Type the correct answer in the box. Jason builds doghouses for a pet store. Each doghouse is a wooden structure with a rectangular base that has an area of 21 square feet and a length that is 4 feet more than its width. If x represents the width of the doghouse, write an equation in the given form that can be used to determine the possible dimensions of the base of the doghouse. ax\textsuperscript{2} + bx = c
The Correct Answer and Explanation is:
Correct Answer:
The correct equation in the form ax2+bx=cax^2 + bx = c is: x2+4x=21x^2 + 4x = 21
Explanation
This problem involves forming a quadratic equation based on a real-world geometric situation.
Jason builds doghouses that have rectangular bases. We’re told:
- The area of each base is 21 square feet.
- The length is 4 feet more than the width.
- We’re asked to write an equation in the form ax2+bx=cax^2 + bx = c, where xx is the width.
Let’s break it down step-by-step:
- Assign variables:
Let xx be the width of the rectangular base (in feet). - Express length in terms of width:
The length is 4 feet more than the width, so the length is x+4x + 4. - Use the area formula:
The area of a rectangle is given by: Area=Length×Width\text{Area} = \text{Length} \times \text{Width} Substituting the known values: 21=x(x+4)21 = x(x + 4) - Simplify the equation:
Multiply the expression: 21=x2+4×21 = x^2 + 4x - Rearrange into standard form:
Bring all terms to one side to match the form ax2+bx=cax^2 + bx = c. Since the prompt asks for the equation in the form ax2+bx=cax^2 + bx = c, we don’t need to rearrange further. We can write: x2+4x=21x^2 + 4x = 21
This equation now models the situation: any solution for xx will give the width of a possible doghouse base that meets the criteria. From there, x+4x + 4 gives the length.
In summary, by translating the word problem into mathematical expressions and applying the formula for area, we derived a quadratic equation that can be used to determine the possible width of the doghouse base.
