Problem 1 - The Calculus with Analytic Geometry

Question

A kite is flying at a height of \(40 \mathrm{ft}\). A boy is flying it so that it is moving horizontally at a rate of \(3 \mathrm{ft} / \mathrm{sec}\). If the string is taut, at what rate is the string being paid out when the length of the string released is \(50 \mathrm{ft}\) ?

Step-by-Step Solution

Answer

The rate at which the string is being paid out is 1.8 ft/sec.

Step 1 Title - Understand the Problem

You need to determine the rate at which the string of the kite is being paid out as the kite is flying horizontally.

Step 2 Title - Recognize Relationships

Identify the variables: let the height of the kite be denoted as 'h', the horizontal distance from the boy to the point directly below the kite be denoted as 'x', and the length of the string be denoted as 'L'. Given: \[ h = 40 \text{ ft} \] \[ \frac{dx}{dt} = 3 \text{ ft/sec} \] \[ L = 50 \text{ ft} \] Asked: \[ \frac{dL}{dt} \]

Step 3 Title - Apply Pythagoras' Theorem

The relationship between the height, horizontal distance, and the string length can be described by the Pythagorean theorem: \[ L^2 = h^2 + x^2 \] Substitute the known value for 'h': \[ L^2 = 40^2 + x^2 \] \[ L^2 = 1600 + x^2 \]

Step 4 Title - Differentiate with Respect to Time

Differentiate both sides of the equation with respect to time 't': \[ \frac{d}{dt}(L^2) = \frac{d}{dt}(1600 + x^2) \] Use the chain rule: \[ 2L \frac{dL}{dt} = 2x \frac{dx}{dt} \] Simplify: \[ L \frac{dL}{dt} = x \frac{dx}{dt} \]

Step 5 Title - Substitute Known Values

Given that when the length of the string is 50 ft, we have: \[ L = 50 \] Using the Pythagorean theorem, solve for 'x': \[ 50^2 = 40^2 + x^2 \] \[ 2500 = 1600 + x^2 \] \[ 900 = x^2 \] \[ x = 30 \text{ ft} \]

Step 6 Title - Solve for Rate of String Being Paid Out

Substitute the values of 'L', 'x', and \( \frac{dx}{dt} = 3 ft/sec \): \[ 50 \frac{dL}{dt} = 30 \times 3 \] \[ 50 \frac{dL}{dt} = 90 \] \[ \frac{dL}{dt} = \frac{90}{50} \] \[ \frac{dL}{dt} = 1.8 \text{ ft/sec} \]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed as:
\[ L^2 = h^2 + x^2 \]
For our kite problem, 'L' represents the length of the string, 'h' is the height at which the kite is flying, and 'x' is the horizontal distance from the boy to the kite's position directly overhead.
Here's how it works:

• Identify the sides: Given the height (\