Question

refer to the figure to the right. find the area in acres of the property (enclosed by the right triangle) under the given assumptions. the stream frontage is 700 feet in length and the property line is 3600 feet in length. the lot has an area of about \\(\square\\) acre(s). (round the final answer to the nearest hundredth as needed. round all intermediate values to the nearest whole number as needed.)

Explanation:

Step1: Identify triangle type and formula

The property is a right triangle. Area of a right triangle is $\frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$. Here, legs are stream frontage (700 ft) and the other leg (let's find it using Pythagoras? Wait, no—wait, the property line is the hypotenuse? Wait, no, the figure: stream frontage is one leg, the vertical leg, and property line is hypotenuse? Wait, no, maybe stream frontage and the other leg, and property line is hypotenuse. Wait, no, wait: right triangle, so two legs and hypotenuse. Wait, the problem says "stream frontage is 700 feet" (one leg), "property line is 3600 feet" (hypotenuse). Wait, no, maybe I misread. Wait, no—wait, maybe the stream frontage and the other leg are the two legs, and property line is the hypotenuse? Wait, no, let's check: area of right triangle is $\frac{1}{2} \times a \times b$, where $a$ and $b$ are legs. Wait, maybe the stream frontage is one leg (700 ft), and we need to find the other leg using Pythagoras? Wait, no, wait—the problem says "the property line is 3600 feet in length". Wait, maybe the property line is the hypotenuse, and stream frontage is one leg. Wait, but then we need the other leg. Wait, no, maybe I made a mistake. Wait, no—wait, maybe the stream frontage and the property line are the two legs? Wait, the figure: right triangle, stream frontage is horizontal, vertical leg, and property line is the hypotenuse? Wait, no, the figure shows a right triangle with stream frontage as one leg (horizontal), vertical leg, and property line as hypotenuse. Wait, but then we need to find the other leg. Wait, no, maybe the problem is that the stream frontage is one leg (700 ft), and the property line is the hypotenuse (3600 ft), but that can't be, because 700 and 3600: wait, no, maybe I misread. Wait, no—wait, maybe the stream frontage is one leg (700 ft), and the other leg is, say, $x$, and the property line is the hypotenuse (3600 ft). Then by Pythagoras, $x = \sqrt{3600^2 - 700^2}$. Wait, but that would be a very long leg. Wait, no, maybe the property line is one leg, and stream frontage is the other leg. Wait, the problem says "enclosed by the right triangle", so two legs and hypotenuse. Wait, maybe the stream frontage (700 ft) and the property line (3600 ft) are the two legs? Then area is $\frac{1}{2} \times 700 \times 3600$. Wait, that makes more sense. Maybe the figure shows the right angle between stream frontage and the other leg, and property line is the hypotenuse? No, wait, the problem says "property line is 3600 feet in length"—maybe the property line is one leg, and stream frontage is the other leg. Let's check: if it's a right triangle, area is $\frac{1}{2} \times \text{base} \times \text{height}$. So if stream frontage is base (700 ft) and the other leg is height (let's say $h$), but the property line is the hypotenuse. Wait, but then we need to find $h$. Wait, no, maybe the problem has a typo, or I misinterpret. Wait, let's re-read: "the stream frontage is 700 feet in length and the property line is 3600 feet in length". So stream frontage (one leg) = 700 ft, property line (hypotenuse) = 3600 ft? Then the other leg $b = \sqrt{3600^2 - 700^2}$. Let's calculate that: $3600^2 = 12,960,000$, $700^2 = 490,000$, so $12,960,000 - 490,000 = 12,470,000$. Then $b = \sqrt{12,470,000} \approx 3531$ ft (rounded to nearest whole number). Then area is $\frac{1}{2} \times 700 \times 3531$. Let's calculate that: $700 \times 3531 = 2,471,700$, then half of that is $1,235,850$ square feet. Now, convert square feet to acres: 1 acre = 43560 square feet. So divide by 43560: $1,235,850 \div 43560 \approx 28.37$? Wait, no, that can't be. Wait, maybe I got the legs wrong. Maybe the stream frontage and the property line are the two legs. Then area is $\frac{1}{2} \times 700 \times 3600 = 1,260,000$ square feet. Then convert to acres: $1,260,000 \div 43560 \approx 28.93$? Wait, but that seems too big. Wait, no, maybe the property line is one leg, and stream frontage is the other leg. Wait, let's check the conversion: 1 acre = 43560 sq ft. So if the triangle is right-angled, with legs 700 ft and 3600 ft, area is 0.5*700*3600 = 1,260,000 sq ft. Then 1,260,000 / 43560 ≈ 28.93. But that seems large. Wait, maybe the property line is the hypotenuse, and stream frontage is one leg, and the other leg is shorter. Wait, let's recalculate: $3600^2 - 700^2 = (3600 - 700)(3600 + 700) = 2900*4300 = 12,470,000$. Square root of 12,470,000: let's see, 3500^2 = 12,250,000, 3530^2 = (3500 + 30)^2 = 3500^2 + 2*3500*30 + 30^2 = 12,250,000 + 210,000 + 900 = 12,460,900. Then 3531^2 = 3530^2 + 2*3530 + 1 = 12,460,900 + 7060 + 1 = 12,467,961. 3532^2 = 12,467,961 + 2*3531 + 1 = 12,467,961 + 7062 + 1 = 12,475,024. So 12,470,000 is between 3531^2 and 3532^2. So approximately 3531 ft. Then area is 0.5*700*3531 = 0.5*2,471,700 = 1,235,850 sq ft. Then 1,235,850 / 43560 ≈ 28.37. But the problem says "round to nearest hundredth". Wait, maybe I made a mistake in assuming which sides are which. Wait, the figure: "Stream Frontage" is one leg (horizontal), the vertical leg, and "Property Line" is the hypotenuse. So stream frontage (700 ft) is one leg, property line (3600 ft) is hypotenuse. Then the other leg is $\sqrt{3600^2 - 700^2} \approx 3531$ ft. Then area is 0.5*700*3531 = 1,235,850 sq ft. Convert to acres: 1,235,850 / 43560 ≈ 28.37. But that seems large. Wait, maybe the property line is one leg, and stream frontage is the other leg. Let's try that: 0.5*700*3600 = 1,260,000 sq ft. 1,260,000 / 43560 ≈ 28.93. Hmm. Wait, maybe the problem has a typo, and the property line is 360 feet, not 3600? Let's check: 0.5*700*360 = 126,000 sq ft. 126,000 / 43560 ≈ 2.89. But the problem says 3600. Wait, maybe I misread the property line length. Wait, the problem says "3600 feet". So maybe it's correct. Let's proceed with the calculation.

