To tackle this problem, let's go through the details step by step:
### Part (a)
First, we need to establish an equation that relates the area of the rectangle to the square of the length of its shorter side. We've been given that the area varies directly with the square of its shorter side. Mathematically, this can be expressed as:
[tex]\[ \text{Area} = k \times (\text{shorter side})^2 \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
Given:
- The length of the shorter side is [tex]\( 8 \)[/tex] cm.
- The area is [tex]\( 384 \)[/tex] cm².
Using these values, we can find [tex]\( k \)[/tex] by substituting into the equation:
[tex]\[ 384 = k \times (8)^2 \][/tex]
[tex]\[ 384 = k \times 64 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{384}{64} \][/tex]
[tex]\[ k = 6 \][/tex]
Therefore, the equation relating the area of the rectangle to the square of the length of the shorter side is:
[tex]\[ \text{Area} = 6 \times (\text{shorter side})^2 \][/tex]
### Part (b)
Now, we need to find the area of the rectangle if the length of the shorter side is 12 cm.
Using the equation from Part (a):
[tex]\[ \text{Area} = 6 \times (\text{shorter side})^2 \][/tex]
Substitute [tex]\( 12 \)[/tex] cm for the shorter side:
[tex]\[ \text{Area} = 6 \times (12)^2 \][/tex]
[tex]\[ \text{Area} = 6 \times 144 \][/tex]
[tex]\[ \text{Area} = 864 \text{ cm}^2 \][/tex]
Therefore, the area of the rectangle when the length of the shorter side is 12 cm is [tex]\( 864 \)[/tex] cm².
