Question: Shadow Speed Related Rates

[Karol]

Homework Statement

2. Homework Equations
Similar triangles

The Attempt at a Solution

$$\frac{y}{30}=\frac{50}{x}~\rightarrow~x=\frac{1500}{y}$$
$$\frac{dx}{dt}=-\frac{1500}{y^2}$$
$$s=16t^2=16\frac{1}{4}=4$$
$$\frac{dx}{dt}=-\frac{1500}{16}$$
The answer should be $~\displaystyle \frac{dx}{dt}=1500$

 

[Dick]

Homework Statement

View attachment 211572
View attachment 211573

2. Homework Equations
Similar triangles

The Attempt at a Solution

View attachment 211574
$$\frac{y}{30}=\frac{50}{x}~\rightarrow~x=\frac{1500}{y}$$
$$\frac{dx}{dt}=-\frac{1500}{y^2}$$
$$s=16t^2=16\frac{1}{4}=4$$
$$\frac{dx}{dt}=-\frac{1500}{16}$$
The answer should be $~\displaystyle \frac{dx}{dt}=1500$

You want to differentiate $x$ with respect to $t$, not $y$. Remember the chain rule?

 

[MarkFL]

When you differentiated with respect to $t$, you didn't apply the chain rule...

edit: Oops...beaten to the punch!

 

[Karol]

$$\frac{dx}{dt}=-\frac{1500}{y^2}\frac{dy}{dt}$$
$$y=16t^2~\rightarrow~\frac{dy}{dt}=32t$$
$$\frac{dx}{dt}=-\frac{1500}{16}\cdot 32\cdot \frac{1}{2}=1500$$