Awoman 5 ft tall walks at the rate of 3.5 ft/sec away from a streetlight that is 12 ft above the ground. at what rate is the tip of her shadow moving? at what rate is her shadow lengthening? ⇒

Refer to the diagram shown below.

The woman walks at a rate of 3.5 ft/s away from the streetlight. Therefore


The length of the shadow changes at the rate


Because triangles ACE and BCD are similar  (AAA), therefore

12x = 5a + 5x
7x = 5a

Therefore


2.5 ft/s

y=-sqrt(x^2+9)

Step-by-step explanation:

This problem can be solved with the help of separating the variables.

dy/dx=x/y

Multiply both sides by y

y dy/dx=x

Multiply both sides by dx

y dy=x dx

Integrate

y^2/2=x^2/2+c

Let's find c using the condition y(0)=-3.

(-3)^2/2=(0)^2/2+c

9/2=0+c

9/2=c.

The equation is y^2/2=x^2/2+9/2.

Let's make this a bit prettier multiplying both sides by 2:

y^2=x^2+9

Taking square root of both sides gives:

y=+/- sqrt(x^2+9)

Since we want y(0)=-3, then we will choose

y=-sqrt(x^2+9).