Question

1. a teacher wants to put a decorative border around his bulletin board. the bulletin board is a square and has an area of 19 square feet. what is the length of border that the teacher needs? approximate your answer to the nearest tenth of a foot.
2. error analysis a carpenter is measuring and placing trim around a square window with an area of 411 square inches. a classmate says that a piece of trim that is 20 inches long can be trimmed and used as one side of the window trim. how do you respond?

reflect
how can you explain to a classmate what an irrational number is?

Explanation:

Question 12

Step1: Recall area of square formula

The area \( A \) of a square is given by \( A = s^2 \), where \( s \) is the side length. We know \( A = 19 \) square feet, so we need to solve for \( s \).
\( s = \sqrt{A} = \sqrt{19} \)

Step2: Calculate side length

\( \sqrt{19} \approx 4.3589 \) feet (using a calculator to find the square root of 19).

Step3: Find perimeter of square

The perimeter \( P \) of a square is \( P = 4s \). Substitute \( s \approx 4.3589 \) into the formula.
\( P = 4\times4.3589 \approx 17.4356 \)

Step4: Round to nearest tenth

Rounding \( 17.4356 \) to the nearest tenth gives \( 17.4 \).

1. First, find the side length of the square window using the area formula \( A = s^2 \), where \( A = 411 \) square inches. So \( s=\sqrt{411}\).
2. Calculate \( \sqrt{411}\approx20.27 \) inches (using a calculator).
3. Compare this side length with the length of the trim (20 inches). Since \( 20.27>20 \), the trim that is 20 inches long is too short to be used as one side of the window trim without trimming (but actually, it's shorter than the required side length, so it can't be used as is; if we trim it, it would be even shorter, so the classmate is incorrect).

1. Start with the definition: An irrational number is a real number that cannot be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q

eq0 \).

1. Give examples: Numbers like \( \sqrt{2}\approx1.41421356\cdots \), \( \pi\approx3.14159265\cdots \), and \( e\approx2.71828\cdots \) are irrational. Their decimal expansions are non - repeating and non - terminating.
2. Contrast with rational numbers: Rational numbers can be written as fractions (e.g., \( \frac{1}{2}=0.5 \), \( \frac{2}{3}=0.\overline{6} \)) and have either terminating or repeating decimal expansions. So, to distinguish, if a number's decimal goes on forever without repeating, or it can't be written as a simple fraction, it's irrational.

Answer:

The length of the border needed is approximately \( 17.4 \) feet.

Question 13