Arectangle is shown below.
all measurements are to the nearest cm.
21 cm
11 cm
a)
calculate the upper bound for the area of the rectangle.
b)
calculate the lower bound for the perimeter of the rectangle.

  a)  247.25 cm²

  b)  62.0 cm

Step-by-step explanation:

All of these upper-bound/lower-bound problems are worked the same way. The bounds on the measurement are presumed to be half of one unit of the least-significant digit.

Here, the least significant digit of both numbers is in the "units" place, so the maximum error is presumed to be 1/2 unit, or ±0.5 cm.

If the measurement were 7.36 inches, the least significant digit is in the hundredths place, so the maximum error is presumed to be half a hundredth, or ±0.5 × 0.01 inches = ±0.005 inches.

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To find the bounds on the measurement, you add or subtract this maximum error from the measurement. So, for the measurements in this problem, the maximum and minimum they can be are ...

  21 cm: minimum of 21 - 0.5 = 20.5 cm; maximum of 21 + 0.5 = 21.5 cm

  11 cm: minimum of 10.5 cm; maximum of 11.5 cm

If you wanted to express these as an inequality, you'd have to remember that half-units are rounded up, so the inequalities would be ...

  20.5 cm ≤ long side < 21.5 cm

  10.5 cm ≤ short side < 11.5 cm

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a) The area is computed as the product of the rectangle's side length measurements. The area will be a maximum when both measurements are a maximum:

  Amax = (21.5 cm)(11.5 cm)

  Amax = 247.25 cm²

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b) The perimeter is twice the sum of the lengths of two adjacent sides of the rectangle. The minimum perimeter will be the sum using the minimum side lengths, or ...

  Pmin = 2(20.5 cm +10.5 cm) = 2(31.0 cm)

  Pmin = 62.0 cm

Comment on measurement bounds

You have to remember that these concepts are mathematical in nature, so only an approximation of the real world. If you have ever used a tape measure to measure a distance of any length, you will have discovered a couple of things:

So, mathematically, there will be very specific definitions and procedures to follow regarding measurement values. In the real world, you have to remember these are only an approximation to reality, hopefully a useful one.

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)