a vertical pole 35 feet high, standing on sloping ground, is braced by a wire which extends from the top of the pole to a point on the ground 25 feet from the foot of the pole. If the pole subtends an angle of 30 degree at the point where the wire reaches the ground, how long is the wire?

find the angle between the pole and the wire using the law of sines:

35/sin30 = 25/sinθ

Now use the law of sines again to find x, since you can now figure the angle opposite x.

12.4573

To find the length of the wire, we can use trigonometry.

Let's denote the length of the wire as "x".

We can form a right triangle with the vertical pole, the wire, and the ground. The height of the pole can be considered as the opposite side, the distance from the foot of the pole to the point where the wire reaches the ground can be considered as the adjacent side, and the wire itself can be considered as the hypotenuse.

The given angle of 30 degrees is opposite to the height of the pole (opposite side). So, we have:

sin(30°) = opposite/hypotenuse
sin(30°) = 35/x

To find the value of sin(30°), we can refer to the standard values of trigonometric ratios.

The sine of 30 degrees is 0.5.

So, we have:

0.5 = 35/x

Cross multiply:

0.5x = 35

Divide both sides by 0.5:

x = 35 / 0.5

x = 70

Therefore, the length of the wire is 70 feet.

To find the length of the wire, we can use trigonometry. Let's break down the problem and use the information given.

1. The vertical pole is 35 feet high.
2. The wire extends from the top of the pole to a point on the ground, which is 25 feet from the foot of the pole.
3. The pole subtends an angle of 30 degrees at the point where the wire reaches the ground.

To solve this problem, we'll use the trigonometric function tangent (tan). Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.

In this case, the opposite side is the height of the pole (35 feet), and the adjacent side is the horizontal distance from the foot of the pole to the point where the wire reaches the ground (25 feet).

Let's use the tangent formula:

tan(angle) = opposite / adjacent

Plugging in the values from the problem:

tan(30 degrees) = 35 / 25

To solve for the length of the wire, we'll isolate the unknown:

35 / 25 = tan(30 degrees)

Now, we need to find the value of tan(30 degrees). We can use a scientific calculator or lookup tables to get the tangent value for 30 degrees.

tan(30 degrees) = 0.577

Now, we rearrange the equation:

35 / 25 = 0.577

To find the length of the wire (opposite side), we multiply both sides by 25:

35 = 0.577 * 25

35 = 14.425

Finally, we can round the value to the nearest whole number:

The length of the wire is approximately 14 feet.