Hello people ~
The given figure shows a cone of height "h cm" and base-radius "r cm" that is made from a wooden spherical solid of radius 10 cm. The volume of the cone is cm³.
(a) Show that r² =20h-h²
(b) Express V in terms of h
(c) Hence, find the value of h such that the volume of the cone is a maximum.
Answers
Cone details:
height: h cmradius: r cmSphere 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.
Using Pythagoras Theorem
(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
===========================
[tex]\longrightarrow \sf V = \dfrac{1}{3} * \pi * (\sqrt{20h - h^2})^2 \ ( h)[/tex]
[tex]\longrightarrow \sf V = \dfrac{1}{3} * \pi * (20h - h^2) (h)[/tex]
[tex]\longrightarrow \sf V = \dfrac{1}{3} * \pi * (20 - h) (h) ( h)[/tex]
[tex]\longrightarrow \sf V = \dfrac{1}{3} \pi h^2(20-h)[/tex]
To find maximum/minimum, we have to find first derivative.
(c)
First derivative
[tex]\Longrightarrow \sf V' =\dfrac{d}{dx} ( \dfrac{1}{3} \pi h^2(20-h) )[/tex]
apply chain rule
[tex]\sf \Longrightarrow V'=\dfrac{\pi \left(40h-3h^2\right)}{3}[/tex]
Equate the first derivative to zero, that is V'(x) = 0
[tex]\Longrightarrow \sf \dfrac{\pi \left(40h-3h^2\right)}{3}=0[/tex]
[tex]\Longrightarrow \sf 40h-3h^2=0[/tex]
[tex]\Longrightarrow \sf h(40-3h)=0[/tex]
[tex]\Longrightarrow \sf h=0, \ 40-3h=0[/tex]
[tex]\Longrightarrow \sf h=0,\:h=\dfrac{40}{3}[/tex]
maximum volume: when h = 40/3
[tex]\sf \Longrightarrow max= \dfrac{1}{3} \pi (\dfrac{40}{3} )^2(20-\dfrac{40}{3} )[/tex]
[tex]\sf \Longrightarrow maximum= 1241.123 \ cm^3[/tex]
minimum volume: when h = 0
[tex]\sf \Longrightarrow min= \dfrac{1}{3} \pi (0)^2(20-0)[/tex]
[tex]\sf \Longrightarrow minimum=0 \ cm^3[/tex]
Answer:
(a) see step-by-step
[tex]\textsf{(b)}\quad V=\dfrac{20}{3} \pi h^2-\dfrac13 \pi h^3[/tex]
[tex]\textsf{(c)}\quad h=\dfrac{40}{3}[/tex]
Step-by-step explanation:
Part (a)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 cmHeight = height of cone - radius of sphere = (h - 10) cmBase = base radius of cone = r cmUsing Pythagoras' Theorem [tex]a^2+b^2=c^2[/tex]
(where a and b are the legs, and c is the hypotenuse, of a right triangle)
[tex]\implies r^2+(h-10)^2=10^2[/tex]
[tex]\implies r^2+h^2-20h+100=100[/tex]
[tex]\implies r^2+h^2-20h=0[/tex]
[tex]\implies r^2=20h-h^2[/tex]
Part (b)[tex]\textsf{Volume of a cone}=\dfrac13 \pi r^2h[/tex]
(where r is the radius and h is the height)
Substitute the expression for [tex]r^2[/tex] found in part (a) into the equation so that volume (V) is expressed in terms of h:
[tex]\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}[/tex]
Part (c)To find the value of h such that the volume of the cone is a maximum, differentiate V with respect to h:
[tex]\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}[/tex]
Set it to zero:
[tex]\begin{aligned}\dfrac{dV}{dh} & =0\\\\ \implies \pi h\left(\dfrac{40}{3}-h\right) & = 0\end{aligned}[/tex]
Solve for h:
[tex]\begin{aligned} \pi h & = 0 \implies h=0\\ \dfrac{40}{3}-h & =0\implies h=\dfrac{40}{3}\end{aligned}[/tex]
Substitute the found values of h into the equation for Volume:
[tex]\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}[/tex]
[tex]\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}[/tex]
Therefore, the value of h such that the volume of the cone is a maximum is:
[tex]h=\dfrac{40}{3}[/tex]
Related Questions
Fourteen people are entered in a race. If there are no ties, how many ways can the first three places come out
Answers
Answer:
2184 ways because that's how math works
Step-by-step explanation:
14 × 13 × 12
PLS HELP
Find the distance between the two points.
