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Aneli [31]
3 years ago
8

A manufacturer drills a round hole of radius r through the center of a metal sphere of radius r. find the volume of the remainin

g metal “bead.”

Mathematics
1 answer:
Kisachek [45]3 years ago
3 0
The media attached below is the bottom/top view of drilled sphere. If you look it from sideways, you'll see that the drilled hole is in the form of cylindrical shape. So,
Volume of the drilled shape= volume of cylinder = 
πr²h, where h is 2r.
∴Volume of cylinder(V1) = πr²(2r) = 2πr³.
Volume of sphere(V2) = 4πr³/3.

Therefore volume of remaining metal bead = V1 - V2
                                                                         =   2
πr³ - 4πr³/3 
                                                                         =   (6πr³ - 4πr³)/3
                                                                         =   2
πr³/3.

Hence volume of remaining bead is 2
πr³/3

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The sum of the first n consecutive even numbers can be found using S=n2+n, where n≥2. What is the value of n when the sum is 42?
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  • C. 6

Step-by-step explanation:

<u>Given</u>

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Divide through everything by <em>b</em> :

\dfrac{a+c}{b+d} = \dfrac{\dfrac ab + \dfrac cb}{1 + \dfrac db}

Since <em>a/b</em> < <em>c/d</em>, it follows that

\dfrac{a+c}{b+d} < \dfrac{\dfrac cd+\dfrac cb}{1 + \dfrac db}

Multiply through everything on the right side by <em>b/d</em> to get

\dfrac{a+c}{b+d} < \dfrac{\dfrac{bc}{d^2}+\dfrac cd}{\dfrac bd+1} = \dfrac{\dfrac cd\left(\dfrac bd+1\right)}{\dfrac bd+1} = \dfrac cd

and so (<em>a</em> + <em>c</em>)/(<em>b</em> + <em>d</em>) < <em>c/d</em>.

For the other side, you can do something similar and divide through everything by <em>d</em> :

\dfrac{a+c}{b+d} = \dfrac{\dfrac ad + \dfrac cd}{\dfrac bd + 1}

and <em>a/b</em> < <em>c/d</em> tells us that

\dfrac{a+c}{b+d} > \dfrac{\dfrac ad + \dfrac ab}{\dfrac bd + 1}

Then

\dfrac{a+c}{b+d} > \dfrac{\dfrac ab + \dfrac{ad}{b^2}}{1 + \dfrac db} = \dfrac{\dfrac ab\left(1+\dfrac db\right)}{1 + \dfrac db} = \dfrac ab

and so (<em>a</em> + <em>c</em>)/(<em>b</em> + <em>d</em>) > <em>a/b</em>.

Then together we get the desired inequality.

5 0
3 years ago
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