Answers
Hi! Your answer is 125/3 π
Please see an explanation for a clear understanding to the problem.
Any questions about the answer/explanation can be asked through comments! :)
Step-by-step explanation:
For this problem, I'll be demonstrating two methods to solve your problem which are Integration method and Formula method. Let's start with Formula method first!
Formula Method
Volume of Cone Formula
[tex]V=\huge{\frac{1}{3}\pi r^2h}[/tex]
r is a radius. Therefore, r² is a diameter.
h is height of cone.
Our r is 5. Therefore, 5² = 25.
Our h is 5.
Substitute these values in the equation.
[tex]V=\huge{\frac{1}{3}\pi (5)^2(5)}\\V=\huge{\frac{1}{3}\pi (25)(5)} \\V=\huge{\frac{1}{3}\pi (125)}\\V=\huge{\frac{125}{3}\pi}[/tex]
Hence, the volume of a cone is 125/3 pi. If you want in Mixed Number form then It'd be 41 and 2/3 pi [ 41 2/3 pi]
Integration Method - Proof
Let r':r = x:h
Therefore,
[tex]\huge{\frac{r'}{r}=\frac{x}{h}}\\\huge{r'=\frac{x}{h}r[/tex]
r and h both are same (r is still radius while h is still height)
[tex]\huge{S(x)=\pi(r')^2[/tex]
The reason why we add pi because the surface is a circle. If the surface is not a circle, we don't add pi.
[tex]\huge{S(x)=\pi(\frac{x}{h}r)^2}\\\huge{S(x)=\pi(\frac{x^2r^2}{h^2})}\\\huge{S(x)=\frac{\pi r^2}{h^2}x^2}\\[/tex]
The range starts from 0 ≤ x ≤ h
[tex]\huge{V=\int\limits^h_0 {S(x)} \, dx[/tex]
Substitute S(x) in.
[tex]\huge{V=\int\limits^h_0 {\frac{\pi r^2}{h^2}x^2 } \, dx[/tex]
Integrate, separate everything in the integrand out except for x^2
[tex]\huge{V=\frac{\pi r^2}{h^2} \int\limits^h_0 {x^2} \, dx}\\\huge{V=\frac{\pi r^2}{h^2}[\frac{1}{3}x^3]^h__0}}\\\huge{V=\frac{\pi r^2}{h^2}[\frac{1}{3}h^3]}\\\huge{V=\pi r^2[\frac{1}{3}h]}\\\huge{V=\frac{1}{3}\pi r^2h}[/tex]
As you notice, Integration method is pretty useful, but it's mostly for an advanced volume lesson only.
Integration Method - Problem Solving
Let r':r = x:h
r is given as 5
h is given as 5 as well.
Therefore, r':5 = x:5
[tex]\huge{\frac{r'}{5}=\frac{x}{5}}\\\huge{r'=\frac{x}{5}(5)}\\r'=x[/tex]
[tex]\huge{S(x)=\pi(r')^2}\\\huge{S(x)=\pi(x^2)}\\\huge{S(x)=\pi x^2}[/tex]
The range of volume starts from surface to height. Therefore the range is 0 ≤ x ≤ 5
[tex]\huge{V=\int\limits^5_0 {S(x)} \, dx }\\\huge{V=\int\limits^5_0 {\pi x^2} \, dx }\\\huge{V=\pi\int\limits^5_0 {x^2} \, dx }\\\huge{V=\pi[\frac{1}{3}x^3]^5__0}}\\\huge{V=\pi[(\frac{1}{3}(5)^3-\frac{1}{3}(0)^3]}\\\huge{V=\pi[(\frac{1}{3}(125)-0)]}\\ \huge{V=\pi[(\frac{1}{3}(125)]}\\\huge{V=\pi(\frac{125}{3})}\\\huge{V=\frac{125}{3}\pi}[/tex]
Notice that both methods have same answer. Hence, we can conclude that the volume of a cone is 125/3 pi units²
Related Questions
I need a hard 8th-grade math question (ON LEVEL) please don't give me the answer also tell me how I can use it in real life
Answers
35 mm
10 cm
20 mm
45 mm
A large bakery has many different products for sale. Suppose that 200 customers come in between 6 am and 10 am. Of the 200 customers, 140 order donuts, 100 order cinnamon rolls, and 80 order both. Suppose a customer is randomly selected. Find P(no donuts and no cinnamon rolls).
Answers
Out of 200 customers, the probability of a customer not ordering either donuts or cinnamon rolls is 0.2 or 20%.
Use the concept of probability defined as:
For the occurrence of an event, it is a ratio of the number of favorable outcomes to the number of total outcomes.
Given that,
Time frame: 6 am to 10 am
Total number of customers: 200
Number of customers who ordered donuts: 140
Number of customers who ordered cinnamon rolls: 100
Number of customers who ordered both donuts and cinnamon rolls: 80
To find the probability of a customer ordering neither donuts nor cinnamon rolls:
Subtract the probability of a customer ordering either donuts or cinnamon rolls or both from 1.
