Problem Setup: Pillars and Rope

We have two vertical pillars separated by a horizontal distance d = 8 feet. The difference in their heights is Δh = 6 feet. A rope of length L = 15 feet is tied to the tips of both pillars.

Geometric Analysis

Let the height of the shorter pillar be $h_s$ and the taller pillar be $h_t$. We know $h_t - h_s = \Delta h = 6$ feet. The horizontal distance between them is $d = 8$ feet. The straight-line distance (hypotenuse) between the tips of the pillars can be calculated using the Pythagorean theorem:

$D = \sqrt{d^2 + (\Delta h)^2}$ $D = \sqrt{8^2 + 6^2}$ $D = \sqrt{64 + 36}$ $D = \sqrt{100}$ $D = 10$ feet.

The calculated straight-line distance between the pillar tops is 10 feet. The rope length is 15 feet. Since the rope is longer than the distance between the attachment points ($L > D$), the rope will hang loosely.

Calculating Rope's Potential Reach

The question asks for the portion of the taller pillar's length that can be brought in contact with the rope. This implies finding how much the rope's length exceeds the direct distance in terms of vertical reach. We calculate the maximum height difference ($D_{max}$) the rope *could* span if pulled taut between the horizontal distance $d=8$ feet.

Using the rope length $L=15$ ft as the hypotenuse and the distance $d=8$ ft as one leg:

$D_{max}^2 = L^2 - d^2$ $D_{max}^2 = 15^2 - 8^2$ $D_{max}^2 = 225 - 64$ $D_{max}^2 = 161$ $D_{max} = \sqrt{161}$ feet.

Calculating the approximate value: $D_{max} \approx 12.688$ feet.

Determining the Contact Portion

This maximum possible height difference ($D_{max} \approx 12.688$ feet) represents the furthest the rope could reach vertically. The actual height difference between the pillars is $\Delta h = 6$ feet. The difference between the maximum potential vertical reach and the actual height difference shows the excess reach due to the rope's length.

Excess Reach = $D_{max} - \Delta h$ Excess Reach = $\sqrt{161} - 6$ feet Excess Reach $\approx 12.688 - 6$ feet Excess Reach $\approx 6.688$ feet.

This value, approximately 6.688 feet, represents the portion of the taller pillar's length that the rope could potentially contact because its length exceeds the direct distance between the pillar tops.

Comparing this value to the options:

• less than 6 feet
• more than 6 feet but less than 7 feet (6.688 feet falls in this range)
• more than 7 feet but less than 8 feet
• more than 8 feet

Therefore, the portion of the taller pillar's length that can be brought in contact with the rope is more than 6 feet but less than 7 feet.