Building and Checking a Geometric Model

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Learning goals

Choose geometric components and distinguish measured dimensions from dimensions fixed for the calculation
Convert inputs to consistent units and estimate the size of a result
Translate a rounding rule into a measurement interval with the correct endpoints
Bound a model prediction by tracking which dimensions raise or lower it
Compare a prediction with an independent observation and decide what to check or revise

Decide what the shapes need to capture

A geometric model uses familiar shapes and measurements to answer a question about a real object or space. For a first estimate of the garden’s plantable area, you could represent the outline by a rectangle and cut out a smaller rectangle around the tree. That choice straightens the bowed edge and replaces a rounded tree bed with a rectangular one. Those are simplifying assumptions, choices to leave out or idealize details for a stated purpose.

The field sketch records the bow and the rounded bed. The model uses straight sides and a rectangular exclusion. Both panels are schematic.

Worked example 1 Estimate the plantable area

Problem. A garden’s outer spans are reported as $9.7$ m and $5.8$ m. Model the outline by a rectangle with those dimensions and the rounded tree bed by a $2.4$ m by $1.5$ m corner cutout. Use the reported values as inputs. Find the model’s plantable area and name one detail that could change the actual area.

Solution. Multiply the outer dimensions and subtract the area excluded around the tree.

\begin{aligned} A&=9.7(5.8)-2.4(1.5)\\ A&=52.66\text{ m}^2. \end{aligned}

The inward bow could leave less plantable area than the model counts. The rounded bed could also change the result. The decimal is the result of the calculation with those inputs. For a rough estimate, report about $53$ square meters.

The outer spans were measured. A different reading could change them. The corner dimensions were stipulated for the model, meaning they were set for the calculation. A dimension given on a sketch could be a recorded measurement or a fixed input. Ask how the number was obtained before deciding whether it can vary.

Check your understanding

A museum measures a display base as 81 cm wide. Its planning model specifies an exact 12 cm border on each side. Which inputs are measured or stipulated?

A Both are measured. B The base width is measured and the border is stipulated. C The border is measured and the base width is stipulated. D Both are stipulated.

Make the units agree and estimate first

A quick estimate gives you a size to expect. Use easy numbers close to the inputs after converting to consistent units. Then compare the calculated value with that estimate. If the answers are far apart, check the conversions and the formula before you check the physical model.

Worked example 2 Check the size of a floor area

Problem. A rectangular entry mat is measured as $265$ cm long and $1.32$ m wide. Estimate its area in square meters using convenient numbers. Then calculate the model area from the reported readings.

Solution. Convert the length to meters.

265\text{ cm}=2.65\text{ m}

Use $2.7$ m and $1.3$ m for a rough estimate. Their product is $3.51$ square meters, so expect about $3.5$ square meters.

\begin{aligned} A&=(2.65\text{ m})(1.32\text{ m})\\ A&=3.498\text{ m}^2. \end{aligned}

The sizes agree. The multiplication uses meters in both directions, so the result is in square meters.

Check your understanding

A rectangular practice strip is 380 cm long and 2.1 m wide. Which is a reasonable estimate of its area?

A About

0.8\text{ m}^2

B About

80\text{ m}^2

C About

800\text{ m}^2

D About

8\text{ m}^2

Recover the range behind a rounded reading

A report of $1.4$ m to the nearest tenth stands for a range of readings. A reading of $1.36$ m would round to it, and so would $1.44$ m. The edges of the range are halfway from $1.4$ to its neighbors $1.3$ and $1.5$ .

In this lesson, for positive lengths reported to the nearest unit of precision, an exact halfway reading rounds to the larger report. The reading $1.35$ m rounds to $1.4$ m. The reading $1.45$ m rounds to $1.5$ m. The measurement interval is the range of readings allowed by that reporting rule.

A filled dot includes the endpoint. An open dot excludes it. The segment marks readings that round to 1.4 m.

If rounding is the only uncertainty we are tracking, we use that interval for the input. A stretched tape or a poor choice of endpoints can affect the reading before it is rounded. Those effects would need separate information.

Worked example 3 Bound the area from one rounded width

Problem. A rectangular noticeboard model uses a fixed height of $0.72$ m. Its measured width was reported as $1.4$ m to the nearest tenth, with ties rounded to the larger report. Find the measurement interval for the width $w$ and the prediction interval for area $A$ from these inputs. Say which area endpoint can occur.

Solution. Half a tenth is $0.05$ m. With widths in meters, the measurement interval is

1.35\le w<1.45.

Multiply each side by the fixed positive height. Multiplication by a positive number preserves the order, so the area bounds are

0.972\le A<1.044

in square meters. The lower endpoint occurs at $1.35$ m. The area can get arbitrarily close to $1.044$ square meters from below. That number is an upper bound, and the area never attains it with these inputs.

