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Anon25 [30]
3 years ago
11

The volume of a cylinder is 81 cubic feet. What is the volume of a cone that fits exactly inside the cylinder? Provide an explan

ation and proof for your answer.
Mathematics
1 answer:
expeople1 [14]3 years ago
8 0
V ( cylinder ) = r² π h
V ( cone ) = 1/3 r² π h
A cone that fits exactly inside the cylinder has the volume:
V ( cone ) = 1/3 V ( cylinder = 1/3 · 81 = 27 ft³
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Prove by mathematical induction that 1+2+3+...+n= n(n+1)/2 please can someone help me with this ASAP. Thanks​
Iteru [2.4K]

Let

P(n):\ 1+2+\ldots+n = \dfrac{n(n+1)}{2}

In order to prove this by induction, we first need to prove the base case, i.e. prove that P(1) is true:

P(1):\ 1 = \dfrac{1\cdot 2}{2}=1

So, the base case is ok. Now, we need to assume P(n) and prove P(n+1).

P(n+1) states that

P(n+1):\ 1+2+\ldots+n+(n+1) = \dfrac{(n+1)(n+2)}{2}=\dfrac{n^2+3n+2}{2}

Since we're assuming P(n), we can substitute the sum of the first n terms with their expression:

\underbrace{1+2+\ldots+n}_{P(n)}+n+1 = \dfrac{n(n+1)}{2}+n+1=\dfrac{n(n+1)+2n+2}{2}=\dfrac{n^2+3n+2}{2}

Which terminates the proof, since we showed that

P(n+1):\ 1+2+\ldots+n+(n+1) =\dfrac{n^2+3n+2}{2}

as required

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3 years ago
PLS HELP I NEED HELP NOW<br> I BEG
stepan [7]
Hello there! The area of the triangle portion is 11 square units, the area of the rectangle portion is 77 square units, and the area of the entire figure is 88 square units.

To find the area of the triangle, we can follow the formula:

A = LW/2 (which means length x width divided by 2)

Given the formula:

2 • 11 = 22

22 divided by 2 gives us 11 square units.

To find the area of the rectangle portion, we can follow the area formula:

A = LW (which means area = length x width)

Given the formula:

7 • 11 = 77 square units

To find the area of the whole figure, we add the areas of both isolated shapes:

11 + 77 = 88 square units.

Therefore, our area for the entire figure is 88 square units. If you need any extra help, let me know and I will gladly assist you.
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