Question

A wire is stretched from the ground to the top of a pole. The wire is 34 ft long. The height of the pole is 14 ft greater than the distance d from the pole's base to the bottom of the wire. Find the distance d and the height of the pole.

Answer 1

To solve for d and the height of the pole, we can set up an equation based on the given information. Using the Pythagorean theorem, we can solve the equation to find the value of d and the height of the pole.

Step-by-step explanation:

To solve this problem, we can set up an equation based on the given information. Let d represent the distance from the pole's base to the bottom of the wire. We know that the height of the pole is 14 ft greater than d, so the height of the pole can be represented as (d + 14). We also know that the wire is 34 ft long. Using the Pythagorean theorem, we can set up the equation d^2 + (d + 14)^2 = 34^2. Solving this equation will give us the value of d and the height of the pole.

Expanding the equation, we get d^2 + d^2 + 28d + 196 = 1156.

Combining like terms, we have 2d^2 + 28d + 196 = 1156.

Subtracting 1156 from both sides, we get 2d^2 + 28d - 960 = 0.

Simplifying, we have d^2 + 14d - 480 = 0.

Factoring the quadratic equation, we get (d + 30)(d - 16) = 0.

Setting each factor equal to zero, we find d = -30 or d = 16.

Since distance cannot be negative, we can discard the -30 value.

Therefore, the distance d is 16 ft. The height of the pole can be found by adding 14 to d, so the height of the pole is 16 + 14 = 30 ft.

Answer 2

The distance from the base of the pole to the bottom of the wire is d = 20 ft, and the height of the pole is h = 34 ft.

Step-by-step explanation:

To solve this problem, let's denote the distance from the pole's base to the bottom of the wire as d and the height of the pole as h. Given that the wire is 34 feet long and the height of the pole is 14 feet greater than d, we can set up an equation based on the Pythagorean theorem: d^2 + (d + 14)^2 = 34^2.

Expanding and simplifying the equation:

d^2 + d^2 + 28d + 196 = 1156

2d^2 + 28d - 960 = 0

Solving the quadratic equation:

d^2 + 14d - 480 = 0

(d + 30)(d - 16) = 0

Therefore, d = 16 ft or d = -30 ft. Since distance cannot be negative,

d = 16 ft.

Now that we have found d, let's calculate the height of the pole using

h = d + 14:

h = 16 + 14

h = 30 ft

Hence, the distance from the base of the pole to the bottom of the wire is d = 16 ft, and the height of the pole is h = 30 ft.