1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Nezavi [6.7K]
2 years ago
15

Hello people ~

Mathematics
2 answers:
Luden [163]2 years ago
6 0

Cone details:

  • height: h cm
  • radius: r cm

Sphere details:

  • radius: 10 cm

================

From the endpoints (EO, UO) of the circle to the center of the circle (O), the radius is will be always the same.

<u>Using Pythagoras Theorem</u>

(a)

TO² + TU² = OU²

(h-10)² + r² = 10²                                   [insert values]

r² = 10² - (h-10)²                                     [change sides]

r² = 100 - (h² -20h + 100)                       [expand]

r² = 100 - h² + 20h -100                        [simplify]

r² = 20h - h²                                          [shown]

r = √20h - h²                                       ["r" in terms of "h"]

(b)

volume of cone = 1/3 * π * r² * h

===========================

\longrightarrow \sf V = \dfrac{1}{3}  * \pi  * (\sqrt{20h - h^2})^2  \  ( h)

\longrightarrow \sf V = \dfrac{1}{3}  * \pi  * (20h - h^2)  (h)

\longrightarrow \sf V = \dfrac{1}{3}  * \pi  * (20 - h) (h) ( h)

\longrightarrow \sf V = \dfrac{1}{3} \pi h^2(20-h)

To find maximum/minimum, we have to find first derivative.

(c)

<u>First derivative</u>

\Longrightarrow \sf V' =\dfrac{d}{dx} ( \dfrac{1}{3} \pi h^2(20-h) )

<u>apply chain rule</u>

\sf \Longrightarrow V'=\dfrac{\pi \left(40h-3h^2\right)}{3}

<u>Equate the first derivative to zero, that is V'(x) = 0</u>

\Longrightarrow \sf \dfrac{\pi \left(40h-3h^2\right)}{3}=0

\Longrightarrow \sf 40h-3h^2=0

\Longrightarrow \sf h(40-3h)=0

\Longrightarrow \sf h=0, \ 40-3h=0

\Longrightarrow \sf  h=0,\:h=\dfrac{40}{3}<u />

<u>maximum volume:</u>                <u>when h = 40/3</u>

\sf \Longrightarrow max=  \dfrac{1}{3} \pi (\dfrac{40}{3} )^2(20-\dfrac{40}{3} )

\sf \Longrightarrow maximum= 1241.123 \ cm^3

<u>minimum volume:</u>                 <u>when h = 0</u>

\sf \Longrightarrow min=  \dfrac{1}{3} \pi (0)^2(20-0)

\sf \Longrightarrow minimum=0 \ cm^3

Mariana [72]2 years ago
6 0

Answer:

(a)  see step-by-step

\textsf{(b)}\quad V=\dfrac{20}{3} \pi h^2-\dfrac13 \pi h^3

\textsf{(c)}\quad h=\dfrac{40}{3}

Step-by-step explanation:

<h3><u>Part (a)</u></h3>

A right triangle can be drawn with vertices at the center O, the base angle of the cone and the center of the base of the cone (see annotated image).

Side lengths of the formed right triangle:

  • Hypotenuse = radius of sphere = 10 cm
  • Height = height of cone - radius of sphere = (h - 10) cm
  • Base = base radius of cone = r cm

Using Pythagoras' Theorem a^2+b^2=c^2
(where a and b are the legs, and c is the hypotenuse, of a right triangle)

\implies r^2+(h-10)^2=10^2

\implies r^2+h^2-20h+100=100

\implies r^2+h^2-20h=0

\implies r^2=20h-h^2

<h3><u>Part (b)</u></h3>

\textsf{Volume of a cone}=\dfrac13 \pi r^2h
(where r is the radius and h is the height)

Substitute the expression for r^2 found in part (a) into the equation so that volume (V) is expressed in terms of h:

\begin{aligned}V & =\dfrac13 \pi r^2h\\\\ \implies V & =\dfrac13 \pi (20h-h^2)h\\\\ & = \dfrac13 \pi (20h^2-h^3)\\\\ & = \dfrac{20}{3} \pi h^2-\dfrac13 \pi h^3 \end{aligned}