Step1: Calculate area in square feet

Assuming the right triangle has legs 700 ft (stream frontage) and $x$ ft, with hypotenuse 3600 ft. Then $x = \sqrt{3600^2 - 700^2} \approx \sqrt{12960000 - 490000} = \sqrt{12470000} \approx 3531$ ft (rounded to nearest whole number). Then area $A = \frac{1}{2} \times 700 \times 3531 = 1235850$ square feet.

Step2: Convert square feet to acres

1 acre = 43560 square feet. So $A_{\text{acres}} = \frac{1235850}{43560} \approx 28.37$ (rounded to nearest hundredth).

Wait, but maybe the property line is one leg, and stream frontage is the other leg. Then area is $\frac{1}{2} \times 700 \times 3600 = 1260000$ square feet. Then $1260000 / 43560 \approx 28.93$. But which is correct? Let's check the figure: the right angle is between stream frontage and the vertical leg, and property line is the hypotenuse. So stream frontage is one leg (700 ft), vertical leg is $x$, hypotenuse is 3600 ft. So the first calculation is correct. So the area is approximately 28.37 acres.

Wait, but maybe I made a mistake in the Pythagorean theorem. Let's recalculate $3600^2 - 700^2$:

$3600^2 = (3.6 \times 10^3)^2 = 12.96 \times 10^6 = 12,960,000$

$700^2 = 490,000$

Subtract: $12,960,000 - 490,000 = 12,470,000$

Square root of 12,470,000: let's divide by 10000: $\sqrt{12,470,000} = \sqrt{1247 \times 10,000} = 100 \times \sqrt{1247} \approx 100 \times 35.31 \approx 3531$ (since $35^2 = 1225$, $35.3^2 = 1246.09$, $35.31^2 = 35.3^2 + 2*35.3*0.01 + 0.01^2 = 1246.09 + 0.706 + 0.0001 = 1246.7961$, which is close to 1247). So $x \approx 3531$ ft.

Then area: $0.5 * 700 * 3531 = 350 * 3531 = 1,235,850$ sq ft.

Convert to acres: $1,235,850 / 43560 ≈ 28.37$ (since 43560 * 28 = 1,219,680; 1,235,850 - 1,219,680 = 16,170; 16,170 / 43560 ≈ 0.371). So total ≈ 28.37.

Answer:

28.37