Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

An above-ground swimming pool with a square base is to be constructed such that the surface area of the pool is 561 square feet. The height of the pool is to be 4 feet (see figure). What should the dimensions of the base be? (Hint: The surface area is \(\left.S=x^{2}+4 x h .\right)\)

Step-by-Step Solution

Answer

The dimensions of the base should be approximately \(15\) feet by \(15\) feet.

Step 1: Substituting the given values

First, substitute the given values into the surface area equation. This gives us \(561 = x^2 + 4x(4)\). This simplifies to \(561 = x^2 + 16x\).

Step 2: Rearranging the equation

Next, rearrange the equation into a standard quadratic form \(ax^2 + bx + c = 0\). In this case, we get \(x^2 + 16x - 561 = 0\).

Step 3: Solving the equation

We can now solve the quadratic equation using the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / 2a\). Substituting the values of \(a = 1\), \(b = 16\), and \(c = -561\) into the formula, we get \(x = [-16 ± sqrt((16)^2 - 4*1*(-561))] / 2*1\). By simplifying this, we get two possible values for \(x\), but only one of them makes sense in our context as dimensions can't be negative. So, we choose the positive solution.

Key Concepts

Surface Area

In the world of geometry, 'surface area' refers to the total area that the surface of a three-dimensional object occupies. For the swimming pool, the surface area is specifically defined by its geometric shape, a square base with vertical sides. To calculate surface area, you consider both the base and the sides.

Specifically for this pool, the formula given is:

• The area of the square base: \(x^2\) (where \(x\) is the side of the square base)
• The area of the sides: \(4xh\), because there are 4 sides, each of height \(h\) where \(h = 4\) feet.

This formula is crucial for ensuring that all parts of the structure combine to meet the desired surface area. Understanding how to manipulate and rearrange this equation is key when solving problems of this nature.

Substitution Method

The substitution method is a straightforward technique used in algebra to find the solution to equations by replacing variables with their known or estimated values. In our swimming pool problem, we're given the height \( h \) of the pool, so we substitute this known value (4 feet) into the surface area formula.

By substituting \(h = 4\) into the equation for the surface area \( 561 = x^2 + 4x(4) \), it simplifies the equation to \( 561 = x^2 + 16x \). This simplification transforms the problem from dealing with two variables \(x\) and \(h\) to handling a single variable \(x\), making it easier to solve. Using substitution is helpful as it can reduce the complexity and number of steps required to solve an algebraic problem.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula is particularly valuable when factoring the quadratic equation is difficult or impossible. The quadratic formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our swimming pool example, after substituting the values into the quadratic equation \(x^2 + 16x - 561 = 0\), we identify:

• a = 1
• b = 16
• c = -561

Applying these into the quadratic formula gives us two potential solutions for \(x\). These solutions are derived from calculating the value of the discriminant \(b^2 - 4ac\) and checking the resulting square roots. Most importantly, the formula guides us not only in finding solutions but also ensures that we consider the practical dimensions (i.e., only positive solutions make physical sense).

Solving Equations

Solving equations involves finding the value of the unknowns that make the equation true. In this exercise, the goal is to find the dimensions of the pool's base by solving the derived quadratic equation. After setting up the equation \(x^2 + 16x - 561 = 0\) from the given surface area formula, you can use the quadratic formula to identify the possible values for \(x\).

The solving process includes:

• Substitute values into the quadratic formula.
• Simplify to find potential solution(s).
• Interpret these solution(s) within the context; in our case, only a positive length of the side of the pool will be applicable.

Remember that quadratic equations can have two solutions because of the \(\pm\) sign in the formula. It is essential to evaluate both solutions in the context of the problem to determine which is viable.