Optimal Apex Slides Engineering Challenge 402
Optimal Apex
Minimizing Material Surface Area in Industrial Packaging Design
Subject
Advanced Geometry
Unit
Optimization & Surface Area
Warm-Up
The Power of the Cone
5:00
Cones are everywhere in industrial design and nature.
Challenge:
"Brainstorm as many real-world objects as possible that use a conical shape. Why might a designer choose this shape?"
Snacks?
Safety?
Shelter?
Aerodynamics?
Visualizing Surface Area
Refresher: Lateral Area vs. Total Surface Area
3:02
Embedded media
Key Formula
\[SA = \pi r^2 + \pi r l\]
Critical Step
We must find the slant height (l) using the Pythagorean Theorem:
\[l = \sqrt{r^2 + h^2}\]
The "Gotcha" Moment
1
Variable Check
The video gave us \(r=5\) and \(h=12\). Why couldn't we just plug those into the \(SA\) formula immediately?
2
The Base Shape
What shape is the base? What is the formula for the area of that shape?
3
Units Matter
If the answer is \(90\pi\), what are the units if the radius and height were in inches?
Project: Packaging Design
The Efficiency Challenge
Your Goal
Design a conical package that holds exactly \(300\pi\) cm³ of product while using the least amount of material possible.
The Constraints
Fixed Volume: \(V = 314.16\) cm³ (rounded)
Variable: Radius (\(r\)) and Height (\(h\))
Objective: Minimum Surface Area (\(SA\))
Why optimize?
In business, surface area = material cost. Lower surface area = higher profit margins. You are the lead designer!
Grab your project packets now!
Path to the Optimal Design
1
Connect
Relate Height to Radius using the Volume formula.
2
Synthesize
Write the \(SA\) formula in terms of a single variable (\(r\)).
3
Analyze
Use your calculator's Table or Graph to find the minimum.
4
Deliver
Produce a blueprint with final dimensions and SA cost.
Packaging Design Project Handout Project: Packaging Design
Subject: Advanced Geometry | Industrial Optimization
Project ID: APEX-402
NAME:
Mission Briefing
As a lead designer at Apex Packaging Solutions, you have been tasked with designing a new conical shipping container. To maximize profit, we must minimize material waste. Your objective is to find the dimensions (\(r\) and \(h\)) of a cone that holds exactly \(300\pi\) cm³ while using the minimum possible total surface area.
Specs
VOLUME (V): \(300\pi\) cm³
RADIUS (r): Variable
HEIGHT (h): Variable
MAT. COST: $0.05/cm²
Phase 1: Isolating the Variables
We need our Surface Area (\(SA\)) formula to be in terms of just one variable, \(r\). First, use the Volume formula to solve for \(h\) in terms of \(r\).
Step A: Start with Volume formula
\[V = \frac{1}{3}\pi r^2 h\]
Step B: Substitute \(V = 300\pi\) and Solve for \(h\)
Phase 2: Defining the Function
Now, substitute your expression for \(h\) into the Surface Area formula: \(SA = \pi r^2 + \pi r \sqrt{r^2 + h^2}\). Simplified, your function for \(SA\) in terms of \(r\) should look something like this:
\(SA(r) = \pi r^2 + \pi r \sqrt{r^2 + \left(\frac{900}{r^2}\right)^2}\)
Complete a table of values to estimate the minimum \(SA\). Use your calculator's table function.
Radius (\(r\)) Calculation Area / Notes Surface Area (\(SA\)) 4 cm 757.2 6 cm 8 cm 10 cm 12 cm
Phase 3: Final Blueprint
Using your graph or table, identify the optimal radius to the nearest tenth. Then calculate the height and final cost.
Final Optimization Results
Optimal Radius (\(r\)):
Optimal Height (\(h\)):
Minimum Surface Area (\(SA\)):
Total Material Cost:
Reflective Conclusion
Why did we have to use the Pythagorean theorem to find the Surface Area, even though the Volume formula only requires \(r\) and \(h\)?
Design Blueprint Sketch
Sketch your optimized cone and label \(r, h, l\).
Project Scorecard
MATH ACCURACY
DESIGN OPTIMALITY
Optimization Reference Sheet Optimizer's Manual
Technician Field Reference | Ver. 1.2
REF-GEO-CONE-01
Essential Geometry
Volume of a Cone
\[V = \frac{1}{3}\pi r^2 h\]
Total Surface Area
\[SA = \pi r^2 + \pi r l\]
Slant Height Requirement
Surface area calculation always requires the slant height (\(l\)). Use the radius (\(r\)) and altitude (\(h\)) with the Pythagorean Theorem:
\[l = \sqrt{r^2 + h^2}\]
Technical Diagram
h r l
Altitude (h)
Radius (r)
Slant Height (l)
Perimeter (p)
Calculator Strategy
1 Enter Function
Press [Y=] and enter your surface area equation. Use X as your radius (\(r\)).
2 Set Window
Press [WINDOW]. Ensure your X-range covers reasonable radii (e.g., 0 to 20) and Y-range covers large surface areas.
3 Analyze Table
Press [2nd] + [TABLE]. Look for where the Y values stop decreasing and start increasing. This is your minimum.
© 2026 Apex Industrial Design Math Ref 4.1-C Secure Document
Cone Designer Graph Paper Designer's Blueprint
Project: Apex-402 Draft: V1.0
Designer:
Date:
Scale: 1 Unit = 1 cm
Technical Sketch Area
Deliverables
Volume Verified
SA Function Built
Min Found (r)
Height Calculation
Labeled Sketch
Final Specs
Optimal Radius
Optimal Height
Min. Surface Area
APEX DESIGN
