\boxed2520Question: An industrial designer creates a series of modular shelves where each shelf has 4 more panels than the previous one. If the first shelf has 7 panels, how many shelves can be built with a total of 150 panels? - Dachbleche24

Understanding the Context

This scenario follows the arithmetic sequence pattern, where:
- First term \( a = 7 \)
- Common difference \( d = 4 \)
- Total panels for \( n \) shelves is \( S_n = 150 \)

The Formula for the Sum of an Arithmetic Sequence
The total number of panels used for \( n \) shelves is given by:
\[
S_n = \frac{n}{2} (2a + (n - 1)d)
\]

Plug in the known values:
\[
150 = \frac{n}{2} (2 \cdot 7 + (n - 1) \cdot 4)
\]
\[
150 = \frac{n}{2} (14 + 4n - 4)
\]
\[
150 = \frac{n}{2} (4n + 10)
\]
Multiply both sides by 2 to eliminate the denominator:
\[
300 = n (4n + 10)
\]
\[
300 = 4n^2 + 10n
\]
Rewriting into standard quadratic form:
\[
4n^2 + 10n - 300 = 0
\]

Simplify and Solve the Quadratic Equation
Divide all terms by 2:
\[
2n^2 + 5n - 150 = 0
\]

Key Insights

Use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 5 \), \( c = -150 \):
\[
n = \frac{-5 \pm \sqrt{5^2 - 4(2)(-150)}}{2 \cdot 2}
\]
\[
n = \frac{-5 \pm \sqrt{25 + 1200}}{4}
\]
\[
n = \frac{-5 \pm \sqrt{1225}}{4}
\]
\[
n = \frac{-5 \pm 35}{4}
\]

This gives two solutions:
\[
n = \frac{30}{4} = 7.5 \quad \ ext{(not valid, must be integer)}
\]
\[
n = \frac{-40}{4} = -10 \quad \ ext{(invalid, panels can't be negative)}
\]

Since 7.5 is not an integer, test whole numbers near 7.5 (try \( n = 7 \) and \( n = 8 \)) to find the maximum number of shelves within 150 panels.

Step-by-step Testing
- For \( n = 7 \):
\[
S_7 = \frac{7}{2} (2 \cdot 7 + 6 \cdot 4) = \frac{7}{2} (14 + 24) = \frac{7}{2} \cdot 38 = 7 \cdot 19 = 133 \ ext{ panels}
\]

• For \( n = 8 \):
  \[
  S_8 = \frac{8}{2} (14 + 7 \cdot 4) = 4 (14 + 28) = 4 \cdot 42 = 168 \ ext{ panels}
  \]

Final Thoughts

168 exceeds 150 — too many panels. Therefore, only 7 shelves can be built.

Why This Math Matters in Industrial Design
Modular design thrives on predictable, scalable systems. By modeling panel growth with arithmetic sequences, designers optimize material usage, reduce waste, and maintain visual consistency. The math behind shelf construction reflects core engineering principles applied to real-world products — turning thoughtful creativity into efficient, producible form.

Key Takeaway: Even in aesthetic design, mathematical precision enables smarter, more sustainable innovation. The engineer-building shelves uses not just intuition, but the power of algebra to move the project forward—just like every great design.

Relevant SEO Keywords:
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Meta Description:
Discover how industrial designers use arithmetic sequences to calculate modular shelf builds. Learn how 7 shelves (with 7 to 150 panels) are built using website sum formulas — a blend of creativity and math in product design.