2. Suppose the angle of depression from a race car driver's eyes to the bottom of the 3-foot high back end of the car in front of him is 18 degrees.

How far apart, to the nearest foot, are their bumpers? Assume the horizontal distance from the race car driver's eyes to the front of his car is 5 feet.

The bumpers are ? feet apart.

3 feet / d = tan 18°

so
d = 3 feet / tan 18° = a little over 9 feet

Subtracting the 5 feet to the front of the driver's own car
leaves a distance between bumpers of 4 feet.

Well, calculating the distance using trigonometry, we can say that the tangent of the angle of depression is equal to the height of the back end of the car divided by the horizontal distance. So, in this case, tangent(18 degrees) = 3 feet / x feet, where x is the distance between the bumpers.

Using a bit of algebra, we can rearrange this equation to solve for x. So, x = 3 feet / tangent(18 degrees).

Now, to find the value of x, I'm going to need my trusty calculator. *Takes out a giant calculator* Let's see... Tangent of 18 degrees is approximately 0.3249 (please don't hold me responsible for the accuracy of my calculations!).

So, x = 3 feet / 0.3249, which gives us approximately 9.24 feet.

Therefore, the bumpers are approximately 9.24 feet apart. But let's just round it to the nearest foot and say the bumpers are about 9 feet apart. That's assuming no one bumped while I was crunching these numbers!

To find the distance between the bumpers, we can use the tangent function.

Let's denote the distance between the race car driver's eyes to the bottom of the back end of the car in front of him as x.

From the given information, we know that the angle of depression is 18 degrees and the height of the back end of the car is 3 feet.

Using the tangent function, we have:

tan(18 degrees) = 3 / x

To solve for x, we can rearrange the equation:

x = 3 / tan(18 degrees)

Using a calculator, we can find the value of tan(18 degrees) ≈ 0.3249.

Therefore, substituting this value into the equation gives:

x = 3 / 0.3249 ≈ 9.2312

Since we want to find the distance between the bumpers and we are given that the horizontal distance from the race car driver's eyes to the front of his car is 5 feet, we can add the respective distances:

Distance between the bumpers = x + 5 ≈ 9.2312 + 5 ≈ 14.2312

Therefore, the bumpers are approximately 14 feet apart.

To solve this problem, we can use trigonometry and the concept of angles of elevation and depression. The angle of depression is the angle formed when the line of sight is below the horizontal line.

In this case, we have an angle of depression of 18 degrees from the race car driver's eyes to the bottom of the 3-foot high back end of the car in front of him. We are also given that the horizontal distance from the race car driver's eyes to the front of his car is 5 feet.

To find the distance between the bumpers, we need to determine the horizontal distance between the race car driver's eyes and the bottom of the back end of the car in front of him.

Step 1: Draw a diagram to represent the situation:

[Race Car Driver]-------------------> [Front of his car]
|
|
X-------[Back end of the car in front of him]

The line connecting the race car driver's eyes to the bottom of the back end of the car in front of him forms a right angle triangle with the horizontal line.

Step 2: Set up the trigonometric relationship:

In the given triangle, we have the opposite side (height of the back end of the car) and the adjacent side (horizontal distance from the race car driver's eyes to the front of his car).

Since we know the angle of depression (18 degrees) and the adjacent side (5 feet), we can use the tangent function to find the opposite side (height of the back end of the car).

tan(angle) = opposite/adjacent

tan(18 degrees) = height/5

Step 3: Solve for the opposite side (height):

height = tan(18 degrees) * 5

Step 4: Calculate the horizontal distance between the bumpers:

The horizontal distance between the bumpers is equal to the sum of the horizontal distance from the race car driver's eyes to the front of his car (5 feet) and the horizontal distance from the back end of the car in front of him to its bumper.

Since the height of the back end of the car is the same as the height of the front bumper, we can use the same height value calculated in the previous step.

Therefore, the horizontal distance between the bumpers is:

Horizontal distance = 5 + 5 (since both bumpers are at the same height)

Horizontal distance = 10 feet

So, the bumpers are approximately 10 feet apart.