A ladder that is 13 feet long leans against a building .The bottom of the ladder is 5 feeet away from the base of the building .how far up the side of the building does the ladder reach?​

Length of the ladder = 13 m

Foot of the ladder from the wall = 5 m.

The upper end of the ladder is (13^2–5^2)^0.5 = 12 m above the ground.

In 1 second the foot of the ladder is 7 m from the wall, while the top end is (13^2–7^2)^0.5 = (169–49)^0.5 = 120^0.5 = 10.95 m. Then the top moves 1.05 m downwards.

In 2 seconds, the foot of the ladder is 9 m from the wall, while the top end is (13^2–9^2)^0.5 = (169–81)^0.5 = 88^0.5 = 9.38 m. Then the top moves 3.62 m downwards.

In 3 seconds, the foot of the ladder is 11 m from the wall, while the top end is (13^2–11^2)^0.5 = (169–121)^0.5 = 48^0.5 or 6.93 m. Then the top moves 6.07 m downwards.

So, if the foot of the ladder moves at a constant rate of 2 m/s the top end of the ladder moves downwards which varies from second to second. The lader will form the hypotenuse. Length of hypotenuse = 13m.

Let us take the horizontal distance between the wall and the ladder as ‘ground’. This g = 5m

wall ^ 2 + ground ^ 2 = 13^2 = 169

w^2 + g^2 = 169

w^2 + 25 = 169. Thus, w^2 = 144. Thus, w = 12

Rate of change of ground distance = dg / dt = 2m/s (given).

In the equation w^2 + g^2 = 169, differentiate the equation with respect to time.

2 w dw/dt + 2 g dg/dt = 0

2 (12) (dw/dt) + 2 (5) (2) = 0. So, 24 dw/dt + 20 = 0

dw/dt = -20 / 24 = -5 / 6

Thus, the height of the wall is decreasing at a rate of -5/6 m / s (that is, -0.833m/s)

The minus sign denotes that the height of the ladder is falling.

Step-by-step explanation:

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