Question

1. challenge: stacy is renting a booth at the fair to sell her jelly. the fair operator said that she is getting a booth with an area of 260 ft². stacy needs to know exactly what the length and width of the booth are so that she can bring the right tables. find the value of the length and the width of the booth. (represented by the figure below) width: length: x = 2x x + 3 260 ft²

Explanation:

Step1: Recall area formula for rectangle

The area formula for a rectangle is $A = \text{length}\times\text{width}$. Given length $l = 2x$ and width $w=x + 3$, and $A = 260$. So, $2x(x + 3)=260$.

Step2: Expand the left - hand side

Expand $2x(x + 3)$ using the distributive property $a(b + c)=ab+ac$. We get $2x^2+6x = 260$.

Step3: Rearrange to standard quadratic form

Subtract 260 from both sides to get $2x^2+6x−260 = 0$. Divide through by 2 to simplify: $x^2+3x - 130=0$.

Step4: Factor the quadratic equation

We need to find two numbers that multiply to - 130 and add up to 3. The numbers are 13 and - 10. So, $x^2+3x - 130=(x + 13)(x - 10)=0$.

Step5: Solve for x

Set each factor equal to zero: $x+13 = 0$ gives $x=-13$ and $x - 10=0$ gives $x = 10$. Since length and width cannot be negative, we take $x = 10$.

Step6: Find length and width

Length $l=2x$, substituting $x = 10$, we get $l = 2\times10=20$ ft. Width $w=x + 3$, substituting $x = 10$, we get $w=10 + 3=13$ ft.

Answer:

$x = 10$, Length: 20 ft, Width: 13 ft