(1,1)
√ [?]
Enter the number that
goes beneath the
radical symbol.
(3,-4)
Answers
Answer:
The number inside radical is 29
Step-by-step explanation:
Given:
Points (1,1) & (3,-4)To find:
Distance between two pointsDistance Formula:
[tex]\displaystyle \large{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}[/tex]
Determine (x2,y2) = (1,1) and (x1,y1) = (3,-4). Therefore:
[tex]\displaystyle \large{\sqrt{(1-3)^2+(1-(-4))^2}}\\\displaystyle \large{\sqrt{(-2)^2+(1+4)^2}}\\\displaystyle \large{\sqrt{4+5^2}}\\\displaystyle \large{\sqrt{4+25}}\\\displaystyle \large{\therefore \sqrt{29}}[/tex]
Therefore, the number inside the radical is 29
Answer:
[tex]\sqrt{29}[/tex]
Step-by-step explanation:
Hi there!
We are given the points (1, 1) and (3, -4)
We want to find the distance between the points
Distance can be found using the formula [tex]\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex], where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points
We can label the values of the points first, before calculating to help avoid confusion and mistakes.
[tex]x_1 = 1\\y_1 = 1\\x_2 = 3 \\y_2 = -4[/tex]
Now substitute these values into the formula
d = [tex]\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]
d = [tex]\sqrt{(3-1)^2 + (-4-1)^2}[/tex]
Now, using order of operations, first simplify what is in the parentheses:
d = [tex]\sqrt{(2)^2 + (-5)^2}[/tex]
Now raise 2 and -5 to the second power.
d = [tex]\sqrt{4 + 25}[/tex]
Add the numbers together under the radical
d = [tex]\sqrt{29}[/tex]
The distance is [tex]\sqrt{29}[/tex]
Hope this helps!
See more on finding the distance here: https://brainly.com/question/24774555
Permutations and, combinations: problem type 2.
(A) From a collection of 51 store customers, 3 are to be chosen to receive a special gift. How many
groups of 3 customers are possible?
(B) How, many different committees of size 4 can be formed from 14 people ?
Answers
Step-by-step explanation:
Permutations:
nP r = n!/(n + r)!
Combinations:
nCr = n!/(n - r)!r!
c - combination
p - permutations
Since in both the problems they are looking for different groups, it is a combination problem. the order in which the people were selected in each group is not important, what is required are different groups.
A.
n = 51
r = 3
51C 3 = 51! / ((51 - 3) ×3!)
51C3 = 20 825 groups
OR:
[tex]c = \frac{51}{3} \times \frac{50}{2} \times 49[/tex]
[tex]c = 20825[/tex]
B.
[tex]c = \frac{14}{4} \times \frac{13}{3} \times \frac{12}{2} \times 11[/tex]
[tex]c = 1001[/tex]
The width of a rectangle measures (s+5t) centimeters, and its length measures (s+2t)centimeters. Which expression represents the perimeter, in centimeters, of the rectangle?
Answers
Answer:
4s+14t
Step-by-step explanation:
2*[(s+5t)+(s+2t)]
=2*(2s+7t)
=4s+14t
Answer:
Step-by-step explanation:
Remark
The Perimeter of a rectangle is found using
P = 2L + 2w
Givens
L = s+2t
w = s + 5t
Solution
P = 2(s + 2t) + 2(s + t5) Remove the brackets
P = 2s + 4t + 2s + 10t Combine
P = 4s + 14t
Answer P = 4s + 14t This cannot be reduced any further.
Will award brainliest for the correct answer!
Answers
Answer:
2^4ab
Step-by-step explanation:
By factorizing 48ab^2 I get 2*2*2*2*3*a*b*b
By factorizing 32a^2b I get 2*2*2*2*2*a*a*b
All the factors that these have in common is 4 2's, one a, and one b, which is 2^4*ab, or 16ab.
