A. vertical angles are congruent
b. alternate interior angles are congruent
c. if alternate interior angles are congruent, lines are parallel <span />
The roots of the polynomial <span><span>x^3 </span>− 2<span>x^2 </span>− 4x + 2</span> are:
<span><span>x1 </span>= 0.42801</span>
<span><span>x2 </span>= −1.51414</span>
<span><span>x3 </span>= 3.08613</span>
x1 and x2 are in the desired interval [-2, 2]
f'(x) = 3x^2 - 4x - 4
so we have:
3x^2 - 4x - 4 = 0
<span>x = ( 4 +- </span><span>√(16 + 48) </span>)/6
x_1 = -4/6 = -0.66
x_ 2 = 2
According to Rolle's theorem, we have one point in between:
x1 = 0.42801 and x2 = −1.51414
where f'(x) = 0, and that is <span>x_1 = -0.66</span>
so we see that Rolle's theorem holds in our function.
The square root of negative four is 2
To express the height as a function of the volume and the radius, we are going to use the volume formula for a cylinder:

where

is the volume

is the radius

is the height
We know for our problem that the cylindrical can is to contain 500cm^3 when full, so the volume of our cylinder is 500cm^3. In other words:

. We also know that the radius is r cm and height is h cm, so

and

. Lets replace the values in our formula:





Next, we are going to use the formula for the area of a cylinder:

where

is the area

is the radius

is the height
We know from our previous calculation that

, so lets replace that value in our area formula:



By the commutative property of addition, we can conclude that: