A 16-foot ladder is placed against the side of a building, as shown in Figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, the ladder is moved 4 feet closer to the base of the building, as shown in Figure 2 below.

Question

A 16-foot ladder is placed against the side of a building, as shown in Figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, the ladder is moved 4 feet closer to the base of the building, as shown in Figure 2 below.

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To the nearest foot, how much farther up the building does the ladder now reach? Show how you arrived at your answer.

Answer 1

In both figures, we have a right triangle formed by the ladder, the building, and the ground.

In Figure 1, the ladder is 16 feet long, and the distance from the base of the building to the ladder (the adjacent side) is 8 feet. We can use the Pythagorean theorem to find the height of the building (the opposite side):

height^2 + 8^2 = 16^2

height^2 = 16^2 - 8^2

height^2 = 192

height ≈ 13.86 feet

So in Figure 1, the ladder reaches a height of about 13.86 feet up the building.

In Figure 2, the ladder is 4 feet closer to the base of the building, so the distance from the base to the ladder is now 4 feet. We can use the same method to find the new height:

height^2 + 4^2 = 16^2

height^2 = 16^2 - 4^2

height^2 = 240

height ≈ 15.49 feet

So in Figure 2, the ladder reaches a height of about 15.49 feet up the building.

To find how much farther up the building the ladder reaches in Figure 2 compared to Figure 1, we can subtract the two heights:

15.49 - 13.86 ≈ 1.63

So to the nearest foot, the ladder now reaches 2 feet farther up the building (since we must round up when the decimal is 0.5 or greater).

Answer 2

In Figure 1, the ladder forms a right triangle with the building and the ground. Let's call the height of the building H. Since the ladder forms a right triangle, we can use the Pythagorean theorem to find the height of the building:

H^2 + 8^2 = 16^2

H^2 + 64 = 256

H^2 = 256 - 64

H^2 = 192

H = sqrt(192) (approximately)

H ≈ 13.86 feet (approximately)

So, in Figure 1, the ladder reaches approximately 13.86 feet up the building.

In Figure 2, the ladder is moved 4 feet closer to the base of the building. This means that the distance from the base of the building to the ladder is now 8 - 4 = 4 feet.

Using the same logic as before, we can find the height of the building in Figure 2:

H^2 + 4^2 = 16^2

H^2 + 16 = 256

H^2 = 256 - 16

H^2 = 240

H = sqrt(240) (approximately)

H ≈ 15.49 feet (approximately)

So, in Figure 2, the ladder reaches approximately 15.49 feet up the building.

To find how much farther up the building the ladder now reaches, we need to subtract the height in Figure 1 from the height in Figure 2:

15.49 - 13.86 ≈ 1.63 feet

Therefore, the ladder now reaches approximately 1.63 feet farther up the building when it is moved 4 feet closer to the base.