Question

a rectangle has measured dimensions of 2.76 miles and 4.37 miles. taking measurement error into account, what are the minimum and maximum possible areas? minimum possible area = 12.0612 mi² maximum possible area = 12.0 mi²

Explanation:

Step1: Find the maximum and minimum of each dimension

The measurement error for a number \(x\) measured to two decimal places is \(\pm0.005\).
For length \(l = 2.76\) miles, the maximum length \(l_{max}=2.76 + 0.005=2.765\) miles and the minimum length \(l_{min}=2.76- 0.005 = 2.755\) miles.
For width \(w = 4.37\) miles, the maximum width \(w_{max}=4.37+0.005 = 4.375\) miles and the minimum width \(w_{min}=4.37 - 0.005=4.365\) miles.

Step2: Calculate the maximum area

The area of a rectangle \(A=l\times w\). The maximum area \(A_{max}=l_{max}\times w_{max}\).

$$A_{max}=2.765\times4.375=(2 + 0.765)\times(4+0.375)=2\times4+2\times0.375+0.765\times4+0.765\times0.375$$

$$=8 + 0.75+3.06+0.286875=12.096875$$

Step3: Calculate the minimum area

The minimum area \(A_{min}=l_{min}\times w_{min}\)

$$A_{min}=2.755\times4.365=(2+0.755)\times(4 + 0.365)=2\times4+2\times0.365+0.755\times4+0.755\times0.365$$

$$=8+0.73+3.02+0.275575 = 12.025575$$

Answer:

Maximum possible area \(= 12.1\) \(mi^{2}\), Minimum possible area \(=12.03\) \(mi^{2}\)