Gena is working on her math project. She has trimmed the sides of a square cardboard by 7 decimeters. The area of the cardboard then has decreased by 81 square decimeters. What was the original length of a side of the cardboard?

Question

Avery Allen · Accepted Answer

2#sqrt(22)# or approx 9.381 decimeters

#sqrt(7+81)# = #sqrt(88)# = 2#sqrt(22)# or approx 9.381 decimeters You add up all the areas to find the area of the square to get 88 square decimeters. Because the area of a square is #a^2# where a is the side length, we need to square root 88 to get the side length.

HIX Tutor · Answer

undefined Let’s solve this step by step!

1. **Define Variables**: 
   Let $ x $ be the original length of a side of the square cardboard.

2. **Area of Original Square**: 
   The area of the square is $ x \times x = x^2 $.

3. **New Length of Side**: 
   Gena trims 7 decimeters from each side, so the new length is $ x - 7 $.

4. **Area of the New Square**: 
   The area of the new square is $ (x - 7) \times (x - 7) = (x - 7)^2 $.

5. **Area Decrease**: 
   The area decreased by 81 square decimeters. This means:
   $$
   x^2 - (x - 7)^2 = 81
   $$

6. **Expand the Equation**: 
   We will expand $ (x - 7)^2 $.
   $$
   (x - 7)^2 = x^2 - 14x + 49
   $$
   So, we can rewrite the equation:
   $$
   x^2 - (x^2 - 14x + 49) = 81
   $$

7. **Simplify the Equation**: 
   $$
   x^2 - x^2 + 14x - 49 = 81
   $$
   $$
   14x - 49 = 81
   $$

8. **Solve for $ x $**: 
   Add 49 to both sides:
   $$
   14x = 81 + 49
   $$
   $$
   14x = 130
   $$
   Now divide by 14:
   $$
   x = \frac{130}{14}
   $$
   $$
   x = 9.2857
   $$

9. **Estimate of the Answer**: 
   This decimal indicates we may not have rounded from the original conditions, so let’s check for any possible rounding. Verify if you want whole numbers.

10. **Conclusion**: 
   The original side was about 9.3 decimeters (if not needing a whole number).

Does this make sense?