Problem 60 - Algebra 1

Question

The average speed of an object \(S\) (in feet per second) that is dropped a certain distance \(d\) (in feet) is given by the following equation. Falling object model: \(S=\frac{d}{\frac{\sqrt{d}}{4}}\) Rewrite the equation with the right-hand side in simplest form.

Step-by-Step Solution

Answer

The equation in simplest form is \(S = 4d^{\frac{1}{2}}\).

Step 1: Understand the Problem and the Formula

From the problem, it's clear that the task is to simplify the formula for average speed of an object falling a distance \(d\). The formula is given as \(S = \frac{d}{\frac{\sqrt{d}}{4}}\). The main focus will be on simplifying the right-hand side of this equation.

Step 2: Apply the Rules of Fractions

To simplify this expression, it's helpful to remember the rule of dividing fractions: \(a / \frac{b}{c} = a * \frac{c}{b}\). Applying this rule to our formula, we get \(S = d* \frac{4}{\sqrt{d}}\) which is equal to \(S = \frac{4d}{\sqrt{d}}\).

Step 3: Further Simplify the Equation

To further simplify this expression, also recall that \(\sqrt{d} = d^{\frac{1}{2}}\). We can rewrite the expression as \(S = \frac{4d}{d^{\frac{1}{2}}}\). By simplifying this expression, we get \(S = 4d^{\frac{1}{2}}\).

Key Concepts

Fractions Simplification

Fractions are a fundamental part of algebraic manipulation. Simplifying fractions involves reducing them to their simplest form. In many cases, this means performing operations to make the fraction easier to understand or work with. Let's explore this in detail:When given a fraction like \( \frac{a}{\frac{b}{c}} \), the simplification process begins by using the rule for dividing one fraction by another. This rule states: divide by a fraction by multiplying by its reciprocal. Thus, the expression can be rewritten as \( a \times \frac{c}{b} \).

• The reciprocal of a fraction \( \frac{b}{c} \) is simply swapping the numerator and the denominator, resulting in \( \frac{c}{b} \).
• Multiplying a number \( a \) by a fraction means multiplying \( a \) with the fraction's numerator and dividing the result by the fraction's denominator.

In our original problem, applying this principle helps transform a complex division into a straightforward multiplication, simplifying our calculations.

Exponents

Exponents are a key concept in algebra that allow us to express repeated multiplication. When dealing with expressions like \( \sqrt{d} \) in algebra, recognizing that this can be rewritten using exponents is crucial.

• A square root, \( \sqrt{d} \), can be expressed as an exponent: \( d^{\frac{1}{2}} \). This exponent notation makes it easier to manipulate and simplify expressions, especially when dealing with multiplication and division.
• In the equation \( S = \frac{4d}{\sqrt{d}} \), it becomes \( S = \frac{4d}{d^{\frac{1}{2}}} \), thanks to the exponent rule.

Understanding exponent rules simplifies expressions by converting roots and powers into a form that makes further operations more straightforward.

Square Roots

Square roots are special mathematical operations that undo another operation, squaring. This means if you square a number and then take the square root, you end up back at your original number.In algebraic manipulation, recognizing when to use square roots is vital. For example, if you have \( \sqrt{d} \), it implies you are looking for a number which when squared gives \( d \).

• In terms of simplification, when a square root appears in a fraction, like in \( \frac{d}{\sqrt{d}} \), it can sometimes be simplified by multiplying or rationalizing the denominator.
• Square roots can also be expressed as fractions with exponents, which is extremely helpful when simplifying more complex algebraic expressions.

Square roots, combined with factoring and manipulating expressions, provide a comprehensive way to deal with quadratic expressions and highlight the importance of understanding this concept thoroughly.