Mr Dominguez is standing on a 40 ft ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 ft above the ground and the angles of depression to his dogs are 34 degrees and 48 degrees, how far apart are the dogs to the nearest foot?

y = 46 ..

1st dog ... tan 34 = y/x ... x = 46/ tan 34 ... x = 68.198 feet

2nd dog .. tan 48 = y/x ... x = 46/ tan 48 ... x = 41.42 feet

68.198 - 41.42 = 26.8 feet ~ 27 feet << answer

To find the distance between the dogs on the beach, we can use the trigonometric concept of angles of depression.

Step 1: Draw a diagram
- Draw a right triangle to represent the situation, with Mr. Dominguez standing on the ocean bluff and his dogs on the beach.
- Label the height of the ocean bluff as 40 ft and the line of sight as 6 ft above the ground.

Step 2: Identify the angles
- The angle of depression to the first dog is 34 degrees, and the angle of depression to the second dog is 48 degrees.
- These angles are measured downwards from the horizontal line.

Step 3: Calculate the distance between the dogs
- Let's assume that the distance between the dogs on the beach is represented by 'x' feet.
- Using the trigonometric tangent function, we can relate the angles and the sides of the right triangle:
- tan(34 degrees) = (40 ft - 6 ft) / x, for the first dog
- tan(48 degrees) = (40 ft - 6 ft) / x, for the second dog

Step 4: Solve the equations for 'x'
- tan(34 degrees) = 34 degrees in decimal form is approximately 0.67.
- tan(48 degrees) = 48 degrees in decimal form is approximately 1.10.
- Substitute these values into the equations:
- 0.67 = 34 / x
- 1.10 = 34 / (x - 1)
- Cross-multiply and solve for 'x':
- 0.67x = 34
- 1.10x - 1.10 = 34
- Subtract the second equation from the first equation:
- 0.43x = 33
- Divide both sides by 0.43:
- x ≈ 76.74 ft

Step 5: Round the distance to the nearest foot
- Since we are asked to find the distance to the nearest foot, round the calculated value of 'x' to the nearest whole number:
- x ≈ 77 ft

Therefore, the dogs are approximately 77 feet apart on the beach.

To find the distance between the dogs to the nearest foot, we can use trigonometry. Let's break down the problem step by step.

Step 1: Draw a diagram
Draw a diagram to visualize the situation described. Represent the ocean bluff as a vertical line, label it as "40 ft." Mark another horizontal line below it to represent the ground level. Place two points on the ground to represent the dogs. Also, draw two lines of sight from Mr. Dominguez to each dog, with angles of depression labeled as 34 degrees and 48 degrees.

Step 2: Identify the given information
From the problem, we know that Mr. Dominguez is standing at a height of 6 ft above the ground. The height of the ocean bluff is given as 40 ft. The angles of depression to the dogs are 34 degrees and 48 degrees.

Step 3: Apply trigonometry
Based on the diagram and the given information, we can create right triangles to solve the problem using trigonometric ratios.

For the triangle formed by Mr. Dominguez, one of the dogs, and the ground:
- The opposite side is the height of the ocean bluff (40 ft).
- The adjacent side is the distance between Mr. Dominguez and the dog we want to find.
- The angle is the angle of depression (34 degrees).

Using the tangent ratio:
tan(angle) = opposite/adjacent

tan(34 degrees) = 40 ft/adjacent

Solving for the adjacent side (distance):
adjacent = 40 ft / tan(34 degrees)

Using a scientific calculator, calculate the value of the tangent of 34 degrees and divide 40 ft by this value. The result will be the distance between Mr. Dominguez and one of the dogs.

Repeat the same process for the second dog using the angle of depression of 48 degrees.

Step 4: Calculate the distances
Using the trigonometric calculations mentioned above, compute the distances between Mr. Dominguez and each dog separately.

Step 5: Find the distance between the two dogs
Finally, calculate the difference between the distances found in step 4. This will give you the distance between the dogs to the nearest foot.