Question

Find the area of rectangular Field in acres whose sides are 50m and 16m 7dm.

Answer 1

Hello!

Find the area of rectangular Field in acres whose sides are 50 m, 16 m and 7 dm.

Note¹: It is important that the questions are well worked out, or else the problem explanation can be changed.

We have the following data:

$a = 50\:m$

$b = 16\:m$

$c = 7\:dm \to c = 0.7\:m$

$Area = ?\:(in\:acres)$

Note²: As we know the three sides of the triangle, let's apply Heron's formula to find the area.

• [First Step] Let's find the semiperimeter of the triangle, let's see:

$S = \dfrac{a+b+c}{2}$

$S = \dfrac{50+16+0.7}{2}$

$S = \dfrac{66.7}{2}$

$\boxed{S = 33.35\:m}\Longleftarrow(semiperimeter)$

• [Second Step]  Let's find the area of ​​the triangle, let's see:

$Area = \sqrt{S*(S-a)*(S-b)*(S-c)}$

$Area = \sqrt{33.35*(33.35-50)*(33.35-16)*(33.35-0.7)}$

$Area = \sqrt{33.35*(-16.65)*(17.35)*(32.65)}$

$Area = \sqrt{33.35*(-9431.850375)}$

$Area = \sqrt{-314552.21}$

$Area = \sqrt{-1*314552.21}$

$Area = \sqrt{-1}*\sqrt{314552.21}$

$\boxed{Area \approx 560.85\:i\:\:m^2}$

Note³: It was observed that the square root was negative, in this case the set of complex numbers was applied

• If we look at the question, (Find the area of rectangular Field in acres). We will have to convert the measures (m² to acres), see:

(area in m²) _______ (area in acres)

1 m² ------------------------ 4046.9 acres

560.85 i m² ---------------- y

$\dfrac{1}{560.85\:i} = \dfrac{4046.9}{y}$

Apply to y = x + vi

$\dfrac{1}{560.85\:i} = \dfrac{4046.9}{x+vi}$

multiply the means by the extremes

$1*(x+vi) = 560.85\:i*4046.9$

$x+vi = 2269703.865\:i$

Let's prescribe for the standard complex form, let's see:

$x+vi = 0 + 2269703.865\:i$

$\left \{ {{x\:=\:0} \atop {v\:=\:2269703.865}} \right.$

replace in the equation: y = x + vi, we heve:

$\boxed{x = 0}$

$\boxed{y = 2269703.865\:i}$

Answer:

The area of rectangular is approximately 2269703.865i acres

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*** Another way to solve (If the question changed) ***

Find the area of rectangular Field in acres whose sides are 50 m, 45 m and 35 m.

We have the following data:

$a = 50\:m$

$b = 45\:m$

$c = 35\:m$

$Area = ?\:(in\:m^2)$

Note: As we know the three sides of the triangle, let's apply Heron's formula to find the area.

• [First Step] Let's find the semiperimeter of the triangle, let's see:

$S = \dfrac{a+b+c}{2}$

$S = \dfrac{50+45+35}{2}$

$S = \dfrac{130}{2}$

$\boxed{S = 65\:m}\Longleftarrow(semiperimeter)$

• [Second Step]  Let's find the area of ​​the triangle, let's see:

$Area = \sqrt{S*(S-a)*(S-b)*(S-c)}$

$Area = \sqrt{65*(65-50)*(65-45)*(65-35)}$

$Area = \sqrt{65*(15)*(20)*(30)}$

$Area = \sqrt{65*(9000)}$

$Area = \sqrt{585000}$

Let's factor for the least common multiple

585000 | 5

117000 | 5

23400 | 5

4680 | 5

936 | 3

312 | 3

104 | 2

52 | 2

26 | 2

13 | 13

1 \___ = $5^4 * 2^3 * 3^2 * 13 = \boxed{5^2*5^2*2^2*2*3^2*13}$

then:

$Area = \sqrt{585000}$

$Area = \sqrt{ 5^2*5^2*2^2*2*3^2*13}$

$Area = 5*5*2*3\sqrt{2*13}$

$\boxed{\boxed{Area = 150\sqrt{26}\:m^2\:\:\:or\:\:\:Area \approx 764.85\:m^2}}\:\:\:\:\:\:\bf\green{\checkmark}$

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