Question

section 3.5c: quadratic modeling (geometric) & section 3.6a: x - intercept function
score: 15/45 answered: 2/9
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question 3
lucia built a rectangular dog pen with a perimeter of 30 feet and an area of 56 square feet. what are the dimensions of the the dog pen?
the width, or shorter side is
the length, or longer side is

Explanation:

Step1: Set up equations

Let the length of the rectangle be $l$ and the width be $w$. The perimeter formula is $P = 2(l + w)$, so $2(l + w)=30$, which simplifies to $l + w=15$, and $l = 15 - w$. The area formula is $A=lw$, so $lw = 56$.

Step2: Substitute and solve quadratic equation

Substitute $l = 15 - w$ into $lw = 56$, we get $(15 - w)w=56$. Expand it to $15w-w^{2}=56$, or $w^{2}-15w + 56 = 0$. Factor the quadratic equation: $(w - 7)(w - 8)=0$.

Step3: Find the values of width and length

Set each factor equal to zero: $w-7 = 0$ gives $w = 7$; $w - 8=0$ gives $w = 8$. If $w = 7$, then $l=15 - 7=8$; if $w = 8$, then $l=15 - 8 = 7$. Since the width is the shorter side and the length is the longer side, the width is 7 feet and the length is 8 feet.

Answer:

The width, or shorter side is 7 feet long.
The length, or longer side is 8 feet long.