A dentist's office and parking lot are on a rectangular piece of land. The area (in square meters) of the land is represented by x² + x - 30. a. Write a binomial that represents t... A dentist's office and parking lot are on a rectangular piece of land. The area (in square meters) of the land is represented by x² + x - 30. a. Write a binomial that represents the width of the land. b. Find the perimeter of the land when the length of the dentist's office is 20 meters. Read more

Steps to Solve

1. Factor the quadratic expression representing the area The area of the rectangular piece of land is given by $x^2 + x - 30$. We need to factor this quadratic expression to find the binomials that represent the length and width of the land. We look for two numbers that multiply to -30 and add to 1. These numbers are 6 and -5. Therefore, the factored form is $(x+6)(x-5)$.

2. Determine the width From the image we can see that the length is $x+6$. Therefore the width must be $x-5$.

3. Find the value of x when the length is 20 meters We are given that the length of the dentist's office is 20 meters. Since the length of the whole land is represented by $x + 6$, we set $x + 6 = 20$ and solve for $x$. $x + 6 = 20$ $x = 20 - 6$ $x = 14$

4. Calculate the width Now that we know $x = 14$, we can find the width by substituting $x$ into the expression for the width, which is $x - 5$. Width $= x - 5 = 14 - 5 = 9$ meters.

5. Calculate the perimeter The perimeter of a rectangle is given by the formula $P = 2(l + w)$, where $l$ is the length and $w$ is the width. We have $l = 20$ meters and $w = 9$ meters. $P = 2(20 + 9) = 2(29) = 58$ meters.