Cary told Tony that she wants to build a rectangular patio in her backyard with an area of x^2-3x-10 square feet. If the width needs to be at least 4 feet, what are two sets of possible dimensions for Cary's patio?
we have to factor
x^2 - 3x - 10
= (x-5)(x+2) , the width is x-5 and the length is x+2
so if the width is 5 and the length is 7
area = 5^2 - 3(5) - 10 = 0
so x > 5
let x = 6, then width is 1 and length is 8
let x = 7 , then width is 2 and length is 9
etc.
To find the possible dimensions for Cary's patio, we need to express the given expression x^2-3x-10 in the form of (length) * (width).
Step 1: Express the given expression in factored form:
To factorize x^2-3x-10, we can find two numbers whose product is -10 and whose sum is -3 (coefficient of x). The numbers -5 and 2 meet these conditions.
Therefore, we can represent x^2-3x-10 as (x-5)(x+2).
Step 2: Set up the inequality:
Since the width needs to be at least 4 feet, we can set the width (x-5) greater than or equal to 4:
x-5 ≥ 4.
Step 3: Solve the inequality:
Add 5 to both sides of the inequality:
x ≥ 4 + 5
x ≥ 9.
So, the minimum value for x is 9.
Step 4: Find two sets of possible dimensions:
To find two sets of possible dimensions, we need to find two values of x that satisfy the inequality x ≥ 9.
Possible dimensions set 1:
If we take x = 9, then the width (x-5) will be 9-5 = 4 feet.
So, one set of possible dimensions is 9 feet by 4 feet.
Possible dimensions set 2:
If we take x = 10, then the width (x-5) will be 10-5 = 5 feet.
So, another set of possible dimensions is 10 feet by 5 feet.
Therefore, two sets of possible dimensions for Cary's patio are:
1) 9 feet by 4 feet.
2) 10 feet by 5 feet.