Mathematics: ASK YOUR TEACHER A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?

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ASK YOUR TEACHER A street light is mounted at the

Ответ:

Masielovebug

23.01.2022

Mathematics

Tip of the shadow is moving at the rate of $\dfrac{20}{3}\:ft/s$ .

Step-by-step explanation:

Refer to the attachment.

Let AB be the height of the street light. So, AB = 15 ft.

Let DE be the height of the man. So, DE = 6 ft.

Let BE be the distance between the man and light pole. Assume BE = x

Let BC be the distance between tip of shadow and light pole. Assume BC = y

Now height of the shadow of the man is EC which can be calculated as follows,

BC = BE + EC

y = x + EC

Subtracting x,

y – x = EC.

So there are two triangles that is, triangle ABC and triangle DEC. Assume that both the triangles are similar.

So applying similar triangle property,

$\dfrac{AB}{BC}=\dfrac{DE}{EC}$

Substituting the value,

$\dfrac{15}{y}=\dfrac{6}{y-x}$

Cross multiplying,

$15\times \left(y-x\right)=6\times y$

Using distributive property,

15y-15x=6y

Subtracting 6y,

15y-6y-15x=0

Adding 15x,

15y-6y=15x

9y=15x

Dividing by 9,

$y=\dfrac{15}{9}x$

Multiply and divide by 3,

$y=\dfrac{5}{3}x$

Differentiating above equation with respect to t,

$\dfrac{dy}{dt}=\dfrac{d}{dt}\left(\dfrac{5}{3}x\right)$

Applying constant multiple rule,

$\dfrac{dy}{dt}=\dfrac{5}{3}\dfrac{d}{dt}\left(x\right)$

$\dfrac{dy}{dt}=\dfrac{5}{3}\dfrac{dx}{dt}$

Given that man walks away from the pole with a speed of 4 ft/s that is,

$\dfrac{dx}{dt}=4$

Substituting the value,

$\dfrac{dy}{dt}=\dfrac{5}{3}\left(4\right)$

$\dfrac{dy}{dt}=\dfrac{20}{3}\:ft/s$

Therefore, tip of the shadow is moving at the rate of $\dfrac{20}{3}\:ft/s$ .

ASK YOUR TEACHER A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks aw

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ASK YOUR TEACHER A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?

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