Could Someone help me with this-
No pasting links,No blabbering just to get the points.
Answers
Answer:
Volume = 90in^3
Step-by-step explanation:
V = 3in x 15in x 2in = 90in^3
Answer:
[tex]\boxed{\sf{90in^3}}[/tex]Step-by-step explanation:
The volume of the prism must be found in order to solve this problem.
GIVEN:
[tex]\Longrightarrow: \sf{V=L*W*H}[/tex]
[tex]\Longrightarrow: \text{LENGTH: = 3}\\\\\Longrightarrow: \text{WIDTH: = 15}\\\\\Longrightarrow: \text{HEIGHT: = 2}[/tex]
SOLUTION:
Multiply.
[tex]\sf{3\ in*15\ in*2\ in=\boxed{\sf{90\ in^3}}[/tex]
Therefore, the volume of the prism is 90in³, which is our answer.I hope this helps. Let me know if you have any questions.
1)
В
13
C
105°
7
A
a. 36.2
b40.2
c81.6
d43.9
Answers
Answer:
Step-by-step explanation:
Could you update the question? It is not clear enough
254 students study english
280 students study math
280 students study science
73 students study all topics
152 students study english and math
79 students study math and science
139 students study science and english
How much students are in the school?
Answers
Answer:
517
Step-by-step explanation:
We can draw a Venn diagram to represent the data.
First, identify the groups.
Let E = students who study EnglishLet M = students who study MathLet S = students who study ScienceThen draw the Venn diagram to represent the groups.
Given 73 students study all subjects:
E ∩ M ∩ S = 73 (add this to the middle section)
Next, add the numbers for the sections where the students study 2 topics.
E ∩ M but not S: 152 - 73 = 79
M ∩ S but not E: 79 - 73 = 6
S ∩ E but not M: 139 - 73 = 66
Now complete each circle by making sure that the values add up to the total value given.
Only E = 254 - S ∩ E - E ∩ M - E ∩ M ∩ S = 254 - 66 - 79 - 73 = 36
Only M = 280 - E ∩ M - M ∩ S - E ∩ M ∩ S = 280 - 79 - 6 - 73 = 122
Only S = 280 - M ∩ S - S ∩ E - E ∩ M ∩ S = 280 - 6 - 66 - 73 = 135
To find the total number of students in the school, add the numbers in each section of the Venn diagram:
73 + 79 + 6 + 66 + 36 + 122 + 135 = 517
40 feet wide and 96 feet long how far is it from 1 corner to the opposite corner
Answers
Answer:
104 feet
Step-by-step explanation:
The Pythagorean theorem can be used to find the hypotenuse of a right triangle with legs 40 feet and 96 feet. It tells you ...
h² = 40² +96²
h² = 1600 +9216 = 10816
h = √10816 = 104
The distance from one corner to the opposite corner is 104 feet.
______
Additional comment
You may recognize that the ratios of the given lengths are 40:96 = 5:12. These match the legs in the Pythagorean triple (5, 12, 13). This tells you the third side (diagonal) of the triangle is 13 ratio units, or 13×(8 ft) = 104 ft.
please help!!!!
simplify the rational expression. state any restrictions on the variable.
t^2+2t-24/t^2-36
Answers
Answer: hope this helps! :)
t - 4
—————
t - 6
Step-by-step explanation:
t2 + 2t - 24
Simplify ————————————
t2 - 36
Trying to factor by splitting the middle term
Factoring t2 + 2t - 24
The first term is, t2 its coefficient is 1 .
The middle term is, +2t its coefficient is 2 .
The last term, "the constant", is -24
Multiply the coefficient of the first term by the constant 1 • -24 = -24
Find two factors of -24 whose sum equals the coefficient of the middle term, which is 2 .
-24 + 1 = -23
-12 + 2 = -10
-8 + 3 = -5
-6 + 4 = -2
-4 + 6 = 2 That's it
Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and 6
t2 - 4t + 6t - 24
Add up the first 2 terms, pulling out like factors :
t • (t-4)
Add up the last 2 terms, pulling out common factors :
6 • (t-4)
Step-5 : Add up the four terms of step 4 :
(t+6) • (t-4)
Which is the desired factorization
Trying to factor as a Difference of Squares :
1.2 Factoring: t2-36
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 36 is the square of 6
Check : t2 is the square of t1
Factorization is : (t + 6) • (t - 6)
Canceling Out :
Cancel out (t + 6) which appears on both sides of the fraction line.
