A cylinder is a solid figure with straight parallel sides and a circular or oval cross section with a radius (r) and a height (h). The volume of a cylinder is π r²h and the surface area is 2(π r²) + 2π rh.

Angles around a line add up to 180°. Angles around a point add up to 360°. When two lines intersect, adjacent angles are supplementary (they add up to 180°) and angles across from either other are vertical (they're equal).

The first step to solve a quadratic expression that's not set to zero is to solve the equation so that it is set to zero:

b² + 16b + 51 = b - 3
b² + 16b + 51 + 3 = b
b² + 16b - b + 54 = 0
b² + 15b + 54 = 0

Next, factor the quadratic equation:

b² + 15b + 54 = 0
(b + 6)(b + 9) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (b + 6) or (b + 9) must equal zero:

If (b + 6) = 0, b must equal -6
If (b + 9) = 0, b must equal -9

So the solution is that b = -6 or -9

A circle is a figure in which each point around its perimeter is an equal distance from the center. The radius of a circle is the distance between the center and any point along its perimeter. A chord is a line segment that connects any two points along its perimeter. The diameter of a circle is the length of a chord that passes through the center of the circle and equals twice the circle's radius (2r).

To find the diagonal we need to know the length of one of the square's sides. We know the area and the area of a square is the length of one side squared:

a = s²

so the length of one side of the square is:

s = \( \sqrt{a} \) = \( \sqrt{81} \) = 9

The Pythagorean theorem defines the square of the hypotenuse (diagonal) of a triangle with a right angle as the sum of the squares of the other two sides:

c² = a² + b²
c² = 9² + 9²
c² = 162
c = \( \sqrt{162} \) = \( \sqrt{81 x 2} \) = \( \sqrt{81} \) \( \sqrt{2} \)
c = 9\( \sqrt{2} \)