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Problem 555

# Let $$a, b, c$$ be the lengths of the sides of a triangle and let $$\theta$$ be the angle opposite the side of length $$c$$. Find the differential do and approximate c when $$\mathrm{a}=6.20, \mathrm{~b}=5.90$$, and $$0=58^{\circ}$$.

Expert verified
In conclusion, the differential $$dc \approx 6.142$$ and the approximate length of side $$c$$ when $$a = 6.20$$, $$b = 5.90$$, and $$\theta = 58°$$ is $$c \approx 5.42$$ units.
See the step by step solution

## Step 1: Write the Law of Cosines for the given triangle.

Let's write the Law of Cosines for the triangle with sides a, b, and c, and angle θ opposite to side c: $$c^{2} = a^{2} + b^{2} - 2ab\cos \theta$$

## Step 2: Differentiate both sides of the equation with respect to θ.

Now, we differentiate both sides of the equation with respect to θ: $$\frac{d}{d\theta}(c^{2})= \frac{d}{d\theta}(a^{2} + b^{2} - 2ab \cos \theta)$$ Using the chain rule and noting that a and b are constants, we have: $$2c\frac{dc}{d\theta} = 2ab\sin \theta \frac{d\theta}{d\theta}$$ The dθ/dθ term is equal to 1, so our equation becomes: $$2c\frac{dc}{d\theta} = 2ab\sin \theta$$ Now, let's solve for dc: $$\frac{dc}{d\theta} = \frac{ab\sin \theta}{c}$$

## Step 3: Calculate the approximate value of c using the given values of a, b, and θ.

Using the given values of a = 6.20, b = 5.90, and θ = 58°, we can plug these values into the relation found in Step 1: $$c^{2} = 6.20^{2} + 5.90^{2} - 2(6.20)(5.90)\cos(58°)$$ Calculating these values, we get: $$c² = 38.44 + 34.81 - 2(6.20)(5.90) \cos(58°) ≈ 38.44 + 34.81 - 43.879$$ $$c² ≈ 29.371$$ Now, we find the square root of c² to get the value of c: $$c ≈ \sqrt{29.371} ≈ 5.42$$

## Step 4: Calculate the value of dc using the found value of c and the given values of a, b, and θ.

Now we can plug in the values of a, b, c, and θ into the relation found in Step 2 to find the value of dc: $$\frac{dc}{d\theta} = \frac{(6.20)(5.90)\sin(58°)}{5.42}$$ Calculating this value, we get: $$\frac{dc}{d\theta} ≈ \frac{36.58}{5.42} ≈ 6.142$$ In conclusion, the differential dc is approximately 6.142, and the approximate length of side c when a = 6.20, b = 5.90, and θ = 58° is 5.42 units.

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