The dimensions of a certain machine are 48" $\times$ 30" $\times$ 52". If the size of the machine is increased proportionately until the sum of its dimensions equals 156", what will be the increase in the shortest side?

### Concept & Proportional Scaling When an object is increased proportionately, all of its dimensions are multiplied by the same scale factor. The sum of the dimensions will also scale by this exact same factor. ### Step-by-Step Solution 1. **Given:** Original dimensions are 48", 30", 52". New sum of dimensions = 156". 2. **Calculate Original Sum:** $$ \text{Sum} = 48 + 30 + 52 = 130\text{"} $$ 3. **Find the Scale Factor:** $$ \text{Scale Factor} = \frac{\text{New Sum}}{\text{Original Sum}} = \frac{156}{130} = \frac{6}{5} = 1.2 $$ 4. **Identify the Shortest Side:** The shortest original side is 30". 5. **Calculate New Shortest Side:** $$ \text{New side} = 30 \times \frac{6}{5} = 36\text{"} $$ 6. **Calculate the Increase:** $$ \text{Increase} = 36 - 30 = 6\text{"} $$ ### Exam Strategy & Shortcut Instead of finding the new dimension and subtracting, calculate the proportional increase directly: The scale factor is $1.2$, meaning a $20\%$ increase. $20\%$ of 30 is simply 6. ### Common Pitfall Trying to find the volume and scaling that, forgetting that linear dimensions scale directly with their sum, while volume scales cubically. ### Final Answer Therefore, the correct answer is **6"**.