Problem preview

F-BF.1.a Warmup M2-014-A14-V02

Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.

Build an explicit model from context and solve for the input that gives a target output

Problem

A rectangle has width \(x~\text{feet}\) and length \(x+5~\text{feet}\). Build its area model and find the width when the area is \(84~\text{square}~\text{feet}\). Show the quadratic factorization, filter both algebraic roots using \(x>0\), and verify the valid width in the original area equation.

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A(x) = x(x + 5). Setting A(x) = 84 gives x² + 5x − 84 = (x + 12)(x − 7) = 0, so x = −12 or 7; only x = 7 feet is valid, and 7 · 12 = 84.

The target 84 square feet is treated as width x = 84. The check 84 · 89 is not 84.

The solutions are x = −12 and 7 feet. A negative width is outside the physical domain, leaving only 7.

Set A(x) = 84, but no area model or algebraic reduction is supplied, so no width can be determined.

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Curriculum context

Course: Math II
Standard: F-BF.1.a
Category: Functions
Domain: Building Functions
Objective: Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
Problem type: Build an explicit model from context and solve for the input that gives a target output