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4 years ago

Use the quadrant system to find the area of the polygon

Mathematics

1 answer:

AfilCa [17]4 years ago

3 0

Answer:

The area of the polygon is 64 units²

Step-by-step explanation:

* Lets explain how to solve the problem

- The polygon has 5 sides and 5 vertices

- To find its area by easy way split it into two trapezoid by a horizontal

line y = 1

- The vertices of trapezoid (1) are (-5 , 1) , (4 , 1) , (-5 , 5) , (0 , 5)

- The vertices of one of the parallel bases are (-5 , 1) , (4 , 1)

- The length of this base = 4 - (-5) = 4 + 5 = 9 units

- The vertices of the other parallel bases are (-5 , 5) , (0 , 5)

- The length of this base = 0 - (-5) = 0 + 5 = 5 units

- The length of its height = 5 - 1 = 4 units

- The area of the trapezoid = 1/2 (b1 + b2) × h

∴ Area of trapezoid 1 = 1/2 (9 + 5) × 4 = 28 units²

- The vertices of trapezoid (2) are (4 , 1) , (-5 , 1) , (-5 , -5) , (-2 , -5)

- The vertices of one of the parallel bases are (4 , 1) , (-5 , 1)

- The length of this base = 4 - (-5) = 4 + 5 = 9 units

- The vertices of the other parallel bases are (-5 , -5) , (-2 , -5)

- The length of this base = (-2) - (-5) = -2 + 5 = 3 units

- The length of its height = 1 - (-5) = 1 + 5 = 6 units

∴ Area of trapezoid 1 = 1/2 (9 + 3) × 6 = 36 units²

∵ Area of the polygon = Area trapezoid (1) + Area trapezoid (2)

∵ Area trapezoid (1) = 28 units²

∵ Area trapezoid (2) = 36 units²

∴ Area of polygon = 28 + 36 = 64 units²

∴ The area of the polygon is 64 units²

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Step 2: Differentiate

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[Implicit Differentiation] Simplify Exponents: $\displaystyle \frac{1}{2}x^{\frac{-1}{2}} - \frac{1}{2}y^{\frac{-1}{2}}\frac{dy}{dx} = 0$
[Implicit Differentiation] Rewrite [Exponential Rule - Rewrite]: $\displaystyle \frac{1}{2x^{\frac{1}{2}}} - \frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = 0$
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[Implicit Differentiation] Isolate $\displaystyle \frac{dy}{dx}$ : $\displaystyle \frac{dy}{dx} = \frac{2y^{\frac{1}{2}}}{2x^{\frac{1}{2}}}$
[Implicit Differentiation] Simplify: $\displaystyle \frac{dy}{dx} = \frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

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Exponents: $\displaystyle \frac{dy}{dx} = \frac{2}{1}$
Division: $\displaystyle \frac{dy}{dx} = 2$

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Implicit Differentiation

Book: College Calculus 10e

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