The surface area of a cylinder is 2πr² + 2πrh:

sa = 2πr² + 2πrh
sa = 2π(6²) + 2π(6 x 1)
sa = 2π(36) + 2π(6)
sa = (2 x 36)π + (2 x 6)π
sa = 72π + 12π
sa = 84π

The Pythagorean theorem defines the relationship between the side lengths of a right triangle. The length of the hypotenuse squared (c²) is equal to the sum of the two perpendicular sides squared (a² + b²): c² = a² + b² or, solved for c, \(c = \sqrt{a + b}\)

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 = \( \sqrt{a} \) = \( \sqrt{49} \) = 7

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 = 7² + 7²
c² = 98
c = \( \sqrt{98} \) = \( \sqrt{49 x 2} \) = \( \sqrt{49} \) \( \sqrt{2} \)
c = 7\( \sqrt{2} \)

The first step to solve a quadratic equation that's set to zero is to factor the quadratic equation:

b² + 6b - 7 = 0
(b - 1)(b + 7) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (b - 1) or (b + 7) must equal zero:

If (b - 1) = 0, b must equal 1
If (b + 7) = 0, b must equal -7

So the solution is that b = 1 or -7

The perimeter of a triangle is the sum of the lengths of its sides:

p = x + y + z
p = 10cm + 12cm + 7cm = 29cm