2.M Focus on Modeling (Chapter Supplement)
Ideas in this lesson
- Translate geometric dimensions into one-variable models
- Recover dimensions and convert geometric measures
- Build and interpret similar-triangle shadow models
- Model separation in perpendicular motion
- Model constrained triangle and semicircle areas
- Optimize products and sums of squares under a fixed sum
- Optimize area with available fencing
- Minimize fencing and cost with fixed area
- Optimize areas made from split wire
- Optimize rectangle-and-semicircle windows
- Model and optimize corner-cut open boxes
- Minimize material for fixed-volume containers
- Optimize rectangles inscribed under parabolas
- Minimize mixed-speed travel time and energy
- Derive and optimize a kite from its border lengths
Translate geometric dimensions into one-variable models · Worked example
A square-based planter has height 4/2 times its base side. Let \(x\) measure the base side in centimeters. Derive a one-variable model, and state the physically meaningful domain and output units. If every dimension doubles, by what factor does its volume change? Verify with base sides 2 and 4 centimeters. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Model: \(f(x)=2 x^{3}\). Volume in cubic centimeters; \(x>0\). Doubling gives a factor of 8; the two volumes are 16 and 128 cubic centimeters.
Volume is base area times height: \(x^2\) times \(2 x\).
Replacing \(x\) by \(2x\) multiplies the cube by 8; direct substitution confirms it.
Written-work review:
Replacing \(x\) by \(2x\) multiplies the cube by 8; direct substitution confirms it.
Written-work review:
- State the variable, dependent quantity, constraint, and feasible domain with units.
- Produce the requested graph, table, diagram, or written justification, labeling relevant values.
- Show the substitution and check the claimed result against the model and physical constraints.