The problem describes a fish shop with a vertical sign [FH] of height 3m. A cable is tied from the top of the sign, F, to a point G on the ground, 1.4m from the base of the sign H. Part (i) asks to find the length of the cable |GF| using the Pythagorean theorem. Part (ii) asks to show that the angle ∠FGH = 65°, correct to the nearest degree.

1. Problem Description

The problem describes a fish shop with a vertical sign [FH] of height 3m. A cable is tied from the top of the sign, F, to a point G on the ground, 1.4m from the base of the sign H. Part (i) asks to find the length of the cable |GF| using the Pythagorean theorem. Part (ii) asks to show that the angle ∠FGH = 65°, correct to the nearest degree.

2. Solution Steps

(i) We need to find the length of |GF|. Since FGH is a right-angled triangle, we can use the Pythagorean theorem:

$|GF|^2 = |FH|^2 + |GH|^2$

We are given |FH| = 3m and |GH| = 1.4m. Therefore,

$|GF|^2 = 3^2 + (1.4)^2$

$|GF|^2 = 9 + 1.96$

$|GF|^2 = 10.96$

$|GF| = \sqrt{10.96}$

$|GF| \approx 3.310589$

Rounding to 1 decimal place, |GF| = 3.3m.

(ii) We want to find the angle ∠FGH.

Since we have a right-angled triangle, we can use the tangent function:

$tan(∠FGH) = \frac{|FH|}{|GH|} = \frac{3}{1.4}$

$tan(∠FGH) \approx 2.142857$

$∠FGH = arctan(2.142857)$

$∠FGH \approx 64.9835...$

Rounding to the nearest degree, $∠FGH \approx 65°$.

3. Final Answer

(i) 3.3 m

(ii) $∠FGH \approx 65°$