Let x = the length of the ladder.

From Pythagorean theorem,

The hypothenuse (c) is the longest side. The side facing the right-angle (∟)

x² = 6² + 8²

x² = 36 + 64

x² = 100

Take square root of both sides.

x = $\sqrt{\mathrm{100}}$ = 10

The ladder is 10 m long.

In ∆PQR, |QR| is the hypothenuse since ∠ QPR = 90°

From pythagoras theorem, p² = r² + q²

Substituting q² = 25 - r² into the equation

p² = r² + (25 - r²)

p² = r² + 25 - r²

p² = 25

Take square root of both sides

p = √25

p = 5

Let the width of the gutter = w

tan(θ) = $\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

tan 35° = $\frac{15}{\mathrm{w}}$

Multiply both sides by w

w x tan 35° = 15

Divide both sides by tan 35°

w = $\frac{15}{\mathrm{tan\; 35°}}$ = 21.42 cm

Let x = the length of the ladder.

From Pythagorean theorem,

The hypothenuse (c) is the longest side. The side facing the right-angle (∟)

x² = 6² + 8²

x² = 36 + 64

x² = 100

Take square root of both sides.

x = $\sqrt{\mathrm{100}}$ = 10

The ladder is 10 m long.