# Related Rates: Baseball Diamond and Fish Reeling

• F.B
In summary, the first question asks about the speed of a baseball player when she is 30 ft away from first base, and the second question asks about the speed of a fish when the fishing line is 50 m long. Both questions involve using the Pythagorean Theorem and finding the value of dx/dt. The second question may have a different interpretation, so it is best to clarify with the teacher.
F.B
Can someone check if i did these two questions right please?

The questions are:

1. A baseball diamond is a perfect square with each side measuring 90 feet
in length. A player runs from the first base to second base at a speed of
25 ft/s. How fast is she moving from home plate when she is 30 ft away from
first base.

2. A fish is being reeled in a rate of 2 m/s (i.e, the fishing line is
being shortened by 2 m/s by a man. If he is sitting 30 m above water on a
dock, how fast is the fish moving through water when the line is 50 m long.

1. A drew my diamond and figured that i will be using half of the diamond.

Let the distance between home plate and first base be x
Let the distance from first base to second base by y

Given
y = 90 ft
dy/dt = 25 ft/s
x = 90 ft - 30ft
=60 ft
dx/dt = ?

x^2 + y^2 = z^2
90^2 + 90^2 = z^2
z = 127.3 ft

x^2 + y^2 = z^2
2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
sine dz/dt = 0
2(60)(dx/dt) + 2(90)(25) = 0
dx/dt = -37.5 ft/s

Is this correct? and why is my answer negative shouldn't it be positive
because you increasing the distance from home plate or is my answer
negative because the distance to first base is decreasing.

2. Given
y = 30 m
z = 50 m
dz/dt = - 2m/s
x = ?
dx/dt = ?

x^2 + y^2 = z^2
x^2 = 50^2 - 30^2
x = 40 m

x^2 + y^2 = z^2
2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
since dy/dt = 0
2(40)(dx/dt) = 2(50)(-2)
dx/dt = -2.5 m/s

Is this question correct because I am not sure when to make my numbers
negative, i know we have to when something is decreasing but I am not sure.

So can you please check if i did this two questions right?

For the first one, y isn't 90. Also, you won't find x by subtracting. It's a right triangle. How can you find an unknown side of a right triangle? The baseball player is 30 ft. from second base, not 90 ft. You had the general setup right though.

$$x^2=90^2+y^2$$

$$x\frac{dx}{dt}=y\frac{dy}{dt}$$

You know y, dy/dt, and you can solve for x using the Pythagorean Theorem. Now just find dx/dt.

The second one looks good to me. Your sign is correct. This question is worded weirdly. It asks how fast the fish is moving through the water and I interpret that to mean what speed, which is $$|\vec{v}|$$, where v is velocity. I think going by the question it's more correct to put the positive value, only showing magnitude and not direction, but you should ask your teacher.

## 1. What are related rates in the context of baseball?

Related rates refer to the mathematical concept of how two or more variables change with respect to each other. In the context of baseball, this can involve determining how the position, velocity, or acceleration of the ball changes in relation to other factors such as time, distance, or player movements.

## 2. How are related rates useful in understanding the physics of baseball?

Related rates can help us understand the complex interactions between the ball, players, and other factors in a baseball game. By using mathematical equations and models, we can analyze and predict how the ball will move and how players can make strategic decisions based on these calculations.

## 3. Can related rates be used to improve player performance in baseball?

Yes, related rates can be used to help players improve their performance. By understanding the physics of baseball and how related rates affect the game, players can make more informed decisions on the field, such as when to swing a bat or how to position themselves to catch a ball.

## 4. Are there any real-life applications of related rates in baseball?

Related rates have many real-life applications in baseball. For example, they can be used to analyze the trajectory of a ball during a pitch or hit, determine the optimal angle for a player to throw a ball, or predict the time it takes for a runner to reach a base.

There are many online resources available for learning about related rates and their applications in baseball. You can also consult with a physics or math teacher, or attend workshops or seminars on the topic. Additionally, studying the laws of motion and calculus can provide a strong foundation for understanding related rates in baseball.

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