Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. A kite is flying on 50 feet of string. How high is it above the ground if its height is 10 feet more than the horizontal distance from the person flying it? Assume that the string is being held at ground level.

Step-by-Step Solution

Answer

The kite is 40 feet above the ground.

Step 1: Define Variables

Let the height of the kite above the ground be \( h \) feet and the horizontal distance from the person be \( x \) feet. According to the problem, the height is 10 feet more than the horizontal distance, so \( h = x + 10 \).

Step 2: Use the Pythagorean Theorem

The kite, the horizontal distance, and the string form a right triangle. The string is the hypotenuse (50 feet), the height \( h \) is one leg, and \( x \) is the other leg. According to the Pythagorean theorem, we have the equation: \[ h^2 + x^2 = 50^2 \]

Step 3: Substitute the Height Equation

Substitute \( h = x + 10 \) into the Pythagorean theorem equation:\[ (x + 10)^2 + x^2 = 2500 \]Expand the equation: \[ x^2 + 20x + 100 + x^2 = 2500 \]

Step 4: Simplify and Solve for \( x \)

Combine like terms to simplify the equation:\[ 2x^2 + 20x + 100 = 2500 \]Subtract 2500 from both sides:\[ 2x^2 + 20x - 2400 = 0 \]Divide the entire equation by 2:\[ x^2 + 10x - 1200 = 0 \]

Step 5: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = 10 \), and \( c = -1200 \).Calculate the discriminant: \[ b^2 - 4ac = 10^2 - 4 \times 1 \times (-1200) = 100 + 4800 = 4900 \]Substitute into the formula:\[ x = \frac{-10 \pm \sqrt{4900}}{2} \]Simplify:\( \sqrt{4900} = 70 \), so\[ x = \frac{-10 \pm 70}{2} \]This gives solutions \( x = 30 \) or \( x = -40 \). Since distance cannot be negative, \( x = 30 \).

Step 6: Calculate the Height

Now that we know \( x = 30 \), substitute back into the height equation:\[ h = x + 10 = 30 + 10 = 40 \]

Key Concepts

Pythagorean Theorem

In trigonometry, the Pythagorean Theorem is a fundamental principle that deals with right triangles. A right triangle has one angle measuring 90 degrees. This theorem helps connect the three sides of a right triangle. The longest side is called the hypotenuse, while the other two sides are the legs.
The theorem states that the square of the hypotenuse is equal to the sum of the squares of the two legs. Mathematically, this can be expressed as:

• Hypotenuse: \( c \)
• Legs: \( a \) and \( b \)

Thus, the equation is: \[ c^2 = a^2 + b^2 \]In our kite problem, this concept is essential because the kite string acts as the hypotenuse, while the height and horizontal distance from the person are the other two sides.Understanding how to apply this theorem enables us to find relationships between different measures in a triangle, which is useful in many practical situations like surveying and navigation.

Right Triangle

A right triangle is a triangle with one of its angles measuring exactly 90 degrees. This specific triangle shape has unique properties that make it invaluable in trigonometry.
In a right triangle:

• The side opposite the right angle is the hypotenuse, the longest side of the triangle.
• The other two sides are referred to as the legs.

In the context of the kite problem, the right triangle formed involved:

• The hypotenuse: the kite string, measuring 50 feet.
• One leg: the height of the kite.
• The other leg: the horizontal distance from the person to the kite.

These elements form a right triangle, making it possible to apply the Pythagorean Theorem. By utilizing these properties, we can solve for unknown lengths when at least two sides are known, making right triangles very useful in practical measurements and engineering.

Quadratic Equation

A quadratic equation is a type of polynomial equation that can be written in the standard form: \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants. Quadratic equations are critical for solving problems involving parabolic movements or relationships, like in many physics problems.
The general formula for finding the solution to a quadratic equation is the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In the kite problem, after using the Pythagorean theorem and substituting expressions, the equation simplifies into a quadratic form:\[ x^2 + 10x - 1200 = 0 \]Here:

• \( a = 1 \)
• \( b = 10 \)
• \( c = -1200 \)

The discriminant \( b^2 - 4ac \) must be calculated to check if there are real solutions. By solving the quadratic equation, we determine that \( x = 30 \) is a valid solution, indicating the horizontal distance. Solutions like these demonstrate how quadratic equations can reveal critical information about dimensions and relationships in geometric problems.