Related rates: How fast is a runner going to 1st base, as seen from 2nd base?

In summary, the problem involves determining the speed of a runner heading to first base from the perspective of an observer at second base. It utilizes related rates in calculus, where the distances between the bases and the runner's position are analyzed. By applying the Pythagorean theorem and differentiating with respect to time, one can calculate the runner's velocity as they approach first base, factoring in the rate at which they are moving and the changing distance from the observer at second base.
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karush
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TL;DR Summary: find how fast the distance is changing from 3rd base to a runer going to first base

Mentor note: Thread has been moved from a technical math section, so is missing the homework template.
A baseball diamond is a square with side 90 feet. A batter hits the ball and runs
toward first base with a speed of 28 feet per second.

\item [a] At what rate is his distance from second base decreasing when he is halfway to first base?

Draw a picture of the baseball diamond. The baseball player hits from home base and runs counterclockwise
towards first base. Then the player can continue counterclockwise to reach second
base, followed by third base.

so s^2=90^2+x^2

picture of field
 
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Look, I don't want to give you another infraction for no effort shown on your schoolwork. Your link generates an error.

Please type your work into the forum, and use the "Attach files" link to upload the diagram of the question. Thank you.
 
  • #3
berkeman said:
Look, I don't want to give you another infraction for no effort shown on your schoolwork. Your link generates an error.

Please type your work into the forum, and use the "Attach files" link to upload the diagram of the question. Thank you.
infraction????
ws07.10.png
 
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berkeman said:
Look, I don't want to give you another infraction for no effort shown on your schoolwork. Your link generates an error.

Please type your work into the forum, and use the "Attach files" link to upload the diagram of the question. Thank you.
I t tried to post a drawing but it never went thru thats parts of hw
 
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Use the "Attach files" link below the Edit window to upload a PDF or JPEG image. Please do it in a new reply. Thanks.
 
  • #6
karush said:
infraction????
Three strikes and your out!
 
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FAQ: Related rates: How fast is a runner going to 1st base, as seen from 2nd base?

What are related rates in the context of this problem?

Related rates in this context refer to the way the speed of the runner (from home to 1st base) affects the rate at which the distance between the runner and 2nd base changes over time. It involves using calculus to relate the rates of change of different quantities that are connected by a function.

How do you set up the problem using coordinates?

To set up the problem, you can place the baseball diamond on a coordinate plane with home plate at the origin (0,0), 1st base at (90,0), and 2nd base at (90,90) since each side of the diamond is 90 feet. The runner's position can be represented as a point moving along the x-axis from (0,0) to (90,0).

What is the formula for the distance between the runner and 2nd base?

The distance \(D\) between the runner and 2nd base can be found using the Pythagorean theorem. If the runner is at position \((x,0)\), then the distance \(D\) to 2nd base \((90,90)\) is given by \(D = \sqrt{(90 - x)^2 + 90^2}\).

How do you find the rate at which the distance to 2nd base is changing?

To find the rate at which the distance \(D\) to 2nd base is changing, you need to differentiate the distance formula with respect to time \(t\). This involves using the chain rule: \(\frac{dD}{dt} = \frac{dD}{dx} \cdot \frac{dx}{dt}\), where \(\frac{dx}{dt}\) is the runner's speed along the x-axis.

What additional information is needed to solve the problem?

To solve the problem, you need to know the runner's speed \(\frac{dx}{dt}\) and the runner's current position \(x\) along the x-axis. With this information, you can calculate the rate at which the distance to 2nd base is changing using the derived formula.

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