The length of a rectangle is 4 cm more than the width. The area of the rectangle is 77 cm squared. Find the width and length.
A) 7 cm, 11 cm
B) 4 cm, 8 cm
C) 6 cm, 8 cm
D) 6 cm, 10 cm
since 77 = 7*11, I think the answer is clear.
Let's call the width of the rectangle "w" cm.
Since the length is 4 cm more than the width, we can represent the length as "w + 4" cm.
The area of a rectangle is calculated by multiplying the length and the width. So, the equation for the area of this rectangle is:
Area = Length * Width
Given that the area is 77 cm², we can write the equation as:
77 = (w + 4) * w
Now we can solve this equation for the width:
77 = w² + 4w
Rearranging the equation, we have:
w² + 4w - 77 = 0
Now we can factor or use the quadratic formula to solve for w. Factoring this quadratic equation, we have:
(w - 7)(w + 11) = 0
Setting each factor equal to zero, we get:
w - 7 = 0 or w + 11 = 0
Solving for w, we have two possible solutions:
w = 7 or w = -11
Since the width cannot be negative, we discard the solution w = -11.
Therefore, the width of the rectangle is 7 cm.
Since the length is 4 cm more than the width, the length is:
w + 4 = 7 + 4 = 11 cm
So, the width is 7 cm and the length is 11 cm.
The correct answer is A) 7 cm, 11 cm.
To find the width and length of the rectangle, we can use the information given in the question.
Let's assume that the width of the rectangle is "w" cm. According to the problem, the length is 4 cm more than the width, which means the length is "w + 4" cm.
We know that the area of a rectangle is given by the formula: Area = length × width.
In this case, the area is given as 77 cm². Therefore, we can represent this using the equation:
77 = (w + 4) × w
Expanding the equation, we get:
77 = w² + 4w
Rearranging the equation to a quadratic form, we have:
w² + 4w - 77 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula.
Since the equation doesn't factor easily, let's use the quadratic formula:
w = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 1, b = 4, and c = -77. Substituting these values into the formula, we have:
w = (-4 ± √(4² - 4(1)(-77))) / (2(1))
Simplifying further, we get:
w = (-4 ± √(16 + 308)) / 2
w = (-4 ± √324) / 2
w = (-4 ± 18) / 2
Now, let's calculate the two possible values of w:
w₁ = (-4 + 18) / 2 = 14 / 2 = 7
w₂ = (-4 - 18) / 2 = -22 / 2 = -11
Since the width of a rectangle cannot be negative, we discard the negative value of -11.
Therefore, the width of the rectangle is 7 cm.
Now, to find the length, we can substitute the width back into the expression "w + 4":
length = 7 + 4 = 11 cm.
Hence, the correct answer is A) 7 cm, 11 cm.