Question Statement

A suspension cable supports a bridge deck and forms a $45^{\circ }$ angle with the ground. If the deck is 30 meters above the ground, determine the length of the cable needed using the inverse cosine function.

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Background and Explanation

This problem utilizes right-angled trigonometry and the properties of isosceles triangles. By identifying the relationship between the height of the deck, the distance along the ground, and the length of the cable (the hypotenuse), we can apply trigonometric ratios to find the missing length.

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Solution

To solve this, we first define the components of the triangle formed by the deck, the ground, and the cable:

• Let $AB$ be the height of the deck ($30\text{ m}$).
• Let $BC$ be the length of the cable (the hypotenuse).
• Let $AC$ be the distance from the anchor point to the base of the deck (the base).
• Let $\angle ACB=\theta =45^{\circ }$ be the angle the cable makes with the ground.

Step 1: Identify the triangle type Because the angle with the ground is $45^{\circ }$ in a right-angled triangle, the remaining angle is also $45^{\circ }$. This makes it an isosceles triangle where the height and the base are equal: $AB=AC=30\text{ m}$

Step 2: Set up the inverse cosine equation The cosine of an angle is the ratio of the adjacent side (Base) to the hypotenuse. The inverse cosine function relates the angle to this ratio: $\theta ={\cos }^{-1}\left(\frac{\text{Base}}{\text{Hypotenuse}}\right)$

Substituting our known values: $45^{\circ }={\cos }^{-1}\left(\frac{AC}{BC}\right)$ $45^{\circ }={\cos }^{-1}\left(\frac{30}{BC}\right)$

Step 3: Solve for the cable length ($BC$) To isolate $BC$, we take the cosine of both sides: $\cos 45^{\circ }=\frac{30}{BC}$

Now, rearrange the equation to solve for $BC$: $BC\times \cos 45^{\circ }=30$ $BC=\frac{30}{\cos 45^{\circ }}$

Step 4: Calculate the final value Using the value of $\cos 45^{\circ }\approx 0.7071$: $BC=\frac{30}{0.7071}\approx 42.43\text{ m}$

The length of the cable needed is $42.43\text{ m}$.

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Key Formulas or Methods Used

• Inverse Cosine Definition: $\theta ={\cos }^{-1}\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)$
• Cosine Ratio: $\cos (\theta )=\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Isosceles Right Triangle Property: In a $45^{\circ }-45^{\circ }-90^{\circ }$ triangle, the legs are equal in length.

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Summary of Steps

1. Identify the height of the deck ($30\text{ m}$) and the angle of the cable ($45^{\circ }$).
2. Recognize that in a $45^{\circ }$ right triangle, the base ($AC$) equals the height ($AB=30\text{ m}$).
3. Apply the inverse cosine formula: $45^{\circ }={\cos }^{-1}(\frac{30}{BC})$.
4. Rearrange the formula to solve for the hypotenuse: $BC=\frac{30}{\cos 45^{\circ }}$.
5. Calculate the final length to arrive at $42.43\text{ m}$.