Question (4 Marks)
Below is an example of a mathematical wall clock.
In the process of designing a new mathematical wall clock, four values have been inserted in the correct place on Diagram 1 below and there are four more values that need to be placed on the diagram.
Label the mathematical clock by placing the values in the correct places.
▶️Answer/Explanation
Ans:
To label the mathematical wall clock, we need to determine the correct placement for the given values. Let’s analyze each value and find their corresponding locations on the clock.
1. Value: $\left|4 \cos 120^{\circ}\right|$
The expression $\left|4 \cos 120^{\circ}\right|$ represents the absolute value of the cosine function evaluated at an angle of $120^\circ$. The cosine function has a positive value in the second quadrant, so $\cos 120^\circ = -\frac{1}{2}$. Therefore, $\left|4 \cos 120^{\circ}\right| = 2$.
Since we need to place this value on the clock, we can put it at the 2 o’clock position.
2. Value: $10 x^0$
The expression $10 x^0$ simplifies to $10 \cdot 1 = 10$. Any value raised to the power of 0 is equal to 1.
We can place this value at the 12 o’clock position on the clock.
3. Value: $\frac{18 x+9}{6 x+3}$
This expression represents a rational function. To find its correct placement, we need to determine its simplified form or find a suitable equivalent representation.
By factoring the numerator and denominator, we can simplify the expression:
$$\frac{18 x+9}{6 x+3} = \frac{9(2x+1)}{3(2x+1)} = \frac{9}{3} = 3.$$
Therefore, we can place the value 3 at the 3 o’clock position.
4. Value: $3 x-4=17$
This equation needs to be solved for $x$. Let’s simplify it step by step:
$$3 x – 4 = 17.$$
Adding 4 to both sides:
$$3 x = 21.$$
Dividing both sides by 3:
$$x = 7.$$
The solution to the equation is $x = 7$. We can place the value 7 at the 7 o’clock position on the clock.
