Problem 58 - Algebra A Combined Function

Question

Solve. The area of the top of the Ping-Pong table shown is \(\left(49 x^{2}+70 x-200\right)\) square inches. If its length is \((7 x+20)\) inches, find its width.

Step-by-Step Solution

Answer

The width of the table is \(7x - 10\) inches.

Step 1: Understand the Problem

You need to find the width of the Ping-Pong table given its area and length. The relationship between area, length, and width is given by the formula: Area = Length × Width.

Step 2: Set Up the Equation

The expression for the area of the table is given as \(49x^2 + 70x - 200\). We know the length of the table is given by \(7x + 20\). To find the width, use the equation: \(\text{Area} = \text{Length} \times \text{Width}\). Thus, we set up the equation: \(49x^2 + 70x - 200 = (7x + 20) \times \text{Width}\).

Step 3: Solve for Width

To find the width, divide the expression for the area by the length: \(\text{Width} = \frac{49x^2 + 70x - 200}{7x + 20}\). Perform polynomial division to solve for the width.

Step 4: Perform Polynomial Division

Divide \(49x^2 + 70x - 200\) by \(7x + 20\). Start by dividing the first term of the dividend \(49x^2\) by the first term of the divisor \(7x\), which gives \(7x\). Multiply \(7x\) by \(7x + 20\) and subtract the result from the dividend. Simplify the remainder and repeat the process until complete. The resulting quotient is the width of the table.

Step 5: Complete the Division

Completing the polynomial division, you should find that the quotient is \(7x - 10\). This gives the width of the table as \(7x - 10\) inches.

Key Concepts

Area Calculation

To understand the concept of area calculation, it is vital to remember that the area of a shape is its surface measured in square units. For rectangles, the area is found by multiplying the length by the width.
In this exercise, the area of the Ping-Pong table is given as a polynomial: 49x^2 + 70x - 200 \text{ square inches}.
The length of the table is 7x + 20 \text{ inches}.
Since the area equation is \text{Area} = \text{Length} \times \text{Width},
we can rearrange this equation to calculate the width by dividing the area by the length.

Area Formula: \( \text{Area} = \text{Length} \times \text{Width} \)

Calculation of Width: \( \text{Width} = \frac{\text{Area}}{\text{Length}} \)

By breaking down the polynomial representation of the area, we can apply these equations straightforwardly to solve for unknown dimensions, like the width in this problem.

Algebraic Expressions

Algebraic expressions, like \(49x^2 + 70x - 200\), are combinations of constants, variables, and arithmetic operators. Understanding how to manipulate these expressions, particularly through operations like division, is essential for solving problems like this.
This exercise involves the use of polynomial expressions to denote the area of a rectangle; the challenge is to determine the width using known length measurements and the given area.

The polynomial \(49x^2 + 70x - 200\) is a quadratic expression.

The linear expression \(7x + 20\) represents the length of the table.

Using the knowledge that:

Multiplication: When two algebraic expressions are multiplied, each term of one is multiplied by each term of the other.

Division: Polynomial division involves dividing each term in the numerator by each term in the denominator.

Through structured manipulation, these expressions can be rewritten or simplified to determine unknown values, such as calculating the table's width.

Factoring Polynomials

Factoring polynomials sometimes simplifies division, making it a useful method when dividing a polynomial like \(49x^2 + 70x - 200\) by another \(7x + 20\).
Factoring involves breaking down a polynomial into simpler 'factors' or components that, when multiplied together, produce the original polynomial. However, polynomial division in this exercise directly resolves the need to factor, yet understanding this process can be insightful.

Factoring Steps: Identify common factors, factor out the greatest common factor, and apply factoring rules such as grouping or quadratic formulas.

Polynomial Division: Divide the leading term of numerator by the leading term of divisor.

Performing polynomial division requires determining how many times the divisor fits into each term of the dividend. Here it is evident when dividing \(49x^2\) by \(7x\) results in \(7x\), helping systematically reduce our polynomial problem to simpler terms.