A person 1.8m tall is walking away from a lamppost 6m high at the rate 1.3m/s. At what rate is the end of the person's shadow moving away from the lamppost?

Answer 1

#= 13/7 m/s approx 1.86 m/s#

the point of what follows and the choice of variables in the drawing is this. we know that #dot x = 1.3#. we have been asked to find the rate at which the end of the shadow is moving away from the lamppost - that's #dot y# !

so similar triangles tells us that

#6/y = 1.8/(y - x)#

#y = (6x)/4.2 = 10/7 x#

and now for the little bit of calculus, taking the deriv of each side with respect to time

#implies dot y = 10/7 dot x#
#= 10/7 * 1.3 = 13/7 m/s approx 1.86 m/s#

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Answer 2

To find the rate at which the end of the person's shadow is moving away from the lamppost, we can use similar triangles. Let's denote:

( h_p ) = height of the person (1.8m)

( h_l ) = height of the lamppost (6m)

( x ) = distance from the person to the lamppost (the length of the shadow)

Given that the person is walking away from the lamppost at a rate of ( 1.3 , \text{m/s} ), we need to find ( \frac{dx}{dt} ), the rate at which the shadow's end is moving away from the lamppost.

Using similar triangles, we have:

[ \frac{h_l}{x} = \frac{h_p}{x + h_p} ]

Differentiating both sides with respect to time ( t ):

[ -\frac{h_l}{x^2}\frac{dx}{dt} = -\frac{h_p}{(x + h_p)^2}\frac{dx}{dt} ]

Given ( h_l = 6 , \text{m} ), ( h_p = 1.8 , \text{m} ), and ( \frac{dx}{dt} = 1.3 , \text{m/s} ), we can solve for ( \frac{dx}{dt} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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