Problem 74 - Algebra for College Students

Question

Suppose the height of a semielliptical archway is given by the function \(h(x)=\sqrt{64-4 x^{2}}\), where \(x\) is the distance from the center line of the arch. Compute \(h(0), h(2)\), and \(h(4) . \quad 8 ; 4 \sqrt{3} ; 0\)

Step-by-Step Solution

Answer

The heights are 8, \(4\sqrt{3}\), and 0 at \(x = 0, 2, 4\) respectively.

Step 1: Understanding the Function

The function given is a square root function: \( h(x) = \sqrt{64 - 4x^2} \). This represents the height of a semielliptical archway at a distance \( x \) from the center line of the arch.

Step 2: Calculate \( h(0) \)

Substitute \( x = 0 \) into the function: \[ h(0) = \sqrt{64 - 4(0)^2} = \sqrt{64} = 8. \] The height at \( x = 0 \) is 8.

Step 3: Calculate \( h(2) \)

Substitute \( x = 2 \) into the function: \[ h(2) = \sqrt{64 - 4(2)^2} = \sqrt{64 - 16} = \sqrt{48}. \] Simplify \( \sqrt{48} \): \[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}. \]The height at \( x = 2 \) is \( 4\sqrt{3} \).

Step 4: Calculate \( h(4) \)

Substitute \( x = 4 \) into the function: \[ h(4) = \sqrt{64 - 4(4)^2} = \sqrt{64 - 64} = \sqrt{0} = 0. \]The height at \( x = 4 \) is 0.

Key Concepts

Understanding the Square Root Function

The square root function involves finding a number which, when multiplied by itself, gives the original number. In the case of the semielliptical archway, the function is given as \( h(x) = \sqrt{64 - 4x^2} \). This specific function measures the height of an archway at various points from its center.

• At the core of the function, the expression inside the square root \( 64 - 4x^2 \) determines the effective height above zero at any given \( x \) position.
• The function is a way to express how high the archway is at different distances "\( x \)" from the center.

Understanding this concept is crucial because it visually and mathematically describes how an arch, being symmetrical, forms a curve that peaks at the center.

How to Substitute Values

Substituting values into a function is a straightforward method to find specific outputs from given inputs. For our archway problem, we're interested in finding the height for particular \( x \) values such as 0, 2, and 4.

Here's how you do it:

• Identify the value you need to substitute into the equation (e.g., \( x = 0 \)).
• Replace \( x \) in the function \( h(x) = \sqrt{64 - 4x^2} \) with the specific value. For example, when \( x = 0 \), it becomes \( h(0) = \sqrt{64 - 4(0)^2} \).
• Simplify the expression inside the square root to find the resulting height.

Substituting values effectively allows us to see how the height changes with distance, revealing the arch's geometric properties.

Simplifying Radicals in Practice

Simplifying radicals means rewriting a square root expression in its simplest form so that it's easier to understand and use. In our computations for the archway, this is often necessary to provide an exact answer.

The process typically involves:

• Breaking down the number inside the square root into its prime factors, identifying perfect squares where possible.
• Taking the square root of the perfect square component out of the radical. For instance, with \( \sqrt{48} \), it can be rewritten as \( \sqrt{16 \times 3} \), and further simplified to \( 4\sqrt{3} \).
• Simplifying allows for a clearer interpretation of results and often demonstrates an exact numerical answer over an approximate decimal.

This approach of simplifying radicals helps in maintaining precision, especially in geometry-related problems involving symmetrical structures like archways.