Questions: Solve the problem. A boat is anchored off a small island. From the bow of the boat, 36 ft of anchor line is out with 4 ft of line above the water. The angle that the line makes with the water is 26 degrees. How deep is the water? Round to the nearest foot. Select one: a. 29 ft b. 16 ft c. 32 ft d. 14 ft

Solution Steps

To find the depth of the water, we need to determine the vertical component of the anchor line that is submerged. We can use trigonometry, specifically the sine function, which relates the angle of the line with the horizontal and the opposite side (depth of the water). The total length of the line is 36 ft, and 4 ft is above water, so 32 ft is the length of the line in the water. We use the sine of the angle to find the depth.

The total length of the anchor line is given as \( 36 \, \text{ft} \). The length of the line above water is \( 4 \, \text{ft} \). Therefore, the length of the line submerged in water is calculated as: \[ \text{line\_in\_water} = 36 \, \text{ft} - 4 \, \text{ft} = 32 \, \text{ft} \]

The angle that the line makes with the water is given as \( 26^{\circ} \). To use trigonometric functions, we convert this angle to radians: \[ \text{angle\_radians} = \frac{26 \times \pi}{180} \approx 0.4538 \]

Using the sine function, we can find the depth of the water. The depth \( d \) can be expressed as: \[ d = \text{line\_in\_water} \times \sin(\text{angle\_radians}) = 32 \, \text{ft} \times \sin(0.4538) \approx 14.0279 \, \text{ft} \] Rounding this value to the nearest foot gives: \[ d \approx 14 \, \text{ft} \]

Final Answer