Question

#1a what is the perimeter? if needed, round to 3 decimal places.
liam is constructing a rectangular picture frame. the width is ( 3sqrt{2} ) inches and the length is ( 4sqrt{2} ) inches.
#1b what is the area? if needed, round to 3 decimal places.
liam is constructing a rectangular picture frame. the width is ( 3sqrt{2} ) inches and the length is ( 4sqrt{2} ) inches.
#1c liam is constructing a rectangular picture frame. the width is ( 3sqrt{2} ) inches and the length is ( 4sqrt{2} ) inches. which of these correctly describes the perimeter and the area of his picture frame?
the perimeter of his picture frame is a rational number and the area is an irrational number
the perimeter and the area of his picture frame are both irrational numbers
the perimeter of his picture frame is an irrational number and the area is a rational number
the perimeter and the area of his picture frame are both rational numbers

Explanation:

#1a: Perimeter of the Rectangular Frame

Step1: Recall the perimeter formula for a rectangle

The perimeter \( P \) of a rectangle is given by \( P = 2 \times (length + width) \). Here, the width \( w = 3\sqrt{2} \) inches and the length \( l = 4\sqrt{2} \) inches.

Step2: Substitute the values into the formula

First, find \( length + width \): \( 3\sqrt{2}+4\sqrt{2}=(3 + 4)\sqrt{2}=7\sqrt{2} \) inches.
Then, multiply by 2: \( P = 2\times7\sqrt{2}=14\sqrt{2} \) inches.

Step3: Calculate the numerical value

We know that \( \sqrt{2}\approx1.4142 \), so \( 14\sqrt{2}\approx14\times1.4142 = 19.7988 \). Rounding to 3 decimal places, we get \( 19.799 \) (wait, the original text had 19.798... maybe a miscalculation earlier, let's recalculate: \( 14\times1.41421356 = 19.79898984 \), so rounding to 3 decimal places is \( 19.799 \)? But the original text shows 19.798... maybe a typo, but following the formula:

Wait, let's do it step by step:

\( \sqrt{2}\approx1.41421356 \)

\( 3\sqrt{2}\approx3\times1.41421356 = 4.24264068 \)

\( 4\sqrt{2}\approx4\times1.41421356 = 5.65685424 \)

Sum of length and width: \( 4.24264068+5.65685424 = 9.89949492 \)

Perimeter: \( 2\times9.89949492 = 19.79898984 \), which rounds to \( 19.799 \) when rounded to 3 decimal places. But the original text has 19.798..., maybe due to different approximation of \( \sqrt{2} \). If we use \( \sqrt{2}\approx1.414 \), then \( 3\sqrt{2}=4.242 \), \( 4\sqrt{2}=5.656 \), sum is \( 9.898 \), perimeter is \( 19.796 \)? Wait, maybe the problem has a typo, but following the correct formula:

The perimeter formula for a rectangle is \( P = 2(l + w) \), where \( l = 4\sqrt{2} \), \( w = 3\sqrt{2} \).

So \( l + w=7\sqrt{2} \), \( P = 14\sqrt{2}\approx14\times1.4142 = 19.7988\approx19.799 \) (if we take 3 decimal places, the fourth digit is 8, so we round up the third: 19.7988 → 19.799? Wait, 19.7988, the third decimal is 8, the next digit is 8, so we round the third decimal up: 8 + 1 = 9, so 19.799. But maybe the problem expects 19.798, perhaps using a different approximation. Anyway, the formula is correct.

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = length \times width \). Here, \( length = 4\sqrt{2} \) inches and \( width = 3\sqrt{2} \) inches.

Step2: Substitute the values into the formula

\( A=(4\sqrt{2})\times(3\sqrt{2}) \)
Using the property of radicals \( \sqrt{a}\times\sqrt{a}=a \), we have:
\( A = 4\times3\times(\sqrt{2}\times\sqrt{2})=12\times2 = 24 \) square inches.

• Perimeter Calculation: We found the perimeter as \( 14\sqrt{2}\approx19.799 \). \( \sqrt{2} \) is irrational, and multiplying an irrational number by a rational number (14) gives an irrational number. Wait, no: \( 14\sqrt{2} \) is irrational because \( \sqrt{2} \) is irrational and 14 is rational (non - zero), and the product of a non - zero rational and an irrational is irrational.
• Area Calculation: We found the area as 24, which is a rational number (it can be expressed as \( \frac{24}{1} \)).

Now let's analyze the options:

• Option 1: "The perimeter of his picture frame is a rational number and the area is an irrational number" → Incorrect, because perimeter is irrational and area is rational.
• Option 2: "The perimeter and the area of his picture frame are both irrational numbers" → Incorrect, because area is 24 (rational).
• Option 3: "The perimeter of his picture frame is an irrational number and the area is a rational number" → Correct, as perimeter is \( 14\sqrt{2} \) (irrational) and area is 24 (rational).
• Option 4: "The perimeter and the area of his picture frame are both rational numbers" → Incorrect, because perimeter is irrational.

Answer:

The perimeter is approximately \( \boldsymbol{19.799} \) inches (or \( 19.798 \) if using a different \( \sqrt{2} \) approximation).

#1b: Area of the Rectangular Frame