A prediction interval from these inputs gives the model’s range using the stated input bounds and assumptions. Its endpoints come from those inputs. They carry no probability or confidence level.

Check your understanding

A length in centimeters is reported as 86 to the nearest centimeter. Ties go to the larger report. Which inequality describes the possible reading $x$ ?

85.5\le x<86.5

85.5<x\le86.5

85\le x<87

86\le x<86.5

Track which way each input moves the result

For an outer rectangle minus a cutout, a larger outer width adds area and a larger cutout width takes area away. The input choices for a lower bound can therefore go in opposite directions.

Worked example 4 Find both bounds on a paved area

Problem. A courtyard model subtracts a rectangular drain opening from a containing rectangle. The outer length is fixed at $6$ m, and its width can be anywhere from $3.8$ to $4.2$ m, inclusive. The opening has fixed length $2$ m, and its width can be anywhere from $1.1$ to $1.3$ m, inclusive. Assume it fits inside for every allowed pair and the two widths can vary independently. Find the least and greatest paved area and justify both choices.

Solution. At fixed lengths, increasing the outer width increases the area of the whole rectangle. Increasing the opening width increases the area taken away.

Least area. Pair the smallest outer width with the largest opening width.

\begin{aligned} A_{\mathrm{least}}&=6(3.8)-2(1.3)\\ A_{\mathrm{least}}&=20.2\text{ m}^2. \end{aligned}

Greatest area. Pair the largest outer width with the smallest opening width.

\begin{aligned} A_{\mathrm{greatest}}&=6(4.2)-2(1.1)\\ A_{\mathrm{greatest}}&=23\text{ m}^2. \end{aligned}

Both pairs are allowed, so the two endpoints can occur.

If two components share a measurement, choose one value for it. For instance, suppose a $6$ m long rectangle has a $2$ m long strip removed across its full width $w$ . The model’s paved area is

A=(6-2)w.

The remaining length is $4$ m. Increasing $w$ makes both the whole rectangle and the removed strip larger, and the net effect is to increase the paved area. Giving the two widths different values would describe a different layout.

Check your understanding

A cover is modeled by a fixed outer area minus a rectangular opening. Only the opening width can vary. Its length is fixed. Which opening gives the lowest cover area?

A The narrowest opening B The midpoint opening C The widest opening D Every opening gives the same cover area.

Compare with an observation from another method

An independent observation measures or estimates the target quantity without using the same inputs to compute it in the same way. For an area predicted from tape measurements, a separate estimate from a grid placed over the actual outline can help. Repeating the original multiplication is an arithmetic check.

Worked example 5 Recheck the curved garden edge

Problem. A rectangular-outline garden model with a tree-bed cutout predicts between $52.3$ and $53.0$ square meters of plantable area from the input bounds. A team that did not use those inputs estimates the same plantable area by counting grid squares. Its estimate is between $50.8$ and $51.2$ square meters. Another field measurement suggests the model includes an extra patch roughly shaped like a triangle with a $6.2$ m base and $0.50$ m perpendicular height. Treat its estimated area as fixed in the revised model. Compare the original prediction with the grid result, revise the prediction interval, and interpret the comparison.

Solution. The original prediction interval lies above the grid estimate. Under the stated bounds they disagree. Check the tape measurements, grid estimate, and the model’s treatment of the bowed edge.

The extra patch has estimated area

\begin{aligned} A_{\mathrm{patch}}&=\tfrac12(6.2)(0.50)\\ A_{\mathrm{patch}}&=1.55\text{ m}^2. \end{aligned}

Subtract $1.55$ square meters from both original bounds. The revised prediction runs from $50.75$ to $51.45$ square meters. The grid estimate lies inside it, so the two are consistent under the revised assumptions. The triangle is still an approximation. A fresh observation could check how well the revised model works beyond this comparison.

When the observation is outside the predicted range, the discrepancy gives you a place to investigate. It could come from the inputs, the model, or the observation. When the observation falls inside, the result is consistent with the prediction under the assumptions. Other models could give the same agreement.

Check your understanding

A flat-roof model predicts 32 to 35 square meters. A separate survey reports 34 square meters for the same surface. What does the comparison establish?

A The survey result is consistent with the model under its assumptions. B The physical roof is exactly flat. C Every input reading must be exact. D Any other roof model must give a different area.

Building and Checking a Geometric Model

Learning goals

Decide what the shapes need to capture

Make the units agree and estimate first

Recover the range behind a rounded reading

Track which way each input moves the result

Compare with an observation from another method

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Core practice

Your progress