<span>12.3
Volume function: v(x) = ((18-x)(x-1)^2)/(4pi)
Since the perimeter of the piece of sheet metal is 36, the height of the tube created will be 36/2 - x = 18-x.
The volume of the tube will be the area of the cross section multiplied by the height. The area of the cross section will be pi r^2 and r will be (x-1)/(2pi). So the volume of the tube is
v(x) = (18-x)pi((x-1)/(2pi))^2
v(x) = (18-x)pi((x-1)^2/(4pi^2))
v(x) = ((18-x)(x-1)^2)/(4pi)
The maximum volume will happen when the value of the first derivative is zero. So calculate the first derivative:
v'(x) = (x-1)(3x - 37) / (4pi)
Convert to quadratic equation.
(3x^2 - 40x + 37)/(4pi) = 0
3/(4pi)x^2 - (10/pi)x + 37/(4pi) = 0
Now calculate the roots using the quadratic formula with a = 3/(4pi), b = -10/pi, and c = 37/(4pi)
The roots occur at x = 1 and x = 12 1/3. There are the points where the slope of the volume equation is zero. The root of 1 happens just as the volume of the tube is 0. So the root of 12 1/3 is the value you want where the volume of the tube is maximized. So the answer to the nearest tenth is 12.3</span>
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Answer:
Proved.
Step-by-step explanation:
The functions are:
1.) f(x) = 3x - 27 (* <em>I am giving an answer using this equation. Perhaps you did't copy the question well!</em>)
2.) g(x) = 
+ 9
If two functions are inverses of each other, then:
f(g(x)) = x and g(f(x)) = x situation must be satisfied.
f(g(x)) = 3(
) + 27
We simply it to get;
f(g(x)) = x - 27 + 27 = x (*<em>This is correct)</em>
<em />
g(f(x)) = 
(3x - 27) + 9 = x - 9 + 9 = x (* <em>This is also correct!</em>)
Answer:
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