Now calculate:
Probability of ordering donuts:
140 out of 200 customers ordered donuts,
The probability of ordering donuts is,
140/200 = 0.7.
Probability of ordering cinnamon rolls:
100 out of 200 customers ordered cinnamon rolls,
The probability of ordering cinnamon rolls is,
100/200 = 0.5.
Probability of ordering both donuts and cinnamon rolls:
80 out of 200 customers ordered both,
The probability of ordering both is,
80/200 = 0.4.
Now, to find the probability of a customer ordering neither donuts nor cinnamon rolls,
Subtract the probability of ordering either or both from 1:
P(no donuts & no cinnamon rolls) = [tex]1 - P( \text{donuts}) - P(\text{cinnamon rolls}) + P(\text{donuts & cinnamon rolls})[/tex]
[tex]\text{P(no donuts & no cinnamon rolls)}= 1 - 0.7 - 0.5 + 0.4[/tex]
[tex]\text{P(no donuts & no cinnamon rolls)} = 0.2[/tex]
Hence,
The probability of a customer not ordering either donuts or cinnamon rolls is 0.2 or 20%.
Learn more about the probability visit:
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One side of a rectangle is 3 yards longer than twice the length of another side. If the perimeter is 27 yards, what is the length of the longer side
Answers
Answer:
10 yards
Step-by-step explanation:
;)
The required length of the longer side is 10 yards.
What is the perimeter?Perimeter, is the measure of the figure on its circumference.
Here,
Let's assume that the length of one side of the rectangle is x yards. According to the problem, the other side of the rectangle is 3 yards longer than twice the length of this side. So, the length of the other side can be expressed as 2x + 3 yards.
The perimeter of a rectangle is the sum of the lengths of all its sides. So, we can write an equation to represent the problem:
Perimeter = 2 × length + 2 × width
Substituting the expressions for length and width, we get:
27 = 2(x) + 2(2x + 3)
27 = 2x + 4x + 6
27 = 6x + 6
21 = 6x
x = 3.5
So, the length of one side of the rectangle is 3.5 yards. The other side of the rectangle is 2(3.5) + 3 = 10 yards.
Therefore, the length of the longer side is 10 yards.
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what 2 numbers multiply to 77 and add to 18
Answers
Answer:
11 and 7
Step-by-step explanation:
11+7=18, and 11 x 7=77
Hope I helped!
Answer:
i think it is 11 and 7
Step-by-step explanation:
help? thank you have a great day
Answers
Answer:
12 square units
50 square feet
77 square meters.
Step-by-step explanation:
cut of the part that's sticking out and put it on the other side to get a rectangle.
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Answer:
49
Step-by-step explanation:
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49
Answer:
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Step-by-step explanation:
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Answers
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Step-by-step explanation:
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Answers
Answer:Percent Increase n the number of parts usually made = 80%
Step-by-step explanation:
Percent Increase = (Increase / Initial Number of parts made daily) X 100/1
Initial Number of parts made daily = 45
Current number of parts ,made daily = 81
Percent Increase = 81-45/ 45 x 100
=36/45 x100/1
= 0.8 x 100
= 80%
Answer:
80 percent
Step-by-step explanation:
What is value of x?
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Enter your answer in the box.
x = ?
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Answers
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27
Step-by-step explanation:
Verified correct with test results.
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Choose 1 answer:
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Answers
Step-by-step explanation:
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Answer:
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Step-by-step explanation:
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Answers
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Step-by-step explanation:
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Step-by-step explanation:
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Answers
Answer:
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Step-by-step explanation:
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Answers
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Step-by-step explanation:
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match the rule to the sequence
n + 5 8, 13, 18, 23, 28
5n 6, 7, 8, 9, 10
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Answers
Answer:
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b) aₙ = 5n → 5,10,15,20,25...
c) aₙ = 3n+5 → 8,11,14,17,20
d) aₙ = 5n+3 → 8,13,18,23,28
Step-by-step explanation:
Explanation
Given [tex]n^{th}[/tex] term of the sequence
aₙ = n+5
Put n =1
a₁ = 1+5 =6
put n =2
a₂ = 2+5 =7
Put n=3
a₃ = 3+5 =8
and so on
The sequence 6,7,8,9,10....
b)
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Put n=1 ⇒ a₁ = 5
put n=2 ⇒ a₂ = 10
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c)
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put n=1 ⇒ a₁ =8
Put n=2 ⇒ a₂ =11
Put n=3 ⇒ a₃ = 14
and so on
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d)
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Put n=1 ⇒ a₁ = 8
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Put n=3 ⇒a₃ = 18
and so on
The sequence 8,13,18, 23..
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Answers
Answer:
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Answer:
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It is right