Question: 12 m
65°
X
Ground
Z
Y
(a) Calculate YZ, the vertical height of the signboard above the ground.
Answer .................................... m [2]
After a month, Jafar noticed that the signboard was placed at the wrong position.
He shifted the signboard horizontally to the right by 25 m as shown below.
There was no change in the vertical height of the signboard to the ground.
Signboard
25 m
Signboard
12 m
65°
X
Ground
Z
B
Y
A
(i) Jafar needs to replace the string of coloured flags.
Calculate the new length of string XA, needed to attach to the position of the signboard at A.
Answer .............................. m [4]
(ii) Calculate the angle the string now makes with the ground.
Answer .............................. [2]
1. **Problem statement:**
Calculate the vertical height $YZ$ of the signboard above the ground given the hypotenuse $XY=12$ m and angle $\angle X=65^\circ$.
2. **Formula and rules:**
In a right triangle, the vertical height opposite the angle is given by:
$$ YZ = XY \times \sin(65^\circ) $$
3. **Calculation of $YZ$:**
$$ YZ = 12 \times \sin(65^\circ) $$
Using $\sin(65^\circ) \approx 0.9063$,
$$ YZ = 12 \times 0.9063 = 10.8756 \approx 10.88 \text{ m} $$
4. **After shifting the signboard horizontally by 25 m to the right, the vertical height remains the same:**
$$ YZ = 10.88 \text{ m} $$
5. **Calculate the new length of string $XA$:**
The horizontal distance $XA$ is the original horizontal distance $XZ$ plus the shift $25$ m.
First, find $XZ$ using:
$$ XZ = XY \times \cos(65^\circ) $$
Using $\cos(65^\circ) \approx 0.4226$,
$$ XZ = 12 \times 0.4226 = 5.0712 \text{ m} $$
6. **New horizontal distance $XA$:**
$$ XA = XZ + 25 = 5.0712 + 25 = 30.0712 \text{ m} $$
7. **Calculate the length of string $XA$ using Pythagoras theorem:**
$$ XA = \sqrt{(YZ)^2 + (XA)^2} = \sqrt{(10.88)^2 + (30.0712)^2} $$
$$ = \sqrt{118.3744 + 904.277} = \sqrt{1022.6514} = 31.97 \text{ m} $$
8. **Calculate the angle $\theta$ the string makes with the ground:**
$$ \tan \theta = \frac{YZ}{XA} = \frac{10.88}{30.0712} = 0.3619 $$
$$ \theta = \arctan(0.3619) = 19.9^\circ $$
**Final answers:**
- Vertical height $YZ = 10.88$ m
- New length of string $XA = 31.97$ m
- Angle with ground $\theta = 19.9^\circ$