Question

1. kassandra has a rectangular patio in her backyard. the patio is 12.74 meters long and 5.45 meters wide.

image of a rectangle labeled 12.74 m (length) and 5.45 m (width)
a. round the length and width to the nearest whole number. then estimate the perimeter of kassandra’s patio. write an equation to model your work. 5.ns0.1.5
b. round the length and width to the nearest tenth. then estimate the perimeter of kassandra’s patio. write an equation to model your work. 5.ns0.1.5
c. find the exact perimeter. which estimate is closer? explain why you think that estimate is closer. 5.gr.2.1

Explanation:

Part A

Step1: Round length and width

Length: \(12.74\) rounded to nearest whole number is \(13\) (since \(0.74\geq0.5\)).
Width: \(5.45\) rounded to nearest whole number is \(5\) (since \(0.45<0.5\)? Wait, no—\(5.45\) to nearest whole: look at tenths place, \(4 < 5\)? Wait, no, \(5.45\): the rule is if the decimal part is \(\geq0.5\), round up. \(0.45 < 0.5\)? Wait, no, \(0.45\) is less than \(0.5\)? Wait, \(5.45\) rounded to nearest whole: the whole number part is \(5\), decimal is \(0.45\). Since \(0.45 < 0.5\), we round down to \(5\)? Wait, no—wait, \(5.45\): the tenths digit is \(4\), which is less than \(5\), so we keep the whole number as \(5\). Wait, but \(5.45\) is closer to \(5\) or \(6\)? \(5.45 - 5 = 0.45\), \(6 - 5.45 = 0.55\), so closer to \(5\). So length \(L = 13\), width \(W = 5\).

Step2: Perimeter of rectangle formula

Perimeter of rectangle: \(P = 2(L + W)\). Substitute \(L = 13\), \(W = 5\).
\(P = 2(13 + 5) = 2\times18 = 36\) meters.

Step1: Round to nearest tenth

Length: \(12.74\) to nearest tenth: look at hundredths place (\(4\)), which is \(< 5\), so \(12.7\).
Width: \(5.45\) to nearest tenth: look at hundredths place (\(5\)), which is \(\geq5\), so round up tenths place: \(5.5\).

Step2: Perimeter formula

\(P = 2(L + W)\), substitute \(L = 12.7\), \(W = 5.5\).
\(P = 2(12.7 + 5.5) = 2\times18.2 = 36.4\) meters.

Step1: Exact perimeter

Exact length \(L = 12.74\), width \(W = 5.45\).
Perimeter \(P = 2(L + W) = 2(12.74 + 5.45) = 2(18.19) = 36.38\) meters.

Step2: Compare estimates

• Part A estimate: \(36\), difference: \(|36.38 - 36| = 0.38\).
• Part B estimate: \(36.4\), difference: \(|36.4 - 36.38| = 0.02\).

Since \(0.02 < 0.38\), the estimate from Part B (rounding to nearest tenth) is closer. This is because rounding to the nearest tenth uses more precise values (closer to the actual length and width) than rounding to the nearest whole number.

Answer:

Equation: \(P = 2(13 + 5)\), Estimated perimeter: \(36\) meters.

Part B