From the top of a 96 m tower, the angles of depression of two cars, on the same side of the tower are `alpha" and "beta`respectively. If `tanalpha=1/4" and "tanbeta=1/7`, then find the distance between two cars.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem and set up the scenario
We have a tower of height 96 meters. From the top of the tower, two cars are positioned at angles of depression α and β respectively. We know the values of tan α and tan β.

### Step 2: Define the distances
Let:
- Distance from the base of the tower to the first car (at angle α) = x
- Distance from the base of the tower to the second car (at angle β) = y

### Step 3: Use the tangent function for the first car
From the angle of depression α, we can use the tangent function:
\[ \tan \alpha = \frac{\text{height of tower}}{\text{distance to car}} = \frac{96}{x} \]
Given that \(\tan \alpha = \frac{1}{4}\), we can set up the equation:
\[ \frac{1}{4} = \frac{96}{x} \]

### Step 4: Solve for x
Cross-multiplying gives:
\[ x = 96 \times 4 = 384 \text{ meters} \]

### Step 5: Use the tangent function for the second car
Similarly, for the second car at angle β:
\[ \tan \beta = \frac{\text{height of tower}}{\text{distance to car}} = \frac{96}{y} \]
Given that \(\tan \beta = \frac{1}{7}\), we can set up the equation:
\[ \frac{1}{7} = \frac{96}{y} \]

### Step 6: Solve for y
Cross-multiplying gives:
\[ y = 96 \times 7 = 672 \text{ meters} \]

### Step 7: Find the distance between the two cars
The distance between the two cars is given by:
\[ \text{Distance between cars} = y - x = 672 - 384 \]

### Step 8: Calculate the final distance
Calculating the above expression gives:
\[ \text{Distance between cars} = 288 \text{ meters} \]

### Final Answer
The distance between the two cars is **288 meters**.

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