Answer:
t⁴ + t³ - 36t - 24Step-by-step explanation:
Given :
t² + 2t - 24/t² - 36Solving :
Multiply throughout with t² so the polynomial is easier to simplifyt² (t² + 2t - 24/t² - 36)t⁴ + t³ - 24 - 36t²t⁴ + t³ - 36t - 24
Juan earns 20% commission as a salesperson. He sold a trampoline that cost $86.40 How
much commission did Juan earn?
Answers
Answer:
$17.28
Step-by-step explanation:
20% of $86.40
= [tex]\frac{20}{100}[/tex] × $86.40
= 0.2 × $86.40
= $17.28 ← commission
Alyssa buys cheese and apples at the store.
• She pays a total of $58.61.
She pays a total of $3.09 for the cheese.
She buys 8 bags of apples that each cost the same amount.
How much does each bag of apples cost?
Answers
Will someone pls mind helping me with these math problems ill mean a lot to me or at least just look at it
Answers
The equation that represents the number of T-shirt , number of colours of shirt and total order is 3x + 50xy.
The number of colours and T- shirts are the variable quantity while the cost of order is the fixed quantity.
How to represent an equation ?The number of members in the school choir = 75
The charge for one shirt = $3
Let the number of T-shirts ordered be x
Cost of ordering x shirts = 3x
Therefore,
Charge for each colour to be printed = $50
Let the number of colours to be printed on each T- shirt be represented by y
Therefore,
Cost of printing y colours on each T- shirt = 50y
Cost of printing y colours on x number of T-shirts = x(50y) = 50xy
Hence, the equation that represents the number of T-shirt , number of colours of shirt and total order is as follows:
Total cost = 3x + 50xy
Hence, the fixed quantity and variable quantity are as follows:
The number of colours on the shirts is a variable quantity
The number of shirts ordered is a variable quantity
The cost of order is fixed quantity
learn more on equation here: https://brainly.com/question/12289422
#SPJ1
which expression is equivalent to 5(x+7)?
Answers
Answer:
5x +35
Step-by-step explanation:
you woul use the distrubitive property
so multiply by x and and 7
so it becomes 5x +35
brainliest if right please
Answers
Answer:
yes
Step-by-step explanation:
Answer:
A
Step-by-step explanation:
a^2+b^2=c^2
The sides are greater than the longest side, so that is right but
81+144 should equal c^2 or 18^2, but it is not equal to it.
how are corresponding sides related in similar triangles??
Answers
Answer:
In a pair of similar triangles, the corresponding sides are proportional. Corresponding sides touch the same two angle pairs. When the sides are corresponding it means to go from one triangle to another you can multiply each side by the same number
What decimal part of 560 is 70
Answers
Answer: 0.125
Step-by-step explanation: 70/560
Find the volume of a cone with a base with an area of 43 mm² and a height of 17 mm. Round your answer to the nearest tenth.
Answers
Please helpppppp and noooo linkssss
Find the value of x.
Answers
Answer:
x is 25˚
Step-by-step explanation:
Opposite angles are equal[tex]5x+7=132[/tex]Subtract 7 from both sides[tex]5x=125[/tex]Divide both sides by 5[tex]x=25[/tex]Answer:
25 = x
Step-by-step explanation:
Since both angles are vertical angles, they are eqaul to each other and hence we can conclude that;
132° = (5x + 7)°
132 - 7 = 5x
125 = 5x
125/5 = x
25 = x
Hope this helps!
Can someone help me please?!?!?!
