Square Herb Plot: Fencing, Irrational Side Length, and Cost
The community garden committee plans a square herb plot with an area of 42 square feet. Fencing is sold only in half-foot sections, and the committee must buy enough to go around the entire plot plus a 3-foot gate opening that does not need fencing. The fence costs $3.25 per foot, and the supplier adds 8% sales tax. A volunteer writes that the side length is 6.5 feet because 6.5^2 = 42.25, and orders 26 feet of fencing. The treasurer questions whether 6.5 feet is exact, whether buying 26 feet leaves enough after the gate, and whether the total cost will stay under the $100 raised. The committee also wants to know if the exact side length is a rational or irrational number so the order form can describe it correctly.
1. Is the exact side length of the square plot rational or irrational? Justify your answer using the definition of a rational number, and explain why 6.5 feet is not the exact side length.
Read the worked answer
Let (s) be the side length in feet. Since the plot is a square with area 42 square feet, (s^2=42), so (s=\sqrt{42}).
Suppose (\sqrt{42}) were rational. Then (\sqrt{42}=\frac{p}{q}) for integers (p) and (q), with (q\ne 0), written in lowest terms. Squaring gives (42=\frac{p^2}{q^2}), so (p^2=42q^2=2\cdot 3\cdot 7\cdot q^2). In a perfect square, every prime exponent is even. The right side has an odd exponent of 2, 3, and 7, while the left side (p^2) must have even exponents. This contradiction means (\sqrt{42}) is irrational.
The volunteer used (6.5) because (6.5^2=42.25), but (42.25\ne 42). Also, (6.48^2=41.9904) and (6.49^2=42.1201), so the exact side length is between 6.48 and 6.49, about 6.4807 feet. Thus 6.5 feet is an approximation, not the exact side length.
2. The volunteer orders 26 feet of fencing. The gate opening is 3 feet and needs no fencing. Determine the minimum number of half-foot fencing sections actually needed for the plot, and the total cost at $3.25 per foot plus 8% tax. Will the $100 raised cover it?
Read the worked answer
The exact perimeter is (4\sqrt{42}\approx 4(6.48074)=25.923) feet. The 3-foot gate opening needs no fencing, so the fencing length required is
[25.923-3=22.923\text{ feet}.]
Fencing is sold in half-foot sections. Divide by 0.5:
[22.923\div 0.5=45.846.]
Round up to 46 half-foot sections, or 23 feet of fencing. The cost before tax is (23(3.25)=74.75) dollars. The tax is (0.08(74.75)=5.98) dollars, so the total is (74.75+5.98=80.73) dollars.
Yes, $100 covers the minimum order. The volunteer's 26-foot order would cost (26(3.25)=84.50) dollars before tax, or (84.50+6.76=91.26) dollars with tax. That also stays under $100, but it is not the minimum order and leaves extra fencing.
3. The order form asks for the side length to the nearest hundredth of a foot. Compute that value and explain why writing 6.48 feet is a rounded measurement, not an exact value.
Read the worked answer
Using a calculator, (\sqrt{42}\approx 6.480740698). To round to the nearest hundredth, look at the third decimal digit, which is 0. Since 0 is less than 5, the hundredths digit stays 8:
[\sqrt{42}\approx 6.48\text{ feet}.]
This is not exact because (6.48^2=41.9904), not 42. The value (6.49^2=42.1201), so the exact side length lies between 6.48 and 6.49. Writing 6.48 feet is a rounded measurement with a small error; the exact value is the irrational number (\sqrt{42}).