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erik [133]
2 years ago
6

A right circular cylinder is inscribed in a sphere with diameter 4cm as shown. If the cylinder is open at both ends, find the la

rgest possible surface area of the cylinder

Mathematics
1 answer:
SOVA2 [1]2 years ago
4 0

Answer:

8\pi\text{ square cm}

Step-by-step explanation:

Since, we know that,

The surface area of a cylinder having both ends in both sides,

S=2\pi rh

Where,

r = radius,

h = height,

Given,

Diameter of the sphere = 4 cm,

So, by using Pythagoras theorem,

4^2 = (2r)^2 + h^2   ( see in the below diagram ),

16 = 4r^2 + h^2

16 - 4r^2 = h^2

\implies h=\sqrt{16-4r^2}

Thus, the surface area of the cylinder,

S=2\pi r(\sqrt{16-4r^2})

Differentiating with respect to r,

\frac{dS}{dr}=2\pi(r\times \frac{1}{2\sqrt{16-4r^2}}\times -8r + \sqrt{16-4r^2})

=2\pi(\frac{-4r^2+16-4r^2}{\sqrt{16-4r^2}})

=2\pi(\frac{-8r^2+16}{\sqrt{16-4r^2}})

Again differentiating with respect to r,

\frac{d^2S}{dt^2}=2\pi(\frac{\sqrt{16-4r^2}\times -16r + (-8r^2+16)\times \frac{1}{2\sqrt{16-4r^2}}\times -8r}{16-4r^2})

For maximum or minimum,

\frac{dS}{dt}=0

2\pi(\frac{-8r^2+16}{\sqrt{16-4r^2}})=0

-8r^2 + 16 = 0

8r^2 = 16

r^2 = 2

\implies r = \sqrt{2}

Since, for r = √2,

\frac{d^2S}{dt^2}=negative

Hence, the surface area is maximum if r = √2,

And, maximum surface area,

S = 2\pi (\sqrt{2})(\sqrt{16-8})

=2\pi (\sqrt{2})(\sqrt{8})

=2\pi \sqrt{16}

=8\pi\text{ square cm}

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Answer:

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vertex (h , k): h = -2 , k = -162

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3 years ago
A car travels 182 kilometers in 4 hours with the constant speed. how much time will it take traveling 42 kilometers
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Answer:

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Step-by-step explanation:

A car travels,

182 km ⇒ 4 hours

And we have to find the time that it takes to travel 42 km .

Lets make a equation.

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Now use cross multiplication.

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Answer:

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Insert x + 3 instead of x into the equation of the function f(x):

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use (a + b)² = a² + 2ab + b²

= 3 - 6(x² + 2(x)(3) + 3²) = 3 - 6(x² + 6x + 9)

use distributive property

= 3 + (-6)(x²) + (-6)(6x) + (-6)(9) = 3 - 6x² - 36x - 54

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