Answers
Answer:
see explanation
Step-by-step explanation:
you are looking for perfect square factors of the numbers under the radical sign. both numbers and the sum are irrational because they contain √5
125 = 5×25
80 = 5×16
3√125 = 3√(25×5) = 3×5√5 = 15√5
√80 = √(16×5) = 4√5
1) 15√5 + 4√5 = 19√5 the √5 is the same square root so you add the coefficients
perfect square factors are bold
4×√(25×2)×√(9×2) =
4×5×3×√(2×2) =
60×√4 =
2) 60 × 2 = 120
the numbers are both irrational but the answer is rational because √2 ×√2 = √4 = 2
What is the quotient for the expression 2,314 divided by 4
Answers
Answer:
578.5
Step-by-step explanation:
Hello people ~
find the area of a circle with radius 13 m .
Answers
Answer:
Area of a circle
=πr^2
therefore...
radius r =13
but π =22/7
so..
=22/7 × 13^2
=22/7 ×169
=531.14m
Area of circle
πr²π(13)²169π530.66m²Drag each set of coordinates to the correct location on the table. Calculate the distance between the pairs of coordinates, and classify them according to the distance between them. (3, 4) and (2, 1) (3, 7) and (5, 2) (5, -2) and (3, 3) (-2, 3) and (1, 2) (-4, -2) and (-3, 1) (4, -1) and (-1, 1)
Answers
Answer: i need time to work on this but i think its 3,3
Step-by-step explanation:
Answer:
Square Root of 10: (3, 4) and (2,1)
(-4,-2) and (-3, 1) (-2, 3) and (1,2)
Square Root of 29: (3,7) and (5,2)
(5, -2) and (3,3) (4,-1) and (-1,1)
A bag contains 8 red marbles, 4 blue marbles and 2 green marbles. If two marbles are
drawn out of the bag, what is the exact probability that both marbles drawn will be
red?
Answers
Answer:
Step-by-step explanation:
Find the values of the variables:
Answers
Answer:
x = 150
y = 30
Step-by-step explanation:
x° = 5y° (corresponding angles) (1)
x° + y° = 180° (Linear pair angles) (2)
From equations (1) and (2), we find:
5y° + y° = 180°
-> 6y° = 180°
-> y° = 180°/6
-> y° = 30°
-> y= 30
-> x° = 5(30°) = 150° (From equation 1)
-> x = 150
what is the area of a circle with a diameter of 4cm
please respond with full work too! pls and ty
{for a BIG test}
for algebra topic: reflections
Answers
Answer:
Are of a Circle = pie x radius squared
Step-by-step explanation:
A = 3.14*2squared
A = 3.14*4
A = 12.56
ok, A movie theater has a seating capacity of 277. The theater charges $5.00 for children, $7.00 for students, and $12.00 for adults. There are half as many adults as there are children. If the total ticket sales were $ 2002, How many children, students, and adults attended?
__ children attended.
__ students attended.
__ adults attended. brainliest for it
Answers
Using a system of equations, it is found that:
126 children attended.88 students attended.63 adults attended.What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the variables are:
Variable x: Number of children that attended.Variable y: Number of students that attended.Variable z: Number of adults that attended.The theater was full, hence:
x + y + z = 277.
There are half as many adults as there are children, hence:
x = 2z.
The theater charges $5.00 for children, $7.00 for students, and $12.00 for adults. The total ticket sales were of $2002, hence:
5x + 7y + 12z = 2002.
Replacing the second equation into the other two:
y + 3z = 277 -> y = 277 - 3z
7y + 22z = 2002.
Then:
7(277 - 3z) + 22z = 2002
z = 63
y = 277 - 3z = 277 - 3 x 63 = 88
x = 2z = 2 x 63 = 126.
Hence:
126 children attended.88 students attended.63 adults attended.More can be learned about a system of equations at https://brainly.com/question/24342899
Help a b c d or e what one
Answers
Answer: D
Step-by-step explanation: The others dont have A m or an s
Can someone please help me
Answers
Answer:
x = 0.71; y = 2.43
Step-by-step explanation:
y = 2x + 1
y = 9x - 4
9x - 4 = 2x + 1
7x = 5
x = 5/7
y = 2x + 1
y = 2(5/7) + 7/7
y = 10/7 + 7/7
y = 17/7
x = 0.71; y = 2.43
WILL GET BRAINIEST!!!! I-ready!!! PLEASE HELP!!!
Answers
Answer:
i am not sure but i think it is 280