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
On the blueprint of a house, 38 millimeters represents 6 meters. the length of the living room is 57 millimeters on the blueprin
Pie
For this case we can make the following rule of three:
 38 milimeters ---------> 6 meters
 57 milimeters ---------> x
 Clearing x we have:
 x = (57/38) * (6)
 x = 9 meters
 Answer:
 
the current length of the living room is:
 
x = 9 meters
3 0
3 years ago
Trailer will be used to transport 40 kg crate to store. The greatest amount of weight that can be loaded onto the trailer is 105
Scorpion4ik [409]
You can still transport 2040 KG crates to the store.
3 0
3 years ago
At a bake sale, as student spent $11.00 buying 3 brownies and 5 cookies. His friend spent $3.95 buying 1 brownie and 2 cookies.
riadik2000 [5.3K]

Answer:

$$2.25

Step-by-step explanation:

Let b = the cost of one brownie, c = cost of one cookie

3b + 5c = 11

b+2c = 3.95

Times 3 on both sides on the second equation above to get the following:

3b + 6c = 11.85

3b+6c - (3b+5c) = 11.85-11

3b+6c-3b-5c = 0.85

c = 0.85

To find the value of b, use substitution:

b+2c = 3.95

b = 3.95-2c

= 3.95 - 2\times0.85

= 2.25

\therefore the cost of a brownie is $$2.25

Hope this helps :)

4 0
3 years ago
Solve |2x - 5| = 4.
vladimir1956 [14]
Decimal form: 4.05, 0.5
Exact form: 9/2, 1/2
8 0
3 years ago
Read 2 more answers
36m⁴-√16 pls help asap​
shusha [124]

Answer:

Step-by-step explanation:

(a^2-b^2)=(a-b)(a+b)\\ \\ 36m^4-4\\ \\ (6m^2-2)(6m^2+2)\\ \\ 4(3m^2-1)(3m^2+1)

3 0
3 years ago
Other questions:
  • 3. Timmy has a total of 15 dimes and nickels in his pocket. He has 7 more
    15·1 answer
  • I need help with this STAAR practice
    14·2 answers
  • Write the fractions in order from least to greatest 1/4 3/6 1/8
    8·2 answers
  • The​ 18th, 19th and 20th numbers in the Fibonacci sequence​ are, respectively,​ 2584, 4181, and 6765. What is the 21st​ number?
    9·1 answer
  • GEOMETRY HELP!
    6·1 answer
  • Solve the inequality! PLEASE HELP I’LL MARK BRAINLIEST!!
    14·1 answer
  • If a card is drawn at random from a standard 52-card deck, what is the probability that
    11·1 answer
  • The value v of your quarters in cents) is a function of n, the number of quarters you have.
    9·1 answer
  • A group of randomly selected Clyde Marketing employees were asked what their most common form of transportation is. The bar grap
    14·1 answer
  • Points, in case you need to ask a question.
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!