Question

a football field is 120 yards by 53.333 yards. what is the area of the football field in acres if 1 acre=43560 ft²? use correct significant figures when answering this question.

Explanation:

Step1: Calculate area in square yards

The area of a rectangle is length times width. So, area \( A = 120 \, \text{yards} \times 53.333 \, \text{yards} \)
\( A = 120\times53.333 = 6400 \, \text{square yards (approx, keeping track of sig figs)} \)

Step2: Convert square yards to square feet

Since 1 yard = 3 feet, 1 square yard = \( 3^2 = 9 \, \text{square feet} \). So, convert area to square feet: \( 6400 \, \text{yd}^2\times9 \, \text{ft}^2/\text{yd}^2 = 57600 \, \text{ft}^2 \) (Wait, actually, let's do it more accurately. First, 120 yards is exact? Wait, 120 has two or three sig figs? Wait, 120 - if the zero is significant? Wait, the problem says 120 yards (maybe two sig figs? Or three? Wait, 53.333 has five sig figs. Wait, 120 - maybe it's three sig figs (if the zero is a placeholder, but sometimes trailing zero without decimal is ambiguous, but maybe here 120 is three sig figs (1,2,0) and 53.333 is five. But let's proceed step by step.

First, calculate area in square yards: \( 120 \times 53.333 = 6400 \, \text{yd}^2 \) (exactly, 120*53.333 = 120*(160/3) = 120*160/3 = 40*160 = 6400. Oh, right! 53.333... is 160/3, so 120*(160/3) = 6400 exactly. So area in square yards is 6400 \( \text{yd}^2 \).

Now, convert square yards to square feet: 1 yard = 3 feet, so 1 \( \text{yd}^2 = 3 \times 3 = 9 \, \text{ft}^2 \). So \( 6400 \, \text{yd}^2 \times 9 \, \text{ft}^2/\text{yd}^2 = 57600 \, \text{ft}^2 \).

Step3: Convert square feet to acres

We know that 1 acre = 43560 \( \text{ft}^2 \). So, number of acres \( = \frac{57600 \, \text{ft}^2}{43560 \, \text{ft}^2/\text{acre}} \)
Calculate that: \( \frac{57600}{43560} \approx 1.322 \) acres. Now, check significant figures. The length is 120 yards (let's assume it's three significant figures, since 120 could be three if the zero is significant, or two if it's not. But 53.333 has five, but 120 - if it's 120. yards, it's three, but here it's 120 yards. However, since 120*53.333 = 6400 (exact, because 53.333 is 160/3, so 120*(160/3)=6400 exactly). Then 6400 square yards is exact? Wait, no, 120 is a measured value? Wait, maybe 120 has two significant figures (the trailing zero is not significant). Wait, the problem says "120 yards by 53.333 yards". 53.333 has five significant figures, 120 - if it's written as 120 without a decimal, it's ambiguous, but often trailing zeros in whole numbers without decimals are not significant, so 120 has two significant figures. But wait, 53.333 is 53.333 (five sig figs). But when multiplying, the result should have the least number of sig figs. But in our case, 120*53.333 = 6400 (exact, because 53.333 is 160/3, so 120*(160/3)=6400 exactly). So maybe 120 is exact? Wait, a football field's length: actually, a standard football field (American) is 120 yards long (100 yards playing field plus 10 yards end zones on each side), so 120 yards is exact. 53.333 yards is 160/3 feet? Wait, no, 53.333 yards is 160 feet (since 1 yard = 3 feet, 53.333*3=160 feet). So the width is 160 feet, length is 360 feet (120 yards*3). Wait, maybe the problem is using 120 yards (length) and 53.333 yards (width, which is 160/3 yards, or 53.333... yards). So the area in square yards is 120*(160/3)=6400 square yards, which is 6400*9=57600 square feet. Then, 57600 square feet divided by 43560 square feet per acre:

\( \frac{57600}{43560} = \frac{5760}{4356} = \frac{480}{363} = \frac{160}{121} \approx 1.3223 \) acres. Now, considering significant figures: 120 yards (if it's exact, then we go by 53.333, which has five, but 120 - maybe it's three sig figs (1,2,0). Wait, but 120 is a defined value (football field length), so maybe it's exact. But the problem says "120 yards by 53.333 yards" - 53.333 has five sig figs, 120 - let's assume 120 has three sig figs (the zero is significant). Then, when multiplying 120 (three sig figs) by 53.333 (five sig figs), the result has three sig figs. Then, converting to square feet (exact conversion, since 1 yard=3 feet is exact), then dividing by 43560 (which is a defined conversion, so exact? Wait, 1 acre is defined as 43560 square feet, so that's exact. So the limiting factor is the 120 and 53.333. Wait, 53.333 is 53.333... (repeating), which is 160/3, so exact. 120 is exact (football field length). So maybe the area is exact, but when we do 57600 / 43560, let's compute that:

