# **A ladder resting on a vertical wall makes an angle whose tangent is 2.5 with the...**

### Question

A ladder resting on a vertical wall makes an angle whose tangent is 2.5 with the ground. If the distance between the foot of the ladder and the wall is 60 cm, what is the length of the ladder?### Options

The correct answer is B.

### Explanation:

To solve this problem, we need to use the concept of trigonometry, specifically the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the angle whose tangent is 2.5 is the angle between the ground and the ladder.

We are given that the tangent of the angle is 2.5 and the distance between the foot of the ladder and the wall (the adjacent side) is 60 cm. We can use the tangent function to find the length of the opposite side (the height of the ladder on the wall). The formula for the tangent function is:

\[\text{Tangent} \ (\angle A) = \frac{\text{opposite side}}{\text{adjacent side}}\]

Plugging in the given values, we get:

\[2.5 = \frac{\text{opposite side}}{60}\]To find the opposite side, we can multiply both sides of the equation by 60:

\[\text{opposite side} = 2.5 \times 60\]Calculating this, we get:

\[\text{opposite side} = 150\]Now we have the lengths of the adjacent and opposite sides of the right triangle formed by the ladder, the ground, and the wall. To find the length of the ladder (the hypotenuse), we can use the Pythagorean theorem:

\[a^2 + b^2 = c^2\]Where a and b are the lengths of the legs (adjacent and opposite sides) and c is the length of the hypotenuse (the ladder). Plugging in the values we found, we get:

\[(60)^2 + (150)^2 = c^2\]Solving for c, we get:

\[c = \sqrt{(60)^2 + (150)^2}\]Calculating this, we get:

\[c \approx 160\]Since the length is given in centimeters, we need to convert it to meters by dividing by 100:

\[\text{Length of the ladder} = \frac{160}{100} = 1.6\ \text{m}\]Therefore, the length of the ladder is approximately **1.6 meters**, which corresponds to **Option B**.

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