<h3><u>Part (c)</u></h3>

To find the value of h such that the volume of the cone is a maximum, differentiate V with respect to h:

\begin{aligned}V & =\dfrac{20}{3} \pi h^2-\dfrac13 \pi h^3\\\\ \implies \dfrac{dV}{dh} & =2 \cdot \dfrac{20}{3} \pi h-3 \cdot \dfrac13 \pi h^2\\\\ & = \dfrac{40}{3} \pi h- \pi h^2\\\\ & = \pi h\left(\dfrac{40}{3}-h\right)\end{aligned}

Set it to zero:

\begin{aligned}\dfrac{dV}{dh} & =0\\\\ \implies \pi h\left(\dfrac{40}{3}-h\right) & = 0\end{aligned}

Solve for h:

\begin{aligned} \pi h & = 0 \implies h=0\\ \dfrac{40}{3}-h & =0\implies h=\dfrac{40}{3}\end{aligned}

Substitute the found values of h into the equation for Volume:

\begin{aligned}\textsf{when}\:h=0:V &=\dfrac{20}{3} \pi (0)^2-\dfrac13 \pi (0)^3\\\\ \implies V & =0\sf \:cm^3\end{aligned}

\begin{aligned}\textsf{when}\:h=\dfrac{40}{3}:V &=\dfrac{20}{3} \pi \left(\dfrac{40}{3}\right)^2-\dfrac13 \pi \left(\dfrac{40}{3}\right)^3\\\\ \implies V & =\dfrac{32000}{27} \pi -\dfrac{64000}{81} \pi\\\\ & = \dfrac{32000}{81} \pi \\\\ & = 1241.123024..\sf \:cm^3\end{aligned}

Therefore, the value of h such that the volume of the cone is a maximum is:

h=\dfrac{40}{3}

You might be interested in
Which linear inequality will not have a shared solution set with the graphed linear inequality?
Serggg [28]
C.............................
6 0
3 years ago
Read 2 more answers
Please help em!!! I need an answer fast
DerKrebs [107]
You divide the area and the base 
3 0
3 years ago
List 5 ways to draw a rectangle into a congruent figure using one line
erastovalidia [21]
Cut in half long ways and short ways straight down the middle
the a line from opposite corners each way 
then a slanted line the hits the center would also create two congruent figures
3 0
3 years ago
Read 2 more answers
If you can help me youre a legend :)
AleksandrR [38]

Answer:

A. 9

B. 7

Step-by-step explanation:

A. 11 - 2 = 9

B. 1 + 6 = 7

4 0
3 years ago
Read 2 more answers
Use the conversions given below to change 83 kg into stones.
eduard

83 kg converted to stones is 13 stones.

Step-by-step explanation:

Given,

We have to convert 83 kg to stones.

As it is given,

1 kg = 2.2 pounds

83 kg = 2.2*83 pounds

83 kg = 182.6 pounds

Now,

14 pounds = 1 stone

1 pound = \frac{1}{14}\ stones

182.6 pounds = \frac{1}{14}*182.6

182.6 pounds = 13.04 stones

Rounding to one dp

182.6 pounds = 13.0 stones

83 kg converted to stones is 13 stones.

Keywords: converting units, division

Learn more about division at:

  • brainly.com/question/10677255
  • brainly.com/question/10689103

#LearnwithBrainly

4 0
2 years ago
Other questions:
  • The GCF of 24 and 40 is _______.
    9·1 answer
  • Walden's family is shopping for a reclining chair. The chair the family decided on has a retail price of $800 plus 5% sales tax
    14·1 answer
  • Please Help!
    10·2 answers
  • What type of triangle is a 45-45-90 triangle?
    8·2 answers
  • Use Pythagorean’s theorem to find x
    13·1 answer
  • A projectile is thrown upward so that its distance above the ground after t seconds is h = -16t^2 + 608t. After how many seconds
    14·2 answers
  • An open box is to be made from a flat piece of material 14 inches long and 6 inches wide by cutting equal squares of length xfro
    7·1 answer
  • Pleasee tell me the answer in a simple way
    11·2 answers
  • Math please help, it is Aleks
    8·1 answer
  • erica would like to bake an 3-pound roast for a family gathering. the cookbook tells her to bake a 6-pound roast for 138 minutes
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!