Question Statement

A cable supporting a 15-meter high tower forms a $60^{\circ }$ angle with the ground. Determine the length of the cable needed to support the tower using the inverse cosine function.

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Background and Explanation

To solve this problem, we use right-angle trigonometry. We represent the tower and the cable as a right triangle where the tower is the opposite side, the ground is the adjacent side, and the cable is the hypotenuse. We utilize trigonometric ratios like tangent and cosine to find missing side lengths.

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Solution

Based on the problem description, let's define the components of our right triangle:

• Let $AB$ be the height of the tower: $AB=15\text{ m}$
• Let $BC$ be the length of the cable (hypotenuse): $BC=r$
• Let $AC$ be the length of the base on the ground: $AC=x$
• Let $\theta$ be the angle the cable makes with the ground: $\theta =60^{\circ }$

Step 1: Find the length of the base ($AC$)

To use the inverse cosine function later, we first need the length of the adjacent side ($AC$). We use the tangent formula because we know the opposite side ($AB$) and the angle ($\theta$):

$\tan \theta =\frac{AB}{AC}$

Substituting the known values:

$\tan 60^{\circ }=\frac{15}{AC}$

Rearranging to solve for $AC$:

$AC=\frac{15}{\tan 60^{\circ }}$

Since $\tan 60^{\circ }=\sqrt{3}$:

$AC=\frac{15}{\sqrt{3}}=5\sqrt{3}$

Step 2: Use the inverse cosine function to find the cable length ($BC$)

The definition of the cosine function is the ratio of the adjacent side to the hypotenuse. The inverse cosine function relates the angle to this ratio:

$\theta ={\cos }^{-1}\left(\frac{AC}{BC}\right)$

Substituting our known values:

$60^{\circ }={\cos }^{-1}\left(\frac{5\sqrt{3}}{BC}\right)$

To solve for $BC$, we take the cosine of both sides:

$\cos 60^{\circ }=\frac{5\sqrt{3}}{BC}$

Now, isolate $BC$:

$BC=\frac{5\sqrt{3}}{\cos 60^{\circ }}$

Since $\cos 60^{\circ }=0.5$ (or $\frac{1}{2}$):

$BC=\frac{5\sqrt{3}}{0.5}=10\sqrt{3}\text{ m}$

Step 3: Final Calculation

Converting the exact value to a decimal:

$BC\approx 10\times 1.732=17.32\text{ m}$

The length of the cable needed is approximately $17.32\text{ m}$.

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Key Formulas or Methods Used

• Tangent Ratio: $\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$
• Cosine Ratio: $\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Inverse Cosine Function: $\theta ={\cos }^{-1}\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)$
• Pythagoras Theorem: $(BC{)}^2=(AC{)}^2+(AB{)}^2$

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Summary of Steps

1. Identify the given values: height ($AB=15$) and angle ($\theta =60^{\circ }$).
2. Calculate the length of the base ($AC$) using the tangent function.
3. Set up the inverse cosine equation using the base ($AC$) and the unknown cable length ($BC$).
4. Solve the equation for $BC$ by applying the cosine of the given angle.
5. Calculate the final numerical value for the cable length.