To solve this problem, we need to use the concept of direct variation. Direct variation implies that if one quantity increases, the other also increases at a constant rate. The relationship can be written as:
[tex]\[ \text{increase in length} = k \times \text{mass} \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
Given:
- An increase in length of 2.5 cm corresponds to a mass of 50 kg.
First, we need to determine the constant [tex]\( k \)[/tex]. We set up our equation with the given values:
[tex]\[ 2.5 = k \times 50 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{2.5}{50} \][/tex]
[tex]\[ k = 0.05 \][/tex]
Now that we have the constant [tex]\( k \)[/tex], we can use it to find the required mass that would produce an increase of 7.5 cm. Set up the direct variation equation with the new increase:
[tex]\[ 7.5 = 0.05 \times \text{mass} \][/tex]
Solving for mass:
[tex]\[ \text{mass} = \frac{7.5}{0.05} \][/tex]
[tex]\[ \text{mass} = 150 \text{ kgs} \][/tex]
So, a mass of 150 kgs would produce an increase of 7.5 cms in length of the spring.
