Sam is hoisting an engine form a pickup truck he is fixing. He hands hoist using 3 rafters on ceiling. Measures the angle the hoist makes at the baseto be 85 degrees and the length across the of chain across rafters to be 80 inches. One length of chain hanging down from ceiling to the hoist to be 70 inches. What is the length of the other piece of chain.?

To solve this problem, we can use the concept of trigonometry and the properties of right triangles. Let's break down the given information:

1. The angle the hoist makes at the base is 85 degrees.
2. The length across the chain across rafters is 80 inches.
3. One length of chain from the ceiling to the hoist is 70 inches.

We can consider the two length of chains as the two sides of a right triangle. The length from the ceiling to the hoist is the vertical side, and the length across the chain across rafters is the horizontal side.

To find the length of the other piece of chain, we need to find the length of the remaining side of the right triangle. Utilizing trigonometric functions, we can use the sine function to calculate this length. The sine function relates the length of the opposite side (vertical in this case) to the length of the hypotenuse (the chain across rafters).

The formula to find the length of the opposite side using the sine function is:

sin(angle) = opposite side / hypotenuse

Plugging in the values we have:

sin(85 degrees) = 70 inches / 80 inches

To solve for the opposite side, we rearrange the equation:

opposite side = sin(85 degrees) * hypotenuse
opposite side = sin(85 degrees) * 80 inches

Using a calculator, we can find the sine of 85 degrees and calculate the length of the opposite side:

sin(85 degrees) ≈ 0.9962

opposite side ≈ 0.9962 * 80 inches
opposite side ≈ 79.7 inches

Therefore, the length of the other piece of chain is approximately 79.7 inches.