A rectangular counter top is 16 inches longer than it is wide. If the area of the counter top is 420 square inches, find the length and the width.
width --- x
length --- x+16
solve
x(x+16) = 420
(it factors quite nicely)
Width = W inches
Length = W + 16 inches.
W(W + 16) = 420
W^2 + 16W - 420 = 0
Use Quadratic Formula:
w = (-16+ / - sqrt(256 + 1680))/2 =
(-16 +- 44)/2 = 14 and -30. use W = 14
= Width.
W +16 = 14 + 16 = 30 in. = Length.
To find the length and width of the rectangular countertop, we start by setting up equations based on the given information.
Let's assume the width of the countertop is 'x' inches. Since the length is given as 16 inches longer than the width, the length can be expressed as 'x + 16' inches.
The area of a rectangle is calculated by multiplying its length by its width. In this case, the area is given as 420 square inches. So, we have the equation:
Area = Length * Width
420 = (x + 16) * x
Now, let's solve this equation to find the values of 'x' and 'x + 16'.
To solve the quadratic equation, we can expand it:
420 = x^2 + 16x
Rearranging this equation to make it equal to zero:
x^2 + 16x - 420 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. For simplicity, let's use factoring.
Taking common factors:
(x + 30)(x - 14) = 0
Setting each factor equal to zero:
x + 30 = 0 or x - 14 = 0
Solving for 'x':
x = -30 or x = 14
Since the width cannot be negative, we can discard the solution x = -30.
Therefore, the width of the countertop is x = 14 inches.
To find the length, we substitute this value of the width back into the equation:
Length = Width + 16
Length = 14 + 16
Length = 30 inches
So, the width of the countertop is 14 inches and the length is 30 inches.