57600 ÷ 43560 = 5760 ÷ 4356 = divide numerator and denominator by 12: 480 ÷ 363 = divide by 3: 160 ÷ 121 ≈ 1.3223. Now, 120 has three significant figures, 53.333 has five, so the area in square yards is 120*53.333=6400 (three significant figures? Wait, 120*53.333=6400. If 120 is three sig figs, then 6400 should be written as 6.40×10³, but here it's 6400. Wait, maybe the problem expects us to use 120 as three sig figs and 53.333 as five, so the product is three sig figs. Then, 6400 square yards (three sig figs) is 6.40×10³ square yards. Then convert to square feet: 6.40×10³ * 9 = 5.76×10⁴ square feet. Then divide by 43560: 5.76×10⁴ / 4.356×10⁴ ≈ 1.322. Now, 5.76×10⁴ has three sig figs, 4.356×10⁴ has four, so the result should have three sig figs. So 1.32 acres? Wait, no, 5.76×10⁴ / 4.356×10⁴ = 5.76 / 4.356 ≈ 1.322, which with three sig figs is 1.32 acres? Wait, 5.76 has three, 4.356 has four, so the result should have three. So 1.32 acres? Wait, but let's check the exact calculation:

120 yards * 53.333 yards = 120 * (160/3) yards² = 6400 yards². 6400 yards² * 9 ft²/yard² = 57600 ft². 57600 ft² / 43560 ft²/acre = 57600 / 43560 = 1.3223... So, considering significant figures: 120 has two or three? If 120 is three (1,2,0) and 53.333 is five, then the product is three sig figs. 6400 (three sig figs) is 6.40×10³. Then 57600 (three sig figs) is 5.76×10⁴. Then 5.76×10⁴ / 4.356×10⁴ = 1.322, which rounds to 1.32 (three sig figs) or 1.3 (two sig figs). Wait, the problem says "120 yards" - if 120 is two sig figs (the zero is not significant), then 120 is two sig figs, 53.333 is five, so the product is two sig figs. 6400 would be 6.4×10³ (two sig figs). Then 57600 would be 5.8×10⁴ (two sig figs). Then 5.8×10⁴ / 4.356×10⁴ ≈ 1.33, which rounds to 1.3 (two sig figs) or 1.33 (three). This is confusing. Wait, maybe the problem considers 120 as three sig figs (since it's written as 120, maybe the zero is significant) and 53.333 as five, so the area in square yards is 120*53.333=6400 (exact, because 53.333 is 160/3, so 120*(160/3)=6400 exactly). So 6400 is exact, then 6400*9=57600 (exact). Then 57600/43560=1.3223... Now, 43560 is exact (since 1 acre=43560 ft² is a definition). So the only measured values are 120 and 53.333. 120: if it's a measured value, maybe three sig figs (1,2,0), 53.333 is five. So the result should have three sig figs. So 1.32 acres? Wait, but 120*53.333=6400 (exact), so maybe the sig figs come from the 120. Wait, 120 could be three sig figs (if the zero is significant) or two (if not). The problem says "use correct significant figures". Let's see: 120 has two significant figures (the trailing zero is not significant, as there's no decimal), and 53.333 has five. When multiplying, the number of significant figures is determined by the least number, so two. So 120*53.333=6400, which with two sig figs is 6.4×10³. Then 6.4×10³ square yards * 9 ft²/yard²=5.76×10⁴ square feet (but we should keep two sig figs, so 5.8×10⁴). Then 5.8×10⁴ / 43560 ≈ 1.33, which with two sig figs is 1.3 acres. But wait, maybe 120 is three sig figs (the zero is significant, like 120.), but it's written as 120. So maybe the problem expects three sig figs. Let's check the exact value: 57600 / 43560 = 1.3223... So, if we take 120 as three sig figs and 53.333 as five, then the answer should have three sig figs, so 1.32 acres. But let's see, maybe the problem is designed so that 120 yards is 360 feet, 53.333 yards is 160 feet (since 53.333*3=160), so area is 360*160=57600 square feet. Then 57600/43560=1.322..., which is approximately 1.32 acres (three sig figs) or 1.3 acres (two). But let's see, 120 has three digits, 53.333 has five, so three sig figs. So the answer is approximately 1.32 acres. Wait, but let's do the calculation again:

1. Calculate area in square yards: \( 120 \times 53.333 = 6400 \, \text{yd}^2 \) (exact, because 53.333 is \( \frac{160}{3} \), so \( 120 \times \frac{160}{3} = 6400 \))

2. Convert square yards to square feet: \( 6400 \, \text{yd}^2 \times 9 \, \text{ft}^2/\text{yd}^2 = 57600 \, \text{ft}^2 \) (exact, since 1 yd = 3 ft, so 1 yd² = 9 ft²)

3. Convert square feet to acres: \( \frac{57600 \, \text{ft}^2}{43560 \, \text{ft}^2/\text{acre}} = \frac{57600}{43560} \approx 1.322 \, \text{acres} \)

Now, considering significant figures: 120 has three significant figures (assuming the zero is significant, as it's a measured value with a trailing zero, maybe indicating precision), and 53.333 has five. The rule for multiplication/division is that the result has the same number of significant figures as the least precise measurement. So 120 has three, so the final answer should have three significant figures. Therefore, 1.32 acres (or 1.32 when rounded to three significant figures). Wait, but 1.322 rounded to three significant figures is 1.32.

Answer:

\boxed{1.32} (or \boxed{1.3} if 120 is considered two sig figs, but likely 1.32)