A rectangular patio is 9 ft by 6 ft. When the length and width are increased by the same amount, the area becomes 88 sq ft. Ginger is using the zero product property to solve the equation $(6+x)(9+x)=88$. What does the value of $x$ in her solutions represent?
A. two possible amounts for the widths
B. one possible length and one possible width
C. two possible amounts for the lengths
D. two possible amounts by which the dimensions changed
Answer (1)
The problem defines a rectangular patio with dimensions 9 ft by 6 ft, which are increased by x to reach an area of 88 sq ft.
The equation ( 6 + x ) ( 9 + x ) = 88 is expanded and simplified to the quadratic equation x 2 + 15 x − 34 = 0 .
The quadratic formula is used to solve for x , resulting in two possible values: x = 2 and x = − 17 .
Since x represents an increase in length and width, the negative solution is discarded, and the final answer is 2 .
Explanation
Understanding the Problem The problem states that a rectangular patio is 9 ft by 6 ft. When both the length and width are increased by the same amount, x , the area becomes 88 sq ft. Ginger is solving the equation ( 6 + x ) ( 9 + x ) = 88 using the zero product property, and we need to determine what the value of x represents.
Setting up the Equation The variable x represents the amount by which both the length and the width of the patio are increased. The equation ( 6 + x ) ( 9 + x ) = 88 models the area of the new patio after the increase. To find the value of x , we need to solve this equation.
Expanding the Equation First, expand the equation: ( 6 + x ) ( 9 + x ) = 88
54 + 6 x + 9 x + x 2 = 88
x 2 + 15 x + 54 = 88
x 2 + 15 x − 34 = 0
Solving for x Now, we need to solve the quadratic equation x 2 + 15 x − 34 = 0 . We can use the quadratic formula or factoring. In this case, let's find the roots using the quadratic formula: x = 2 a − b ± b 2 − 4 a c
where a = 1 , b = 15 , and c = − 34 . x = 2 ( 1 ) − 15 ± 1 5 2 − 4 ( 1 ) ( − 34 )
x = 2 − 15 ± 225 + 136
x = 2 − 15 ± 361
x = 2 − 15 ± 19
So, we have two possible values for x : x 1 = 2 − 15 + 19 = 2 4 = 2
x 2 = 2 − 15 − 19 = 2 − 34 = − 17
Interpreting the Solution Since x represents an increase in length and width, it must be a positive value. Therefore, x = 2 . The value x = − 17 is not physically meaningful in this context because we cannot have a negative increase in length and width.
Final Answer The value of x represents the amount by which both the length and width are increased. In this case, x = 2 ft. Therefore, the dimensions changed by 2 ft.
Examples
Imagine you're designing a garden and want to increase its size while keeping the same shape. This problem helps you determine how much to increase each side to achieve a specific new area. Understanding how to solve quadratic equations in this context allows you to plan your garden expansion efficiently, ensuring you have enough space for all your plants. This approach is also useful in other design and construction projects where you need to adjust dimensions to meet area requirements.
