Math Problem Statement

Budek is flying a kite. The kite is initially directly over her head at a height of 30 m. Then the wind starts carrying the kite in one direction at a rate of 40 m/min while Budek starts running in the opposite direction at a rate of 200 m/min. Assuming that the height of the kite remains constant and that the string forms a straight line, find the rate at which the string is paying out after 10 s.

Solution

To solve the problem, we can use the Pythagorean Theorem and calculus to find the rate at which the kite string is paying out.


Step 1: Define the variables

Let:

  • xx be the horizontal distance between Budek and the kite in meters.
  • zz be the length of the kite string in meters.
  • The height of the kite remains constant at h=30mh = 30 \, \text{m}.

From the Pythagorean theorem: z2=x2+h2z^2 = x^2 + h^2


Step 2: Differentiate with respect to time (tt)

Differentiate both sides of the equation z2=x2+h2z^2 = x^2 + h^2 with respect to tt: 2zdzdt=2xdxdt2z \frac{dz}{dt} = 2x \frac{dx}{dt} Simplify: zdzdt=xdxdtz \frac{dz}{dt} = x \frac{dx}{dt} Solve for dzdt\frac{dz}{dt}, the rate at which the string is paying out: dzdt=xdxdtz\frac{dz}{dt} = \frac{x \frac{dx}{dt}}{z}


Step 3: Express xx and dxdt\frac{dx}{dt}

  1. Initially, the kite is directly above Budek, so x=0x = 0 at t=0t = 0.
  2. The kite moves at 40m/min40 \, \text{m/min} in one direction, and Budek runs at 200m/min200 \, \text{m/min} in the opposite direction. Therefore, the rate at which the horizontal distance xx increases is: dxdt=40+200=240m/min.\frac{dx}{dt} = 40 + 200 = 240 \, \text{m/min}.

After t=10s=1060mint = 10 \, \text{s} = \frac{10}{60} \, \text{min}, the horizontal distance xx is: x=dxdtt=2401060=40m.x = \frac{dx}{dt} \cdot t = 240 \cdot \frac{10}{60} = 40 \, \text{m}.


Step 4: Find zz at t=10st = 10 \, \text{s}

At t=10st = 10 \, \text{s}, the kite string length zz is: z=x2+h2=402+302=1600+900=2500=50m.z = \sqrt{x^2 + h^2} = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \, \text{m}.


Step 5: Calculate dzdt\frac{dz}{dt}

Substitute x=40mx = 40 \, \text{m}, dxdt=240m/min\frac{dx}{dt} = 240 \, \text{m/min}, and z=50mz = 50 \, \text{m} into the equation: dzdt=xdxdtz=4024050=960050=192m/min.\frac{dz}{dt} = \frac{x \frac{dx}{dt}}{z} = \frac{40 \cdot 240}{50} = \frac{9600}{50} = 192 \, \text{m/min}.


Final Answer:

The rate at which the kite string is paying out after 10 seconds is: 192m/min\boxed{192 \, \text{m/min}}


Would you like a detailed explanation of any step or further clarification?
Here are 5 related questions for deeper understanding:

  1. How would the solution change if Budek ran in the same direction as the wind?
  2. What happens if the kite's height changes over time instead of staying constant?
  3. How do you derive the formula for related rates using the Pythagorean theorem?
  4. How does the rate of payout change if Budek’s speed doubles?
  5. How can you use this method to solve real-world problems involving distances and rates?

Tip: Always identify the changing variables and their relationships when solving related rates problems.

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Math Problem Analysis

Mathematical Concepts

Calculus
Related Rates
Pythagorean Theorem

Formulas

Pythagorean theorem: z^2 = x^2 + h^2
Rate of change: dz/dt = (x * dx/dt) / z

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 10-12