A boat drops an anchor to the bottom of a lake. The anchor rope makes a 15 degree angle with the boat. The anchor rope is 24 feet long. How deep is the lake to the nearest foot?
d/24 = cos 15°
To find the depth of the lake, we can use trigonometry. Here's how it works:
Step 1: Draw a diagram. Draw a right triangle where the length of the anchor rope is the hypotenuse, the depth of the lake is the opposite side, and the distance from the boat to the anchor point is the adjacent side. Label the appropriate angles and sides.
Step 2: Use the given information. The anchor rope is 24 feet long, and it forms a 15-degree angle with the boat. Therefore, we have the hypotenuse (24 feet) and one angle (15 degrees).
Step 3: Apply trigonometric ratios. We can use the sine ratio in this case, since we have the opposite side (depth) and the hypotenuse. The sine ratio is defined as follows: sin(angle) = opposite/hypotenuse.
Step 4: Substitute the values into the formula. Let's call the depth of the lake "x". The equation becomes sin(15 degrees) = x/24.
Step 5: Solve for x. Rearrange the equation to isolate x: x = sin(15 degrees) * 24.
Step 6: Calculate the value of x. Using a calculator, find the sine of 15 degrees, which is approximately 0.2588, and multiply it by 24. This gives us x ≈ 6.21 feet.
Therefore, the depth of the lake to the nearest foot is 6 feet.