Answer:
2.29 ft of side length and 1.14 height
Step-by-step explanation:
a) Volume V = x2h, where x is side of square base and h is hite.
Then surface area S = x2 + 4xh because box is open.
b) From V = x2h = 6 we have h = 6/x2.
Substitude in formula for surface area: S = x2 + 4x·6/x2, S = x2 + 24/x.
We get S as function of one variable x. To get minimum we have to find derivative S' = 2x - 24/x2 = 0, from here 2x3 - 24 = 0, x3 = 12, x = (12)1/3 ≅ 2.29 ft.
Then h = 6/(12)2/3 = (12)1/3/2 ≅ 1.14 ft.
To prove that we have minimum let get second derivative: S'' = 2 + 48/x3, S''(121/3) = 2 + 48/12 = 6 > 0.
And because by second derivative test we have minimum: Smin = (12)2/3 + 4(12)1/3(12)1/3/2 = 3(12)2/3 ≅ 15.72 ft2
Answer:
The volume of the ball with the drilled hole is:

Step-by-step explanation:
See attached a sketch of the region that is revolved about the y-axis to produce the upper half of the ball. Notice the function y is the equation of a circle centered at the origin with radius 15:

Then we set the integral for the volume by using shell method:

That can be solved by substitution:

The limits of integration also change:
For x=5: 
For x=15: 
So the integral becomes:

If we flip the limits we also get rid of the minus in front, and writing the root as an exponent we get:

Then applying the basic rule we get:

Since that is just half of the solid, we multiply by 2 to get the complete volume:


Answer:
4
Step-by-step explanation:
Divide 28 by 7 and get 4.
Step-by-step explanation:
2πr
2x 3.14 x 8.1 = 50.22in
Answer:
b. circle; 
Step-by-step explanation:
The given conic has equation;

We complete the square to obtain;

This is a circle with center;

This implies that;

When the circle is rotated through an angle of
,
The new center is obtained using;
and 
We plug in the given angle with x and y values to get;
and 
This gives us;

The equation of the rotated circle is;

Expand;

Multiply through by 4; to get

Write in general form;

Divide through by 2.
