Problem 35 - Precalculus: Functions and Graphs

Question

A right triangle has area \(30 \mathrm{ft}^{2}\) and a hypotenuse that is 1 foot longer than one of its sides. (a) If \(x\) denotes the length of this side, then show that \(2 x^{3}+x^{2}-3600=0\). (b) Show that there is a positive root of the equation in part (a) and that this root is less than 13. (c) Find the lengths of the sides of the triangle.

Step-by-Step Solution

Answer

The sides are approximately 12 ft, 2.5 ft, and 13 ft.

Step 1: Establish the Area Condition

The area of a right triangle can be expressed as \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{30}{2} = 15\). Let the base and height be \(x\) and \(h\) respectively, so \(x \times h = 30\). This can be rewritten as \(h = \frac{30}{x}\).

Step 2: Establish the Hypotenuse Condition

The hypotenuse of the triangle is given as 1 foot longer than one side, chosen here as \(x\). Thus, the hypotenuse \(c = x + 1\). By the Pythagorean theorem, \(x^2 + h^2 = (x+1)^2\). Substitute \(h = \frac{30}{x}\) to get \(x^2 + \left(\frac{30}{x}\right)^2 = (x+1)^2\).

Step 3: Simplify the Pythagorean Equation

Substitute and simplify: \[x^2 + \frac{900}{x^2} = x^2 + 2x + 1\]. Remove \(x^2\) from both sides, leading to \(\frac{900}{x^2} = 2x + 1\). Multiply through by \(x^2\) to clear the fraction: \(900 = 2x^3 + x^2\). Rearranging gives \(2x^3 + x^2 - 900 = 0\).

Step 4: Show Root Existence and Bound

We can apply the Intermediate Value Theorem between \(x = 0\) and \(x = 13\) to verify the existence of a root. Evaluate \(f(x) = 2x^3 + x^2 - 900\). At \(x = 0\), \(f(0) = -900\) and at \(x = 13\), substitute into the equation to confirm \(f(13) > 0\). Therefore, since \(f(x)\) moves from negative to positive, a root exists less than 13.

Step 5: Solve for the Side Lengths

The cubic equation is \(2x^3 + x^2 - 900 = 0\). By using methods like numerical calculations or graphing to estimate, or iterative methods like Newton's method, you find \(x \approx 12\). The hypotenuse is \(x + 1 = 13\). Thus, the other side \(h = \frac{30}{x} \approx \frac{30}{12} = 2.5\). Ensure these satisfy the conditions: \(x = 12\), \(h = 2.5\), and \(c = 13\).

Key Concepts

Right Triangle Properties

Right triangles, a cornerstone of geometry, are triangles where one of the angles is exactly 90 degrees. This unique property lends several special characteristics:

• They are defined by their three sides: two shorter sides, named the 'legs,' and the longest side, called the hypotenuse.
• The base and height of a right triangle are typically the two legs, which form the right angle.
• The area of a right triangle can be calculated as half the product of its base and height: \(\frac{1}{2} \times \text{base} \times \text{height}\).
• The hypotenuse is always opposite the right angle and is the longest side.

In our problem, we use these properties to relate the area to one side of the triangle and express it in terms of a variable \(x\). Therefore, this concept is pivotal for forming the fundamental equations needed to describe the triangle's dimensions.

Pythagorean Theorem

The Pythagorean theorem is crucial for solving right triangle problems. Stated simply, for a right triangle:- The square of the hypotenuse is equal to the sum of the squares of the other two sides.Mathematically, this is expressed as:\[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs.
In our example, this theorem helps us to relate the given hypotenuse condition to the sides of the triangle. By substituting expressions derived from the area equation, we find:\[x^2 + \left(\frac{30}{x}\right)^2 = (x+1)^2\]Using algebraic manipulation, these expressions aid in forming a cubic equation, a higher-level algebraic expression that helps further dissect the problem.

Cubic Equations

A cubic equation is a polynomial equation of the form:\[ax^3 + bx^2 + cx + d = 0\]This type of equation can have up to three real roots. Finding the roots involves understanding polynomial behavior. In this exercise, re-arranging and simplifying the right triangle's properties and Pythagorean geometry ends in the cubic equation:\[2x^3 + x^2 - 900 = 0\]
Such an equation may not be solvable through simple algebra alone, so numerical methods or graphing are often employed. For example, techniques like Newton's method or software tools that perform these calculations can help approximate the solutions when solving real-world problems involving cubic equations.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) provides powerful insights when working with continuous functions. It states if a function \(f(x)\) is continuous on an interval \([a, b]\), and \(N\) is any number between \(f(a)\) and \(f(b)\), then there is at least one \(c\) in \([a, b]\) such that \(f(c) = N\).
In the context of our problem, IVT helps confirm the existence of a positive root. By evaluating the function \(f(x) = 2x^3 + x^2 - 900\) at specific values within the range, like \(x = 0\) and \(x = 13\):

• \(f(0) = -900\), clearly negative.
• \(f(13)\) is positive.

Thus, by passing from a negative to a positive value, IVT assures us there must be a root within \((0, 13)\), providing a useful check for our solution, thereby guiding us to find the actual root that makes sense in our right triangle context.