# Question: Shadow Speed Related Rates

• Karol
In summary, the content of the conversation pertains to finding the derivative of x with respect to t using similar triangles and the chain rule. The correct answer is 1500.
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##

Karol said:

## 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?

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

edit: Oops...beaten to the punch!

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

## 1. What is the concept of related rates in relation to shadow speed?

The concept of related rates involves analyzing the rate of change of one variable with respect to another. In the context of shadow speed, this means studying how the speed of a moving object affects the rate of change of the length of its shadow.

## 2. How is the speed of the object related to the shadow speed?

The speed of the object and the shadow speed are related through the use of similar triangles. As the object moves, its shadow also moves and the ratio of their speeds is equal to the ratio of their distances from the light source.

## 3. What is the role of trigonometry in related rates of shadow speed?

Trigonometry is used to calculate the angles and sides of the similar triangles formed by the object and its shadow. This allows us to determine the relationship between their speeds and distances.

## 4. How can the related rates formula be applied to determine shadow speed?

The related rates formula can be applied by taking the derivative of the shadow length with respect to time and setting it equal to the product of the object speed and the derivative of the distance between the object and the light source with respect to time.

## 5. What are some real-life applications of related rates in shadow speed?

Related rates in shadow speed can be applied in various fields such as astronomy, engineering, and physics. For example, it can be used to calculate the speed of a planet's rotation based on the length of its shadow, or to determine the speed of a car based on the length of its shadow on the ground.

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