Question

a vineyard has 145 acres of chardonnay grapes. a particular soil supplement requires 5.50 g for every square meter of vineyard. how many kilograms of the soil supplement are required for the entire vineyard? (recall that 1 km² = 247 acres) express your answer in kilograms to three significant figures.

Explanation:

Step1: Convert acres to square kilometers

First, we know that 1 km² = 247 acres. So to convert 145 acres to square kilometers, we use the conversion factor. The area in km², \( A_{km^2} \), is given by \( A_{km^2}=\frac{145\ acres}{247\ acres/km^2} \). Calculating this, we get \( A_{km^2}=\frac{145}{247}\approx0.5870445344\ km^2 \).

Step2: Convert square kilometers to square meters

Since 1 km = 1000 m, 1 km² = \( (1000\ m)^2 = 10^6\ m^2 \). So the area in square meters, \( A_{m^2} \), is \( A_{m^2}=0.5870445344\ km^2\times10^6\ m^2/km^2 = 587044.5344\ m^2 \).

Step3: Calculate the mass in grams

The soil supplement required is 5.50 g per square meter. So the total mass in grams, \( m_g \), is \( m_g = 5.50\ g/m^2\times587044.5344\ m^2 \). Calculating this, we get \( m_g = 5.50\times587044.5344 = 3228744.939\ g \).

Step4: Convert grams to kilograms

Since 1 kg = 1000 g, the mass in kilograms, \( m_{kg} \), is \( m_{kg}=\frac{3228744.939\ g}{1000\ g/kg}=3228.744939\ kg \).

Step5: Round to three significant figures

Rounding 3228.744939 to three significant figures, we look at the fourth digit. The number is 3228.74..., the first three significant figures are 3, 2, 3 (wait, no: 3228.74, the first three are 3,2,2? Wait, 3228.74. Let's count: 3 (1st), 2 (2nd), 2 (3rd), 8 (4th). Wait, no, 3228.74 is 3.22874×10³. So to three significant figures, we look at the first three digits: 3,2,3? Wait, no, 3228.74. The first significant figure is 3, second 2, third 2, fourth 8. Since the fourth digit is 8 which is greater than 5, we round the third digit up. So 3230? Wait, no, wait 3228.74. Let's do it properly. 3228.74. The number is 3.22874×10³. To three significant figures, we take 3.23×10³, which is 3230? Wait, no, 3228.74 rounded to three significant figures: the first three are 3,2,2, the next digit is 8, so we round the third digit (2) up by 1, making it 3,2,3. So 3230? Wait, no, 3228.74: 3 (1), 2 (2), 2 (3), 8 (4). So 3228.74 ≈ 3230 when rounded to three significant figures? Wait, no, 3228.74 is 3228.74. Let's calculate 3228.744939. The first three significant figures: 3,2,2. The next digit is 8, which is more than 5, so we add 1 to the third digit: 3230? Wait, no, 3228.74. Let's see: 3228.74. If we round to three significant figures, the number is 3.23×10³, which is 3230? Wait, 3228.74 is 3228.74. Let's check the calculation again. Wait, maybe I made a mistake in the area conversion.

Wait, let's re - check step 1: 145 acres divided by 247 acres per km². 145/247: 247×0.5 = 123.5, 145 - 123.5 = 21.5, 21.5/247≈0.087, so total≈0.587 km². Then step 2: 0.587 km²×10⁶ m²/km² = 587000 m² (approx). Then step 3: 5.50 g/m²×587000 m² = 5.50×587000 = 3228500 g. Then step 4: 3228500 g / 1000 g/kg = 3228.5 kg. Now, 3228.5 rounded to three significant figures: the first three significant figures are 3,2,3? Wait, 3228.5 is 3.2285×10³. The third significant figure is 2, the next digit is 8, so we round up the 2 to 3, so 3.23×10³, which is 3230? Wait, no, 3228.5: 3 (1), 2 (2), 2 (3), 8 (4). So when rounding to three significant figures, we look at the fourth digit. Since 8 > 5, we add 1 to the third digit. So 3228.5 becomes 3230? Wait, but 3228.5 is closer to 3230? Wait, no, 3228.5: if we round to three significant figures, the number is 3230? Wait, maybe my initial calculation of the area was wrong.

Wait, let's do the area conversion more accurately. 1 km² = 247 acres, so 1 acre = 1/247 km². Then 145 acres = 145/247 km². Let's calculate 145÷247: 247×0.5 = 123.5, 145 - 123.5 = 21.5, 21.5÷247≈0.0870445. So total area in km² is 0.5 + 0.0870445 = 0.5870445 km². Then area in m²: 0.5870445×10⁶ = 587044.5 m². Then mass in grams: 5.50×587044.5 = 5.50×587044.5 = 3228744.75 g. Then mass in kg: 3228744.75÷1000 = 3228.74475 kg. Now, rounding 3228.74475 to three significant figures. The first three significant figures are 3,2,2. The next digit is 8, which is greater than 5, so we add 1 to the third significant figure. So 3228.74475≈3230 when rounded to three significant figures? Wait, no, 3228.74475: the number is 3228.74475. The first three significant digits are 3,2,2. The fourth digit is 8, so we round the third digit (2) up by 1, making it 3. So the number becomes 3230? Wait, 3228.74475 is approximately 3.23×10³, which is 3230.

Answer:

\( 3.23\times 10^{3}\) (or 3230)