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Chapter 14: Q. 55 (page 1120)

Mac owns a farm in the Palouse, a large agricultural region in the northwestern United States. One field on his farm can be modeled as the surface $$s(x,y)=0.24\sqrt{x^{2}+y^{2}}$$ on the square $$[0, 0.25]脳[0, 0.25]$$, where all distances are given in miles. Use this information to answer Exercises 55 and 56.

What is the actual area of Mac鈥檚 field?

Short Answer

Expert verified

The actual area of Mac鈥檚 field is equal to $$0.0643$$ miles.

Step by step solution

01

Step 1. Given Information

One field on Mac's farm can be modeled as the surface $$s(x,y)=0.24\sqrt{x^{2}+y^{2}}$$ on the square $$[0, 0.25]脳[0, 0.25]$$, where all distances are given in miles.

02

Step 2. Explanation

Partially differentiating $$z$$ with respect to $$x$$, we get

$$\frac{\partial z}{\partial x}=0.25$$

Partially differentiating $$z$$ with respect to $$y$$, we get

$$\frac{\partial z}{\partial y}=-0.25$$

So, $$ds= \sqrt{0.25^{2}+0.25^{2}+1}dA$$

$$\implies ds=1.0606$$

Hence, evaluating area, we get

$$A= \int 1 \cdot ds$$

$$\implies A=\int_{0.25}^{0.25}\int_{0}^{0.25}1.0606dxdy$$

Solving, we get

$$A=0.0643$$ miles

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