 50° , 75°
Colleen has put $26,000 into a retirement fund. the fund has an estimated annual return of 10%. if colleen doesn't add any more money, how much can she expect to have in the fund in 7 years?
if necessary, round your answer to the nearest cent
the answer is : 50,666.64
Answers
Using an exponential function, it is found that she can be expected to have $50,666.64 in the fund in 7 years.
What is an exponential function?An increasing exponential function is modeled by:
[tex]A(t) = A(0)(1 + r)^t[/tex]
In which:
A(0) is the initial value.r is the decay rate, as a decimal.In this problem, we have that the initial amount and the growth rate are given, respectively, by:
A(0) = 26000.r = 0.1.Hence the amount after t years is given by:
[tex]A(t) = 26000(1.1)^t[/tex]
Then, after 7 years, the amount is:
[tex]A(7) = 26000(1.1)^7 = 50,666.64[/tex]
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Miss Violet is buying granola bars for her students. She already has 16 granola bars in her desk and needs a total of 100 bars. They are sold in packs of 12. Write an equation to show how many more boxes of granola bars she needs.
16 – 12x = 100
16 – 12x = 100
16x + 12 = 100
16x + 12 = 100
16 + 12x = 100
16 + 12x = 100
12x – 16 = 100
12x – 16 = 100
Answers
Answer:
100 - 16 = 84, and 84 divided by 12 is 7. She needs to buy 7 more boxes of granola bars for her students.
Step-by-step explanation:
Answer:
She need 7
Step-by-step explanation:
12x7=84
Then add 16 to 84 to get 100
In an orchard, there are 16 apple trees, 22 orange trees, and a cherry tree. What is the ratio of apple trees to orange trees? OA. 1:22 OB 22:16 OC. 16:22 OD. 38:22
Answers
Answer:
Answer:
C. 16 : 22
Step-by-step explanation:
Given:
Apple trees = 16
Orange trees = 22
Cherry trees = 1
Ratio of apples trees to orange trees:
⇒ apple : orange
⇒ 16 : 22
How many square inches of fabric does Kendall need for a triangle flag with a base of 20 inches a height of 40 inches
Answers
Answer: [tex]400^{2}[/tex]
Step-by-step explanation:
First, write the formula:
A = (b × h) / 2
Next, multiply the base and the height of the triangle.
20 x 40 = 800
Finally, divide 800 by 2.
800/2=400
Therefore, the answer is [tex]400^{2}[/tex].
Select all of the combinations of side lengths below that cannot be used to draw a triangle.
Answers
Answer:
A , C and D cannot be used to drwa a traingle
Step-by-step explanation:
If 50x=500 then….
What would the answer be ?
Answers
[tex]\large\textit{Answer :}[/tex]
x = 10
[tex]\large\textit{Calculations :}[/tex]
The way to solve this one - step equation is to divide by 50 on both sides to get x by itself :
50x=500
x = 10
[tex]\tiny\textsc{HOPE HELPFUL ~}[/tex]
Please help. I don't know any theorems that relate to this
Answers
Answer:
125°
Step-by-step explanation:
In circle with center O.
BC is tangent at point B.Draw segment OBOB is radius[tex]\implies OB\perp AB[/tex] (Tangent theorem)[tex]\implies\red{\bold{ m\angle OBC = 90\degree}}[/tex][tex]m\angle AOB= 90\degree + 20\degree[/tex] (By exterior angle theorem)[tex]\implies m\angle AOB= 110\degree[/tex] OA = OB (radii of same circle)[tex]\implies m\angle OAB = m\angle OBA = x \: (say)[/tex] (By isosceles triangle theorem)[tex]m\angle AOB + m\angle OAB +m\angle OBA=180\degree[/tex] (By interior angle sum theorem of a triangle)[tex]\implies 110\degree+ x +x=180\degree[/tex][tex]\implies 2x =180\degree-110\degree[/tex][tex]\implies 2x =70\degree[/tex][tex]\implies x =35\degree[/tex][tex]\implies \purple{\bold{m\angle OBA =35\degree}}[/tex][tex]m\angle CBA=\red{\bold{m\angle OBC}} +\purple{\bold{m\angle OBA }}[/tex][tex]m\angle CBA=90\degree + 35\degree[/tex][tex]\huge{\orange{m\angle CBA=125\degree}}[/tex]Find the SURFACE AREA of the
cylinder below.
10 yd radius and 15 yd height. (ROUND TO NEAREST TENTH)
Answers
Answer:
1571
Step-by-step explanation:
its 1570.8 but you have to round
8. The model of a building is 32 inches tall. The actual building is 240 feet tall. What scale is being used?
Answers
Answer:
7.5 feet for 1 in.
Step-by-step explanation:
32:240/2
16:120/2
8:60/2
4:30/2
2:15/2
1:7.5
Put these fractions in order of size, smallest to largest.
7/10 2/3 4/5 11/15
Answers
Step-by-step explanation:
2/3
7/10
11/15
4/5
(if you struggle with identifying when one fraction is bigger or smaller than another, just convert it to decimals in a calculator)
Hope this helps
60. If a publisher charges $15 for the first copy of a book
that is ordered and $12 for each additional copy, which
of the following expressions represents the cost of
y books?
F. 12y + 3
G. 12y + 15
H. 15y - 3
J. 15y + 3
K. 15y + 12
Answers
if I have my mean mode median range so how can I construct a frequency table
Answers
Answer:
To find the mean, multiply the values by frequencies, add the subtotals, and divide by the total number of the frequency. Median : To find the median, calculate a running total of the frequencies, which is half the total, it contains the median that corresponds to the value.
Step-by-step explanation:
PLEASE ASAP we need to turn in in 20 mins
Answers
Answer:
14 units^2
Step-by-step explanation:
Hey there!
In order to find the area of this triangle, we need to find the base and height of the triangle
the base of the triangle is 7 units long and the height of the triangle is 4 units high
Formula for area of triangle: Base x Height x 1/2
7 x 4 x 1/2 --> 28 x 1/2 --> 14
So, the area of this triangle is 14
Now the question is asking to find the area by forming a parallelogram
All you have to do for this is duplicate that same triangle, and mirror it upside down
Then all you would have to do is find the Base and the Height and multiply them together
Find the equation of the line that goes through the points (0, 4) and (10, 2).
y = -1/5 x + 4
y = 1/5 x - 4
y = -1/5 x + 2
y = -5x + 4
Answers
Y=-1/5x+4
Y-intercept is (0,4)
Can someone please help me with this question
Answers
Answer:
B)
Step-by-step explanation:
The way I think of it, a fair election comes from all qualifiable candidates, in this case, only 50 girls were allowed to vote, giving the impression that boys aren't qualified, thus not even being able to vote for a candidate. In conclusion, I think that it's option B).