<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>
Answer:
1 1/2 cups of peanut butter
1 cup of vegetable shortening
2 1/2 cups grimly packed light brown sugar
6 tablespoons of milk
5 1/2 teaspoon vanilla extract
2 large eggs
3 cups of flour
1 1/2 teaspoon of baking soda
1/2 teaspoon of salt
Step-by-step explanation:
You just double everything by 2
<em>40</em>
- Step-by-step explanation:
<em>2 + 10 = 24</em>
<em>2(2 + 10) = 2×12 = 24</em>
<em>3 + 6 = 27</em>
<em>3(3 + 6) = 3×9 = 27</em>
<em>7 + 2 = 63</em>
<em>7(7 + 2) = 7×9 = 63</em>
<em>5 + 3 = 40</em>
<em>5(5 + 3) = 5×8 = 40</em>
Enter a problem...
Calculus Examples
Popular Problems Calculus Find the Domain and Range f(x)=5x-3
f
(
x
)
=
5
x
−
3
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
(
−
∞
,
∞
)
Set-Builder Notation:
{
x
|
x
∈
R
}
The range is the set of all valid
y
values. Use the graph to find the range.
Interval Notation:
(
−
∞
,
∞
)
Set-Builder Notation:
{
y
|
y
∈
R
}
Determine the domain and range.
Domain:
(
−
∞
,
∞
)
,
{
x
|
x
∈
R
}
Range:
(
−
∞
,
∞
)
,
{
y
|
y
∈
R
}
Answer:
The answer is a). 5/24.
Step-by-step explanation:
For the seventh graders there are 4 Boys + 2 Girls which totals 6 students.
- The probability that a girl will be picked from the <em>seventh graders is 2/6</em>
For the eighth graders, there are 3 Boys + 5 Girls which totals 8 students
- The probability that a girl will be picked from the <em>eighth graders is 5/8</em>
To find the<u> probability that both are girls</u>, we multiply the two fractions together:
