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.
The correct Answer and Explanation is:
To determine the possible dimensions of the base of the doghouse, we can represent the width of the doghouse as ( x ), and the length as ( x + 4 ), since the problem states that the length is 4 feet more than the width. The area of the base of the doghouse is given as 21 square feet.
The area of a rectangle is calculated by multiplying the length and width, so we can set up the following equation:
[
\text{Area} = \text{Length} \times \text{Width}
]
Substitute the known values:
[
21 = (x + 4) \times x
]
Now, expand the equation:
[
21 = x(x + 4)
]
[
21 = x^2 + 4x
]
To write the equation in standard form, we subtract 21 from both sides:
[
x^2 + 4x – 21 = 0
]
This is a quadratic equation in the form ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = 4 ), and ( c = -21 ).
Explanation
The quadratic equation ( x^2 + 4x – 21 = 0 ) represents the relationship between the width ( x ) and the given area of the rectangular base of the doghouse. In this equation, ( x ) is the width, and the length is ( x + 4 ), which reflects the condition that the length is 4 feet more than the width.
This problem essentially asks you to apply the formula for the area of a rectangle to a situation where the length and width are related. By expressing the length in terms of the width and plugging this into the area formula, you obtain a quadratic equation. Solving this quadratic equation will give the possible values of ( x ), the width of the doghouse. Once you have the width, you can easily determine the corresponding length by adding 4 to the width.
In practical terms, solving this quadratic equation helps Jason figure out the possible dimensions of the doghouse’s base, ensuring that the total area remains 21 square feet.