A 7-foot ladder is leaning against a wall such that the angle relative to the level ground is `70^(@)`. Which of the following expressions involving cosine gives the distance, in feet, from the base of the ladder to the wall?

Options

A. `(7)/(cos 70^(@))`
B. `(cos70^(@))/(7)`
C. `(1)/(7cos 7^(@))`
D. `7cos 70^(@)`

Answer

To solve the problem of finding the distance from the base of the ladder to the wall, we can use trigonometric relationships in a right triangle. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have a right triangle formed by the wall, the ground, and the ladder. The ladder is the hypotenuse, the distance from the base of the ladder to the wall is the base, and the height of the ladder against the wall is the perpendicular.

### Step 2: Identify the Given Information
- Length of the ladder (hypotenuse, AC) = 7 feet
- Angle with the ground (∠CAB) = 70 degrees

### Step 3: Use the Cosine Function
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side (base) to the length of the hypotenuse. Thus, we can write:

\[ \cos(70^\circ) = \frac{\text{Base (AB)}}{\text{Hypotenuse (AC)}} \]

### Step 4: Substitute the Known Values
Substituting the known values into the equation:

\[ \cos(70^\circ) = \frac{AB}{7} \]

### Step 5: Solve for the Base (AB)
To find the distance from the base of the ladder to the wall (AB), we can rearrange the equation:

\[ AB = 7 \cdot \cos(70^\circ) \]

### Conclusion
Thus, the expression that gives the distance from the base of the ladder to the wall is:

\[ AB = 7 \cos(70^\circ) \]

The correct answer is option D: \(7 \cos(70^\circ)\).

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